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Study of the attenuation of elastic waves in metals. Hasegawa, Henry S. 1965

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A STUDY OP THE ATTENUATION OP ELASTIC WAVES IN METALS by Henry S. Hasegawa B . S c , U n i v e r s i t y of A l b e r t a , i960 M.Sc, U n i v e r s i t y of A l b e r t a , 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OP DOCTOR OP PHILOSOPHY i n the Department of GEOPHYSICS We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OP BRITISH COLUMBIA September, 1965 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f g g O P K V g f & S T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL .ORAL. EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of HENRY SUSUMU HASEGAWA B.Sc, The University of Alberta, 1960 M.Sc, The University of Alberta, 1962 FRIDAY, NOVEMBER 12, 1965 AT 3:30 P.M. TN ROOM 10, HEBB BUILDING (PHYSTCS) COMMITTEE LN..GHARGE Chairman: J . D. Chapman R. M. E l l i s W. F. Slawson J. A. Jacobs R. W. Stewart R. D. Russell T„ Watanabe Research Supervisor: R. D. Russell External Examiner: C Lomnitz Department of Geology and Geophysics University of C a l i f o r n i a , Berkeley, C a l i f o r n i a A STUDY OF THE ATTENUATION OF ELASTIC WAVES IN METALS ABSTRACT The primary purpose of this thesis is to deter-mine experimentally i f the attenuation of small -amplitude elastic waves in metals is governed princi-pally by a linear mechanism (i.e. one for which the principle of superposition is valid). The secondary purpose is to interpret the attenuation measurements in terms of existing theories on acoustic dissipation in solids. Attenuation measurements of the Fourier components of a Rayleigh pulse were compared with those of sinu-soidal Rayleigh waves of the same frequency. One copper, one aluminum and two G>L -hrass circular cylindrical shells were used, and Rayleigh waves propagating along the truncated edges of these tubes were studied. Ray-leigh pulses were detected at strain levels of appro-ximately four and forty microstrain in order to test for any amplitude-dependent effects accompanying the attenua-tion. The sinusoidal Rayleigh waves were detected at strain amplitudes between three and ten microstrain. For three out of the four tubes the results indi-cated that the dominant attenuation mechanism is a linear process in the frequency range from 100 to 500 kc/s and in the strain region from four to forty micro-strain. For the copper tube, however, the scatter in the results is such that no definite conclusion could be drawn. For a l l four tubes the internal f r i c t i o n , 1/Q, increases with frequency. For some of them there is evidence of a relaxation peak, probably as a result of the Zener effect, superimposed on the general trend, Dislocation damping, as proposed by Koehler (1952) and lat e r generalized by Granato and Lucke (1956), i s suggested as a dissipative mechanism which could account for this general trend. Most of the internal f r i c t i o n values are found to be between 20 X 10" 5 and 100 X 10" 5. GRADUATE STUDIES Fie l d of Study: Seismology Waves J. C. Savage Advanced Geophysics J„ A. Jacobs Related Studies: Noise i n Physical Systems R. E. Burgess Electromagnetic Theory G. M. Volkoff Electronic Instrumentation F. K. Bowers Computer Programming C„ Froese PUBLICATIONS K. Vozoff, H.. Hasegawa and R. M. E l l i s : Results and li m i t a t i o n s of magnetotelluric surveys i n simple geologic situations. Geophysics 28, 778 (1963). J . C. Savage and H. S. Hasegawa: Some properties of tensi l e fractures inferred from e l a s t i c wave radiation. Journal of Geophysical Research 6£, 2091 (1964). J. C„ Savage and H. S. Hasegawa: A two-dimensional model study of the direct-i v i t y function. B u l l e t i n of the Seismologi-cal Society of America 55, 27 (1965). i i ABSTRACT i Attenuation measurements of the F o u r i e r compo-nents of a Rayleigh pulse were compared wi t h those of s i n u s o i d a l Rayleigh waves of the same frequency. One copper, one aluminum and two OL -brass c i r c u l a r c y l i n -d r i c a l s h e l l s were used, and Rayleigh waves propagating along the truncated edges of these tubes were stu d i e d . Rayleigh p u l s e s were detected at s t r a i n l e v e l s of appro-ximately f o u r and f o r t y m i c r o s t r a i n i n order to t e s t f o r any amplitude-dependent e f f e c t s accompanying the attenua-t i o n . The s i n u s o i d a l Rayleigh waves were detected at s t r a i n amplitudes between three and ten m i c r o s t r a i n . For three out of the f o u r tubes the r e s u l t s i n d i c a t e d that the a t t e n u a t i o n mechanism i s a l i n e a r process i n the frequency range from 100 to 500 kc/s and i n the s t r a i n r egion from f o u r to f o r t y m i c r o s t r a i n . For the copper tube, however, the s c a t t e r i n the r e s u l t s i s such that no d e f i n i t e c o n c l u s i o n could be drawn. For a l l f o u r tubes the i n t e r n a l f r i c t i o n , 1/Q, r i s e s w i t h frequency. For some of them there i s evidence of a r e l a x a t i o n peak, probably as a r e s u l t of the Zener i i i e f f e c t , superimposed on the general trend. D i s l o c a t i o n damping, as proposed by Koehler (1952) and l a t e r gener-a l i z e d by Granato and Lucke (1956), i s suggested as a d i s s i p a t i v e mechanism which could account f o r t h i s gen-e r a l t r e n d . Most of the i n t e r n a l f r i c t i o n values are found to be between 20 X 10" 5 and 100 X 10"^. i v ACKNOWLEDGMENT I t i s w i t h pleasure that I acknowledge the s u p e r v i s i o n received from Dri J . C. Savage. A f t e r suggesting the problem, Dr. Savage devoted much of h i s time a s s i s t i n g with a l l aspects of the i n v e s t i g a t i o n . I am indebted to Dr. R. D. R u s s e l l who has been my supervisor since the departure of Dr. Savage. Thanks are a l s o due to Dr. R. M. E l l i s f o r reading the man-u s c r i p t c r i t i c a l l y . The d i s c u s s i o n s with Dr. E l l i s and D. Weichert on the a n a l y s i s of t r a n s i e n t s i g n a l s have been very b e n e f i c i a l . The constant i n t e r e s t and encouragement shown by Dr. J . A. Jacobs during the experimental work i s g r e a t l y appreciated. The h e l p f u l comments on the fundamental aspects of t h i s study by Dr. R. W. Stewart of the Department of Oceanography are appreciated. I wish to thank the s t a f f members at the Computing Centre of the U n i v e r s i t y of B r i t i s h Columbia f o r t h e i r e x c e l l e n t s e r v i c e . This work was financed through grants r e c e i v e d from the N a t i o n a l Research Council of Canada and the American Petroleum I n s t i t u t e . TABLE OP CONTENTS ABSTRACT i i ACKNOWLEDGMENT i v LIST OP FIGURES v LIST OP TABLES l x CHAPTER I - INTRODUCTION 1.1 General 1 1.2 Outline of Thesis....... 4 CHAPTER I I - THEORY 2.1 General... 6 2.2 The P r i n c i p l e of Sup e r p o s i t i o n 6 2.3 The S p e c i f i c D i s s i p a t i o n Function 1/Q 1 2.4 D e s c r i p t i o n of Acoustic D i s s i p a t i o n 8 2.4.1 The Thermoelastic E f f e c t 8 2.4.2 E l a s t i c A f t e r e f f e c t s 10 2.4.3 D i s l o c a t i o n Damping 12 2.4.4 S l i d i n g F r i c t i o n 14 CHAPTER I I I - SEISMIC MODEL 3.1 General 16 3.2 D i s p e r s i o n of Short Wavelengths 17 3.3 D i s p e r s i o n of Long Wavelengths 18 3.3.1 Curvature of Disk 18 3.3.2 Curvature of Tube 19 3.3.3 Comparison of Tube wit h Disk 26 CHAPTER IV - INSTRUMENTATION AND EXPERIMENTAL PROCEDURE 4.1 General 27 4.2 The Explosive Material. 27 4.3 The Stra i n Gauge. 30 4.4 P i e z o e l e c t r i c Transducers 33 4.4.1 Equivalent C i r c u i t s of Transducers. 34 4.4.2 Importance of Geometry of Transducer 37 4.4.3 C r y s t a l Scattering E f f e c t s 39 4.4.4 Benders or Bimorphs 41 4.4.5 E f f e c t s of Bond between the Transducer and Metal 42 4.5 E l e c t r o n i c C i r c u i t s and Experimental Procedure 43 4.5.1 Recording of Transient Rayleigh Pulses 44 4.5.2 Recording of Time-Harmonic Rayleigh Waves 49 CHAPTER V - ANALYSIS OP RAYLEIGH-WAVE RECORDS 5.1 General 53 5.2 Analysis of a Rayleigh Pulse from a Distant Impulse 53 5.3 Fourier Integral Method which Incorporates F i l o n ' s Method 57 5.4 Amplitude of Signal 62 5.5 Noise Level of Record 64 5.6 Distortions of a Pulse Spectrum During Harmonic Analysis 65 5.7 Attenuation of Rayleigh Waves Close to the Source 69 5 . 8 C a l c u l a t i o n of the Attenuation of the F o u r i e r Components of a Pulse 72 5 . 9 C a l c u l a t i o n of Attenuation of Time-Harmonic Raylelgh Waves 74 5 . 1 0 I n t e r n a l F r i c t i o n . . . . . 76 CHAPTER VI - RESULTS 6 . 1 General 78 6 . 2 Grain Structure...» 78 6 . 3 Sound S c a t t e r i n g - E f f e c t s ; 79 6 . 4 Records of Rayleigh Waves 80 Ir 6 . 5 R e s u l t s f o r Brass Tube of 10 cm. Diameter 80 6 . 6 R e s u l t s f o r Brass Tube of 15 cm. Diameter 83 c 6 . 7 R e s u l t s f o r Aluminum Tube of 1 3 . 7 cm. Diameter v. 84 6 . 8 R e s u l t s f o r Copper Tube of 1 2 . 5 cm. Diameter 85 6 . 9 Curve F i t t i n g 8 7 6 . 1 0 The General Trend of 1/Q 88 6 . 1 1 Anomalous Peaks 89 6 . 1 2 Conclusion 90 APPENDIX 152 BIBLIOGRAPHY 163 V LIST OP FIGURES P i g . 3.1 F i r s t symmetric mode of a f r e e e l a s t i c p l a t e ( a f t e r Tolstoy and Usdia [1953] ) 91 F i g . 3-2 Phase v e l o c i t y of Rayleigh wave propagating around circumference of homogeneous c y l i n d e r ( a f t e r O l i v e r [1954] ) 91 F i g . 3.3 Co-ordinate system of c y l i n d r i c a l s h e l l 92 P i g . 3-4 Phase and group v e l o c i t y of Rayleigh wave propagating along truncated edge of c y l i n d r i c a l s h e l l 92 F i g . 4.1 Comparison of amplitude d e n s i t y spectra of Rayleigh pulse and sine wave 93 P i g . 4.2 S t r a i n as a f u n c t i o n of the product of the wave number of a Rayleigh wave and the distance z from the surface 94 P i g . 4.3 Equivalent c i r c u i t f o r p i e z o e l e c t r i c transducer ( a f t e r Mason [1942J ) 95 P i g . 4.3 Equivalent c i r c u i t f o r p i e z o e l e c t r i c transducer ( a f t e r Mason [1942] ) 96 F i g . 4.4 P i e z o e l e c t r i c bender and equivalent electromechanical c i r c u i t ( a f t e r Mason U942J ) 97 F i g . 4.5 E f f e c t of bonding m a t e r i a l on c r y s t a l performance ( a f t e r Mason [1942] ) 97 P i g . 4.6 Schematic diagram of composite c i r c u i t f o r d e t e c t i o n and rec o r d i n g of a seismic pulse 9^ P i g . 4.7 Schematic diagram of c i r c u i t f o r d e t e c t i o n , a m p l i f i c a t i o n and photographing a seismic pulse 99 F i g . 4.8 Trigger c i r c u i t f o r seismic pulse 100 F i g . 4.9 Voltage b i a s c i r c u i t f o r seismic pulse. 101 Pig.. 4.10 C i r c u i t f o r production, d e t e c t i o n and recor d i n g of a time-harmonic wavetrain. 102 v i P i g . 4.11 Photograph showing e f f e c t of s c a t t e r i n g of a Rayleigh wave by the source F i g . 5.1 Diagrammatic r e p r e s e n t a t i o n of the method of a n a l y s i s of a seismic F i g . 5.2 Diagrammatic r e p r e s e n t a t i o n of the method of a n a l y s i s of a time-harmonic wavetrain. 105 F i g . 5.3 E l e c t r o n i c noise records and the corresponding F o u r i e r spectra 106 P i g . 5.4 D i s t o r t i o n of pulse spectrum when the e n t i r e s i g n a l i s not analyzed 107 Pig« 5-5 Inadequacies of hand d i g i t i z a t i o n ( a f t e r Bogert et a l [1962J ) . . . 108 F i g . 6.1 G r a i n s t r u c t u r e (255*) of a specimen from (a) brass tube of 10 cm. d i a . and (b) brass tube of 15 cm. d i a 109 F i g . 6.2 G r a i n s t r u c t u r e (255x) of a specimen from (a) aluminum tube of 13-7 cm. d i a . and (b) copper tube of 12.5 cm. d i a . . . . 110 F i g . 6.3 S t r a i n record of a Rayleigh pulse I l l F i g . 6.4 S t r a i n record of a time-harmonic Rayleigh wavetrain. 112 F i g . 6.5 S t r a i n records of Rayleigh pulses propagating around edge of brass tube P i g . 6.6 Amplitude density spectra of records shown i n F i g . 6.5 114 F i g . 6.7 Phase d i f f e r e n c e (80 ) p l o t t e d against frequency... 115 F i g . 6.8 A t t e n u a t i o n c o e f f i c i e n t s (<X ) f o r Rayleigh pulses shown i n F i g . 6.5 116 F i g . 6.9 Attenuation c o e f f i c i e n t s f o r s i n u s o i d a l Rayleigh wavetrains propagating along edge of (a) brass tube of 10 cm. d i a . and (b) brass tube of 15 cm. d i a 117 v i i F i g . 6.10 S u p e r p o s i t i o n of a t t e n u a t i o n c o e f f i c i e n t s shown i n F i g . 6.8 and top of F i g . 6.9 118 F i g . 6.11 S t r a i n records of Rayleigh pulses propagating around edge of brass tube of 15 cm. d i a 119 F i g . 6.12 Amplitude d e n s i t y spectra of records shown i n F i g . 6.11. 120 F i g . 6.13 Attenuation c o e f f i c i e n t s f o r Rayleigh pulses propagating around brass tube of 15 cm. d i a 121 F i g . 6.14 S u p e r p o s i t i o n of the a t t e n u a t i o n measurements shown i n F i g . 6.13 and the bottom-half of F i g . 6 . 9 . . 122 F i g . 6.15 S t r a i n records of Rayleigh pulses propagating around edge of aluminum tube 123 F i g . 6.16 Amplitude density spectra of the records shown i n F i g . 6.15 124 F i g . 6.17 A t t e n u a t i o n c o e f f i c i e n t s f o r Rayleigh pulses propagating around edge of aluminum tube 125 F i g . 6.18 A t t e n u a t i o n c o e f f i c i e n t s f o r s i n u s o i d a l Rayleigh waves propagating around edge of (a) aluminum tube and (b) copper tube 126 F i g . 6.19 S u p e r p o s i t i o n of the a t t e n u a t i o n c o e f f i c i e n t s shown i n F i g . 6.17 and t o p - h a l f of F i g . 6 . l 8 127 F i g . 6.20 S t r a i n records of Rayleigh pulses propagating around edge of copper tube. 128 F i g . 6.21 Amplitude d e n s i t y spectra of the records shown i n F i g . 6.20 129 F i g . 6.22 A t t e n u a t i o n c o e f f i c i e n t s - o f Rayleigh pulses propagating around edge of copper tube 130 v i i i F i g . 6.23 S u p e r p o s i t i o n o f the a t t e n u a t i o n c o e f f i c i e n t s shown i n F i g . 6.22 and i n b o t t o m - h a l f o f F i g . 6.18. 131 F i g . 6 . 24 I n t e r n a l f r i c t i o n o f the f o u r tubes s t u d i e d when the type of e x c i t a t i o n i s a s m a l l - a m p l i t u d e R a y l e i g h p u l s e 132 ix LIST OF TABLES 3 . 1 Phase and group v e l o c i t i e s of Rayleigh waves propagating around truncated edge of c y l i n d r i c a l s h e l l (for the case \ = 2f*.\.. 133 4 . 1 Semiconductor s t r a i n gauge 134 4.2 F e r r o e l e c t r i c ceramic 134 6 . 1 Composition and dimensions of metal tubes... 135 6.2 Attenuation c o e f f i c i e n t ( otP ) for sinusoidal Rayleigh waves propagating around edge of brass tube of 10 cm. dia 136 6 . 3 Attenuation c o e f f i c i e n t ( a ) of Rayleigh pulses (3 - ) propagating around edge of brass tube of 15 cm. d i a . . 137 6 . 4 Attenuation c o e f f i c i e n t of Rayleigh pulses (27 - 36fxe ) propagating around edge of brass tube of 15 cm. dia 138 6 . 5 Attenuation c o e f f i c i e n t of sinusoidal Rayleigh waves propagating around edge of brass tube of 15 cm. dia 139 6 . 6 Attenuation c o e f f i c i e n t of Rayleigh pulses ( 2 - 3 M-€ ) propagating around edge of aluminum tube 140 6 . 7 Attenuation c o e f f i c i e n t of Rayleigh pulses (33 - 4 8 ) propagating around edge of aluminum tube '. 141 6 . 8 Attenuation c o e f f i c i e n t of sinusoidal Rayleigh waves propagating around edge of aluminum tube 142 6 . 9 D i s t o r t i o n of Rayleigh pulse spectrum as a r e s u l t of not analyzing entire length of signal 143 6.10 D i s t o r t i o n of Rayleigh pulse spectrum as a res u l t of not analyzing entire length of signal 144 6.11 D i s t o r t i o n of Rayleigh pulse spectrum as a r e s u l t of not a n a l y z i n g e n t i r e length of s i g n a l 145 6.12 D i s t o r t i o n of Rayleigh pulse spectrum as a r e s u l t of not a n a l y z i n g e n t i r e length of s i g n a l 146 6.13 Attenuation c o e f f i c i e n t of Rayleigh pulses propagating around edge of copper tube 147 6.14 Attenuation c o e f f i c i e n t of s i n u s o i d a l Rayleigh waves propagating around edge of copper tube 148 6.15 Measured phase v e l o c i t i e s of Rayleigh waves propagating around edge of tube 149 6.16 I n t e r n a l f r i c t i o n (l/Q) of the four metals studied 150 6.17 T h e o r e t i c a l curves ( a = c f n ) f i t t e d to experimental a t t e n u a t i o n measurements 151 1 CHAPTER I INTRODUCTION 1.1 General There i s s t i l l no agreement as to whether the p r i n c i p a l a t t e n u a t i o n mechanisms i n granular m a t e r i a l s are l i n e a r f o r f r e q u e n c i e s l e s s than 1 Mc/s. This d i v e r -gence of op i n i o n stems from the f a c t that the a t t e n u a t i o n mechanisms are not f u l l y understood. Various l o s s pro-cesses have been proposed by g e o p h y s i c i s t s , m e t a l l u r g i s t s , and s o l i d - s t a t e p h y s i c i s t s . D i s s i p a t i o n has been a t t r i -buted to e l a s t i c a f t e r e f f e c t s , r e l a x a t i o n phenomena, s l i d i n g f r i c t i o n , d i s l o c a t i o n s , and other mechanisms. Some t h e o r i e s , f o r example the thermo e l a s t i c e f f e c t pro-posed by Zener (1938) are w e l l understood] others, f o r example the d i s l o c a t i o n theory proposed by Koehler (1952) which d e s c r i b e s the amplitude-independent l o s s mechanism i n metals, are not f u l l y understood. I n a d d i t i o n there i s a divergence of o p i n i o n as to the r e l a t i v e importance of many of these t h e o r i e s . I t i s appropriate to des c r i b e a convenient mea-sure of the energy d i s s i p a t i o n of a harmonic wave. The 2 s p e c i f i c d i s s i p a t i o n f u n c t i o n 1/Q (which m e t a l l u r g i s t s g e n e r a l l y r e f e r to as the i n t e r n a l f r i c t i o n ) i s defined as the dimensionless r a t i o A E 2TTE ( 1 - D where A E i s the energy d i s s i p a t e d per u n i t volume per c y c l e and E i s the maximum energy d e n s i t y stored dur i n g the c y c l e . Most measurements of 1/Q i n homogeneous s o l i d s show that f o r small-amplitude s i n u s o i d a l waves 1/Q i s approximately independent of frequency over a range of frequencies which extend from the c/s r e g i o n to the Mc/s r e g i o n . For some s o l i d s d i f f e r e n t modes of v i b r a t i o n have been employed to o b t a i n values of 1/Q at i s o l a t e d f r e q u e n c i e s over a wide frequency range; f o r others, the same mode of v i b r a t i o n has been employed to o b t a i n values of 1/Q at c l o s e l y spaced frequencies over a com-p a r a t i v e l y narrow frequency range. An extensive review of a v a i l a b l e data has been given by Knopoff (1964). Several models have been invoked to account f o r the experimental observation on the frequency inde-pendence of 1/Q i n most s o l i d s f o r small-amplitude 3 e l a s t i c waves. Lomnitz (1957, 1 9 6 2 ) and MacDonald ( 1 9 6 1 ) have proposed a s u p e r p o s i t i o n model f o r which the d i s s i -p a t i v e p r o p e r t i e s are a t t r i b u t e d to an e l a s t i c a f t e r -e f f e c t termed t r a n s i e n t creep. Knopoff and MacDonald ( 1 9 5 8 , I 9 6 0 ) have proposed a phenomenological model which i s n o n l i n e a r and f o r which the d i s s i p a t i v e property i s a t t r i b u t e d to a p e c u l i a r type of s l i d i n g f r i c t i o n which v a r i e s as the a c c e l e r a t i o n ( s o l i d f r i c t i o n ) . Thus both l i n e a r and no n l i n e a r models have been able to account f o r the observed frequency independence of 1/Q i n most s o l i d s . I n t h i s t h e s i s an experimental i n v e s t i g a t i o n i s c a r r i e d out to determine whether the a t t e n u a t i o n of small-amplitude Rayleigh waves i n metals i s the r e s u l t of a l i n e a r mechanism ( i . e . one f o r which the p r i n c i p l e of s u p e r p o s i t i o n holds) i n the frequency range from 1 0 0 to 5 0 0 kc/s. The at t e n u a t i o n r a t e s of the F o u r i e r compo-nents of a Rayleigh pulse are compared w i t h the attenua-t i o n r a t e s of s i n u s o i d a l Rayleigh waves of the same f r e -quency. The o v e r a l l agreement i n d i c a t e s that the energy d i s s i p a t i o n process i n metals f o r small-amplitude e l a s -t i c waves i s l i n e a r i n the frequency range i n v e s t i g a t e d . For granular m a t e r i a l s such as g r a n i t e s , shales and limestones, measurements of 1/Q have been obtained 4 by employing e i t h e r pulses or harmonic waves. Unfortu-n a t e l y both types of e x c i t a t i o n have not been employed on the same specimen. However i n v e s t i g a t i o n s of the at t e n u a t i o n of seismic pulses by McDonal e t . a l . (1958) and Knopoff and P o r t e r (1963) suggest that the attenua-t i o n mechanism i s l i n e a r because 1/Q does not vary s i g -n i f i c a n t l y w i t h frequency. 1.2 Outline of Thesis CHAPTER I - INTRODUCTION: a h i s t o r i c a l review of important i n v e s t i g a t i o n s i s presented, the purpose of the present study i s sta t e d and the contents of each chap-t e r are o u t l i n e d . CHAPTER I I - THEORY: the p r i n c i p l e of super-p o s i t i o n i s de f i n e d , p r o p e r t i e s of the s p e c i f i c d i s s i p a -t i o n f u n c t i o n are described and s e v e r a l t h e o r i e s and models of a c o u s t i c d i s s i p a t i o n i n s o l i d s are discussed b r i e f l y . CHAPTER I I I - SEISMIC MODEL: the seismic model used i n the present study i s described, the e f f e c t of the thi c k n e s s of the model upon the propagation c h a r a c t e r i s t i c s of short wavelengths are discussed, and the theory of d i s -p e r s i o n a s s o c i a t e d w i t h curvature of the model i s developed. 5 CHAPTER IV - INSTRUMENTATION AND EXPERIMENTAL PROCEDURE: p r o p e r t i e s of the transducers used i n the present study are described, schematic diagrams of the e l e c t r o n i c c i r c u i t s are presented, and the experimental procedure i s o u t l i n e d . CHAPTER V - ANALYSIS OF RAYLEIGH-WAVE RECORDS: the method used to analyze a pulse i s described, d i s t o r -t i o n s to the pulse spectrum introduced d u r i n g the a n a l y s i s are described, the method used to c a l c u l a t e the attenua-t i o n of the F o u r i e r components of a pulse i s explained, and f i n a l l y the method used to c a l c u l a t e the a t t e n u a t i o n of s i n u s o i d a l wavetrains i s o u t l i n e d . CHAPTER VI - RESULTS: measurements of the at t e n u a t i o n of the F o u r i e r components of a Rayleigh pulse and the corresponding s i n u s o i d a l waves are presented and compared, and the i m p l i c a t i o n s are discussed. 6 CHAPTER I I THEORY 2.1 General The contents of t h i s chapter are organized i n the f o l l o w i n g manner: the p r i n c i p l e of s u p e r p o s i t i o n i s defined, v a r i o u s p r o p e r t i e s of the s p e c i f i c d i s s i p a t i o n f u n c t i o n 1/Q are described, and f i n a l l y a b r i e f d e s c r i p -t i o n i s given of the l o s s processes i n granular m a t e r i a l s which are e i t h e r known to operate or which are suggested as being capable of opera t i n g i n the frequency range of t h i s experiment (100 to 500 kc/s) and at room temperature. In a l a t e r chapter these l o s s processes w i l l be re-examined f o r the purpose of determining which type or types can best account f o r the measurements of 1/Q obtained i n the present study. 2.2 The P r i n c i p l e of S u p e r p o s i t i o n This p r i n c i p l e s t a t e s : i f r-^ i s the response to R^ and r g i s the response to Rg then r ^ + r g i s the response to Rn + Ro • 7 T h i s p r i n c i p l e i s a property of l i n e a r systems only and i s at the heart of t h i s t h e s i s . Thus, i f the a b s t r a c t i o n of energy from an ac o u s t i c wave by a s o l i d i s the r e s u l t of a l i n e a r process, then the F o u r i e r components of an e l a s t i c wave attenuate at the same r a t e as do s i n u s o i d a l waves of the same frequency. 2.3 The S p e c i f i c D i s s i p a t i o n F u n ction 1/Q A convenient measure of the a c o u s t i c d i s s i p a -t i o n of a harmonic wave i s the s p e c i f i c d i s s i p a t i o n func-t i o n 1/Q, which i s a l s o r e f e r r e d to as the i n t e r n a l f r i c -t i o n . Consider a plane s i n u s o i d a l wave of the form where CL i s the a t t e n u a t i o n c o e f f i c i e n t , a) i s the angular frequency and k i s the wave number. The i n t e r n a l f r i c t i o n i s r e l a t e d to the a t t e n u a t i o n c o e f f i c i e n t by the f o l l o w i n g (see Ko l s k y [1953] ) e e (2-1) (2-2) 8 where A E and E are d e f i n e d as i n eq. ( 1 -1 ) and c i s the phase v e l o c i t y . For a given specimen, v a r i o u s types of e x c i t a t i o n such as f l e x u r a l , l o n g i t u d i n a l or shear waves can be employed to o b t a i n values of 1/Q. Experimental evidence seems to i n d i c a t e that 1/Q does depend upon the type of e x c i t a t i o n ; but I t i s not c l e a r what the dependence i s . 2.4 D e s c r i p t i o n of Acoustic D i s s i p a t i o n There i s s t i l l doubt concerning the p r i n c i p a l a t t e n u a t i o n mechanism i n p o l y c r y s t a l l i n e metals f o r small-amplitude a c o u s t i c waves i n the frequency range from 10 kc/s to 1 Mc/s and i n the normal temperature range. Four types of d i s s i p a t i v e mechanisms are discussed: the t h e r m o e l a s t i c e f f e c t , e l a s t i c a f t e r -e f f e c t s , d i s l o c a t i o n damping and s l i d i n g f r i c t i o n . The f i r s t three types are l i n e a r , whereas the f o u r t h i s n o n l i n e a r . 2 . 4 . 1 The Thermoelastic E f f e c t Granular m a t e r i a l s which c o n s i s t of cubic c r y -s t a l s or which are e l a s t i c a l l y a n i s o t r o p i c e x h i b i t the t h e r m o e l a s t i c e f f e c t . The f o l l o w i n g theory d e s c r i b i n g t h i s e f f e c t i s due to Zener ( 1 9 3 8 ) . In response to an 9 o s c i l l a t i n g s t r e s s neighboring g r a i n s experience d i f f e r -ent amounts of s t r a i n and then as a r e s u l t of the thermo-e l a s t i c e f f e c t a temperature d i f f e r e n c e a r i s e s between these g r a i n s . I t i s the d i f f u s i o n of t h i s excess heat to neighboring g r a i n s that c o n s t i t u t e s the source of a c o u s t i c d i s s i p a t i o n . Consider a re g i o n which has re c e i v e d a greater volume s t r a i n than i t s surroundings. Then t h i s process of heat generation and d i f f u s i o n i s de s c r i b e d by - - t : a T ~ywt (2-3) where A T i s the temperature d i f f e r e n c e between the s t r a i n c o n c e n t r a t i o n and i t s surroundings, T £ i s the r e l a x a t i o n time f o r constant s t r a i n , e i s the d i f f e r e n c e between the volume s t r a i n w i t h i n and i n the neighborhood of the s t r a i n c o n c e n t r a t i o n and Y i s the rate of change of temperature w i t h respect to s t r a i n under a d i a b a t i c con-d i t i o n s . The s t r e s s - s t r a i n r e l a t i o n i s given by (2-4) where i s the r e l a x e d or isothermal modulus and \ i s the l i n e a r thermal expansion c o e f f i c i e n t . The a p p l i c a -10 t i o n of (1-1) i n co n j u n c t i o n w i t h (2-3) and (2-4) r e s u l t s i n an expression f o r 1/Q which i s as f o l l o w s : I / C p ~ C \ r pj f f o / Q " c7~ r + c (2-5) where C P and C v are the s p e c i f i c heat at constant pressure and constant volume r e s p e c t i v e l y , R i s the a n i -sotropy f a c t o r and i s tha t f r a c t i o n of the t o t a l s t r a i n energy which i s a s s o c i a t e d w i t h the f l u c t u a t i o n i n d i l a -t a t i o n , f i s the r e l a x a t i o n frequency. The maximum value of 1/Q occurs at the r e l a x a t i o n frequency. This frequency i s r e l a t e d to the thermal and geometric proper-t i e s of the g r a i n s by the f o l l o w i n g : f - 3x _ D _ = 2 c f (2-6) where d i s the mean g r a i n diameter and D, the thermal d i f f u s i v i t y . 2 . 4 . 2 . E l a s t i c A f t e r e f f e c t s S e v e r a l models based on Boltzmann's superposi-t i o n p r i n c i p l e have been proposed to account f o r the ob-served law of v a r i a t i o n of 1/Q w i t h frequency f o r most 11 s o l i d s . According to t h i s p r i n c i p l e the mechanical be-haviour of a substance i s i n f l u e n c e d by i t s e n t i r e pre-vious l o a d i n g h i s t o r y . Thus i f a substance experiences' a number of deformations, then i t s subsequent behaviour i s the sum of the e f f e c t s of each deformation i n the absence of others. A convenient r e p r e s e n t a t i o n of B o l t z -mann1 s equation i s (Lomnitz [1957] ) € ( t ) = M cr t (t) + JCT(T) 0 ( t - T ) d r (a) o r cr( t) = M € ( t ) - J e ( T ) l // ( t - T ) d T (b) (2-7) where <j ( t ) and d ( t ) are the s t r e s s and s t r a i n r e s p e c t i v e l y , M i s the appropriate e l a s t i c modulus, 0 ( 0 i s the creep f u n c t i o n and i s the deformation r e s u l t i n g from a constant f o r c e of u n i t magnitude a p p l i e d suddenly at t = 0 , ^ ( t ) i s the s t r e s s f u n c t i o n and i s the f o r c e which must be a p p l i e d so that the deformation may change a b r u p t l y at t = 0 from zero to u n i t y and remain t h e r e a f t e r at u n i t y . These two expressions have been shown to be mathematically equivalent by Gross (1953). The use of (2-7a) i n c o n j u n c t i o n with a l o g a r i t h m i c creep f u n c t i o n 12 has been shown by Lomnitz (1957* 1962) to r e s u l t i n an expression f o r 1/Q which i s s u b s t a n t i a l l y independent of frequency over a wide i n t e r v a l . Using the same method of approach (with a ge n e r a l i z e d l o g a r i t h m i c creep f u n c t i o n ) MacDonald (1961) has obtained s i m i l a r r e s u l t s ; h i s a n a l y s i s accounts f o r the approximate 2 8 constancy of 1/Q from perhaps 10 c/s to 10 c/s w i t h values of 1/Q of the order of 10"^ to 10" 2 , 2.4.3 D i s l o c a t i o n Damping Many of the l o s s processes i n metals have been i n t e r p r e t e d i n terms of a l i n e i m p e r f e c t i o n termed a d i s -l o c a t i o n . ( D e s c r i p t i o n s of these l o s s processes are given by Mason [ 1 9 5 8 1 .) There are two types of d i s l o c a -t i o n s , an edge and a screw d i s l o c a t i o n . An edge d i s l o c a -t i o n i s a regi o n i n a c r y s t a l where a plane of atoms i s e i t h e r introduced i n t o or a b s t r a c t e d from an otherwise p e r f e c t c r y s t a l . A screw d i s l o c a t i o n i s , i n geophysical terms, the l i n e at the bottom of a s t r i k e s l i p f a u l t . The d i r e c t i o n and amount of t h i s s l i p i s represented by the Burger's v e c t o r . According to Koehler (1952), an o s c i l l a t i n g s t r e s s s e t s l i n e segments of d i s l o c a t i o n s i n motions. These l i n e segments are pinned by i m p u r i t i e s , other d i s -13 l o c a t i o n s , e t c . , and v i b r a t e l i k e s t r e t c h e d s t r i n g s ; the l o s s mechanism i s a t t r i b u t e d to the v i s c o u s - l i k e damping of the o s c i l l a t i n g l i n e s of the d i s l o c a t i o n . The equation of motion of such a l i n e i s of the form ( N i b l e t t and Wilks [196OJ) = b a: si n cut (2-8) where y i s the l a t e r a l displacement at a p o s i t i o n x along the l e n g t h , A the e f f e c t i v e mass per u n i t l e n g t h , B the damping f o r c e per u n i t l e n g t h , C the l i n e t e n -s i o n a s s o c i a t e d w i t h the d i s l o c a t i o n , b the Burger's vector and cr. s i n cot the re s o l v e d component of the a p p l i e d s t r e s s i n the g l i d e plane of the d i s l o c a t i o n . For the frequency range of i n t e r e s t VQ O C L 4 O ) (2-9) where L i s the mean le n g t h of the v i b r a t i n g loop and i s the angular frequency. In order to account f o r the experimental observation that 1/Q i n many metals seems t o vary l e s s r a p i d l y than the f i r s t power of f r e -quency, s e v e r a l m o d i f i c a t i o n s to the above model have been proposed. Wilks (1959) has suggested that may 1 4 decrease w i t h i n c r e a s i n g frequency w i t h the r e s u l t t h a t 1/Q v a r i e s l e s s r a p i d l y than the f i r s t power of the f r e -quency. However, more measurements of the i n t e r n a l f r i c t i o n of the same specimen are req u i r e d over f a i r l y wide ranges of temperature and frequency f o r a b e t t e r i 4 understanding of the dependence of L on frequency. 2 . 4 . 4 S l i d i n g F r i c t i o n A n o n l i n e a r model f o r which the d i s s i p a t i v e mechanism i s a t t r i b u t e d to a s p e c i a l type of s l i d i n g f r i c t i o n ( s o l i d f r i c t i o n ) has been proposed by Knopoff and MacDonald ( 1 9 5 8 * I960). For granular m a t e r i a l s the f r i c t i o n a l f o r c e i s thought to act along the g r a i n boun-d a r i e s . The d i f f e r e n t i a l equation of motion, f o r the one-dimensional case, i s p _ jJ. cfu _ f e b t * x 2. A. i t ' sqn £>u at ( 2 - 1 0 ) where U i s the displacement of a p a r t i c l e , p I s the d e n s i t y , y. i s an e l a s t i c constant, and f o i s a constant and i s a s s o c i a t e d w i t h the l o s s term. The energy 15 l o s t (work done against f r i c t i o n ) per c y c l e i s of the order of f 6 u 9zu 3 t l = fo c o a A & ; the maxi-mum energy stored during the c y c l e i s of the order of ^ p u e = yzpuzr\* (Knopoff [1964] ). Con-sequently 1/Q i s independent of frequency. 1 6 CHAPTER I I I SEISMIC MODEL 3 . 1 General The b a s i s of two-dimensional seismic model-i n g (see O l i v e r e t . a l . [ 1 9 5 4 ] ) i s the observation t h a t the e x t e n s i o n a l p l a t e wave m o d e ) i n t n e model p l a y s the r o l e of the d i l a t a t i o n a l wave i n the plane s t r a i n problem. On t h i s b a s i s O l i v e r e t . a l . ( 1 9 5 4 ) demonstrated the existence of a surface wave which propagated along the edge of the plane two-dimensional model and i s analo-gous to the two-dimensional Rayleigh wave. Seismic models i n the form of d i s k s or c y l i n -d r i c a l s h e l l s provide a convenient means f o r measuring the a t t e n u a t i o n of surface waves. The peri p h e r y of a d i s k and the truncated edge of a c y l i n d r i c a l s h e l l pro-vide c i r c u i t s around which a surface wave can make mul-t i p l e t r i p s without e x p e r i e n c i n g geometric a t t e n u a t i o n . A s i n g l e transducer can be used to detect the same sur-face wave a number of times; consequently a t t e n u a t i o n measurements are f r e e of both instrument c a l i b r a t i o n and d i f f e r e n c e s of c o u p l i n g of the model to the t r a n s -ducer. 1 7 In the f o l l o w i n g s e c t i o n s the e f f e c t s of the thi c k n e s s and the curvature of the model upon the propa-g a t i o n c h a r a c t e r i s t i c s of a surface wave are discussed. The reason f o r s e l e c t i n g the tube over the d i s k f o r the a t t e n u a t i o n measurements i n t h i s study i s given. 3 . 2 D i s p e r s i o n of Short Wavelengths A p l a t e a c t s as a d i s p e r s i v e medium f o r wave-lengths which are comparable to the p l a t e t h i c k n e s s . The dependence of the phase v e l o c i t y of the p l a t e d i l a -t a t i o n a l wave (M-^ mode) upon wavelength i s shown i n P i g . 3 . 1 . For wavelengths greater than the p l a t e t h i c k -ness the phase v e l o c i t y I s approximately constant and can be represented by (see O l i v e r e t . a l . [ 1 9 5 4 ] ) c = _ P ( \ + 2A0_ ft ( 3 - 1 ) where \ and u^. are the Lame e l a s t i c parameters, and p i s the d e n s i t y . But, as the wavelength becomes comparable to the p l a t e t h i c k n e s s , the phase v e l o c i t y s t a r t s to decrease. The propagation c h a r a c t e r i s t i c s of the Rayleigh wave and the p l a t e d i l a t a t i o n a l wave are i n t e r r e l a t e d . Consider the c h a r a c t e r i s t i c equation f o r 18 Rayleigh waves ( O l i v e r e t . al. [ 1 9 5 4 ] ) : ( ( 2 (3-2) where C^, $ and C are the phase v e l o c i t i e s of the Rayleigh wave, the shear wave and the p l a t e d i l a t a t i o n a l i n C r e s u l t s l n a change i n C ^ . Because d i s p e r s i o n introduces c o m p l i c a t i o n s i n t o a t t e n u a t i o n measurements, we s h a l l r e s t r i c t our study to those wavelengths which are greater than the p l a t e t h i c k n e s s ; f o r most m a t e r i a l s a v a i l a b l e commercially t h i s t h i c k n e s s i s approximately 2 mm. By so doing we circumvent d i s p e r s i o n a s s o c i a t e d w i t h t h i s source. 3.3 D i s p e r s i o n of Long Wavelengths The curvature of a d i s k or a tube can i n t r o -duce d i s p e r s i o n i n t o the propagation c h a r a c t e r i s t i c s of surface waves f o r which the wavelengths are comparable to the r a d i u s of the model. wave r e s p e c t i v e l y . Since i s a constant a change 3.3.1 Curvature of Disk The d i s p e r s i o n of Rayleigh waves propagating 1 9 along the edge of a t h i n c i r c u l a r d i s k was both observed and explained t h e o r e t i c a l l y by O l i v e r (see Ewing e t . a l . [ l 9 5 7 ] ) . P i g . 3 . 2 shows the v a r i a t i o n of the phase v e l o -c i t y of the Rayleigh wave w i t h the dimensionless para-meter , where AR i s the wavelength and a i s the r a d i u s of the d i s k . D i s p e r s i o n a s s o c i a t e d w i t h curvature s t a r t s to become appreciable f o r ^ R > -y^j— 3 . 3 . 2 Curvature of Tube The e f f e c t of curvature upon the propagation c h a r a c t e r i s t i c s of a Rayleigh wave t r a v e l l i n g along the truncated edge of a c y l i n d r i c a l s h e l l i s i n v e s t i g a t e d i n t h i s study. The r a d i u s of curvature i n t h i s case i s p e r p e n d i c u l a r to the surface of the model r a t h e r than p a r a l l e l to the s u r f a c e . The theory of the e l a s t i c v i b r a t i o n s of an i n f i n i t e l y l o n g c y l i n d r i c a l s h e l l has been developed by R a y l e i g h ( 1 8 9 4 ) under the assumption t h a t the w a l l t h i c k n e s s of the s h e l l i s a r b i t r a r i l y s m a l l . The w a l l s of the c y l i n d e r s are considered to be f r e e surfaces and consequently, f o r s u f f i c i e n t l y long wavelengths (com-pared w i t h the t h i c k n e s s ) , the s t r e s s e s P , P „ it r r z r a n d a r e n e g l i g i b l e throughout the c y l i n d r i c a l s h e l l , ro Although we seek a s o l u t i o n somewhat d i f f e r e n t 20 from that of Rayleigh, we w i l l f o l l o w h i s n o t a t i o n as c l o s e l y as p o s s i b l e . P i g . 3 .3 shows the co-ordinate system used. The s t r a i n s , i n terms of the displacements u r > u», u t and t h e i r d e r i v a t i v e s , are given by c _ 9 u* C = _Ur , J_ l U f i f - §LUft , _L 3 u » t l " 3 Z ' l > 0 T 0 3 9 » C t ( 9Z 0 3 6 (3-3) Prom Hooke 1s Law, _ 4 M . U + M ) / f , K e \ P,e = (3-*) where V and u^. are the Lame parameters. S o l u t i o n s of the form u. - Ue""*p ev" u. = -Ve i , 5 S*p t' eJ* W Usa + pt) J I / O L e e (3-5) 21 f o r the a x i a l , azimuthal and r a d i a l displacements r e s p e c t i v e l y , w i l l be assumed. Three l i n e a r equations i n U, V and W are obtainable by u s i n g Lagrange»s equations, which are d , dT s 3T 3 S d\\ a n i 9U / 9 U 3 U (3-6) w i t h two s i m i l a r equations i n V and W. T = k i n e t i c energy per u n i t area S = p o t e n t i a l energy per u n i t area N O W S = H ( p r t + P 6 e ta 4 - p r e tie ) T _ £ ± L [ { ^ f + ( | ^ + (-f^ )<) ( 3 - 7 ) where 2H - th i c k n e s s of p l a t e p = d e n s i t y of p l a t e g = a c c e l e r a t i o n due to g r a v i t y The three l i n e a r equations i n U, V and W (with P r r = p r e = p r z = °) a r e ( - 2 \ N + I) J 2 + s* - k\ Q- ) U + i ( 2 N + I) J s V t i 2 N J W = 0 22 i ( 2 N + l ) J s U + (-J*+ 2 ( N + I ) s x - k ^ V + 2(N + l)sW= 0 i 2 N J U + 2(N +l)sV + ( 2(N + I) - k£ a*) W = 0 (3-8) where i s the de n s i t y and p i s the angular frequency. In order that these equations may possess a n o n t r i v i a l s o l u t i o n f o r U, V and W, the se c u l a r determinant of the equations must vanish. The c h a r a c t e r i s t i c equation i s (k ia 1 + J 1 - s 1 ) kV- 2(N + Dk'aV J + s*+ I ) - 4 ( 2 N + I) J - 4 ( 2 N + | ) J s = 0 (3-9) Equation (3-9) being a quadratic i n J , pos-sesses f o u r r o o t s . By the s u b s t i t u t i o n of any one of these roots i n t o (3-8) that set of equations i s s o l v a b l e f o r the r a t i o s V/U and W/U as s o c i a t e d w i t h that p a r t i c u -l a r r o o t . Hence V % - 1 % and W / u = 1 V / A x - - { 2( N + I ) -. K£a*} + 4 J : N 2 3 A = V ( 2 ( N +1) - k^a'J - 4 s ' ( N •+ I) y = 2 { - N J l+ ( N + l ) s 4 ] 1 2. T Z z. Z V = s - J - k . a (3-10) The roo t s w i l l be l a b e l e d w i t h numerical sub-s c r i p t s (e.g. J, ) and the same s u b s c r i p t w i l l be a t -tached to the a s s o c i a t e d q u a n t i t i e s (e.g. U 1, x, etc) Now ,a new s o l u t i o n can be constructed by the s u p e r p o s i t i o n of two such elementary s o l u t i o n s ; thus u, = ( U, e** + U e** ) e i (sa 4 -p t ) U . = - ( V A , ) U , e ^ + ( Y A J U £ e ^ i ( S9 + pt) U r = K s e + p t ; (3-11) For p o s i t i v e r e a l values of J 1 and Jg these equations describe surface waves of the type sought. Since the truncated edge of the c y l i n d r i c a l s h e l l i s a l s o f r e e of t r a c t i o n , the s t r e s s e s P z e and 24 ? z z must vanish at the edge z = 0, r = a . (The s t r e s s P z r has been assumed to be n e g l i g i b l e throughout t h i s treatment.) Theso c o n d i t i o n s imply J i l N + + (sx. + y,)/ /N / A - s + J ; K , / u, + / A , u, + J.(N +j)/ + (sx f t + yft)/ / N A a - s + X x , U 4 = 0 U = 0 (3-12) The c o n d i t i o n f o r a n o n t r i v i a l s o l u t i o n f o r and U 2 i s J, ( N + I )A,/ + sx, + y, / N Jz ( N + I) A a / + s x £ + y, / N •.SA, + J, X, = 0 (3-13) The r e q u i r e d d i s p e r s i o n r e l a t i o n i s now o b t a i n -able from (3-13)* (3-10) and the c h a r a c t e r i s t i c equation (3-9). To ensure t h a t the s o l u t i o n be s i n g l e valued, only i n t e g r a l values are s e l e c t e d f o r s . For each such value of a, the values of which s a t i s f y (3-13) are sought by a t r i a l and e r r o r method, and among these s o l u t i o n s only that value which leads to two p o s i t i v e r e a l r o o t s of (3-9), J-^ and J g , i s s e l e c t e d . This r e s t r i c t i o n i s necessary i n order to ensure that the 25 s o l u t i o n s (3-11) have the character of surface waves. The numerical solxitions for v a r i o u s values of s are shown In Table 3.1 f o r the special case X = 2/I . Instead of tabulating k0 i t s e l f , k»cy - c/p 1 8 shown. The quantity c so defined i s the phase velocity of the surface wave and £ ( h / p ) z i s the v e l o c i t y of the transverse wave i n the medium. Other q u a n t i t i e s may be i d e n t i f i e d as f o l l o w s : 2 n o ^ s = the wavelength, s/a = the wave number and p = /3 i s the angular frequency. Also Q A p , where A p i s the change i n the angular frequency p when s inc r e a s e s by u n i t y , may be represented by the symbol U. For la r g e values of s, where the d i s c r e t e spectrum approaches a continuum, U w i l l be the group v e l o c i t y . The d i s p e r s i o n introduced by curvature may be judged by n o t i n g the rat e at which the var i o u s tabu-l a t e d values approach the l i m i t i n g value given f o r s = o o . i f the s u b s t i t u t i o n y = 0 9 i s made and both a and s a r e allowed to approach i n f i n i t y so that the r a t i o s/a remains f i n i t e , the values a s s o c i a t e d with s = o o are seen to correspond to those f o r f i n i t e wavelengths i n a plane sheet and a l s o to those a s s o c i a t e d with a plane Rayleigh wave i n a n e l a s t i c medium i n which \ = / J l 26 A graph of and vs = s i s shown i n P i g . 3 . 4 . D i s p e r s i o n a s s o c i a t e d w i t h curva-ture of the tube s t a r t s to become appreciable f o r wave-lengths greater than the r a d i u s of the tube. 3 . 3 . 3 Comparison of tube with d i s k Consider a tube and a d i s k of i d e n t i c a l t h i c k -ness, r a d i u s and composition. Let 2H represent the t h i c k -ness, a the r a d i u s , and X * the Ra y l e i g h wavelength . Then the wavelength range over which the propagation char-a c t e r i s t i c s of the Rayleigh wave are a p p r e c i a b l y f r e e of d i s p e r s i o n a s s o c i a t e d w i t h the t h i c k n e s s and the curvature of the seismic model i s (a) f o r the d i s k 2 H < X * and (b) f o r the tube 2 H « = X* Q ( 3 - 1 4 ) Because t h i s wavelength range i s greater f o r the tube than f o r the d i s k , the tube was s e l e c t e d as the seismic model i n t h i s study. 27 CHAPTER IV INSTRUMENTATION AND EXPERIMENTAL PROCEDURE 4 . 1 General Rayleigh pulses and s i n u s o i d a l Rayleigh waves were generated on the truncated edge of a m e t a l l i c tube. The instrumentation was designed to detec t , a m p l i f y and record these waves. The Rayleigh pulse was generated by detonating (through the a p p l i c a t i o n of heat) a small charge placed on the edge of the tube. The time-harmonic wave was generated by p u l s i n g (through the a p p l i c a t i o n of a s i n u s o i d a l voltage) a p i e z o e l e c t r i c c r y s t a l glued onto the edge of the tube. The s i g n a l d e t e c t o r s were miniature semiconductor s t r a i n gauges glued onto the edge of the tube. The recordings were made on p o l a r o i d f i l m . F i r s t the p r o p e r t i e s of the m a t e r i a l s and t r a n s -ducers used to generate and to detect both the Rayleigh pulse and the time-harmonic Rayleigh wave w i l l be presented. Then the various e l e c t r o n i c c i r c u i t s used i n t h i s e x p e r i -ment w i l l be described and i l l u s t r a t e d . 4 . 2 The E x p l o s i v e M a t e r i a l Because values of the a t t e n u a t i o n c o e f f i c i e n t s ( f o r the f o u r specimens tested) were r e q u i r e d over the 2 8 frequency range from 1 0 0 to 5 0 0 kc/s, an e x p l o s i v e was de s i r e d f o r which the d u r a t i o n of detonation was approxi-mately a few microseconds. F o u r i e r a n a l y s i s of the pulse generated from such an e x p l o s i v e would have a s p e c t r a l d e n s i t y w i t h a maximum value i n the range from 200 to 300 kc/s. The reason f o r t h i s i s t h a t the waveform from such a pulse resembles that of a s i n u s o i d a l pulse of one c y c l e , of p e r i o d T say. F o u r i e r a n a l y s i s of the l a t t e r r e s u l t s i n a s p e c t r a l d e n s i t y g ( f ) which contains the f a m i l i a r d i f f r a c t i o n f u n c t i o n . Hence f o r a sine wave of one c y c l e gif) = sin(2n( - f - f )\) _ sm{27t(4: + f) Ij L 2it( ± - f ) 2TI(=L + f ) . ( 4 - 1 ) when the o r i g i n i s taken at the centre of the sine wave. In F i g . 4 . 1 ( a ) i s shown a t y p i c a l s t r a i n record of a Rayleigh wave from a d i s t a n t e x p l o s i o n ; superimposed on t h i s curve i s a sine wave of one c y c l e . The correspond-i n g amplitude d e n s i t y spectrums are shown i n F i g . 4 . 1 ( b ) . The s u p e r p o s i t i o n of d i f f r a c t i o n curves ( 1 ) and ( 2 ) shown i n F i g . 4 . 1 ( c ) r e s u l t s i n the amplitude d e n s i t y spectrum f o r the sine wave; the corresponding spectrum f o r the cosine wave i s obtained by the s u p e r p o s i t i o n of curves ( 1 ) and (3). 29 The explosion-generated pulse has s l i g h t l y sharper peaks than the corresponding s i n u s o i d a l wave of one c y c l e . This r e s u l t s l n l a r g e r values of g ( f ) at freque n c i e s above the dominant one and hence the f i r s t zero c r o s s i n g i s separ-ated much f u r t h e r from the dominant frequency than i n the corresponding s i n u s o i d a l case. I n a d d i t i o n the e x p l o s i o n -generated pulse has a t a i l or wake which enhances the low frequency end of the spectrum. The e x p l o s i v e used to generate a t r a n s i e n t Ray-l e i g h pulse was s i l v e r a c e t y l i d e , AggCg.AgNO^. This ex-p l o s i v e was s e l e c t e d because of the short d u r a t i o n of deto-n a t i o n and ease of p r e p a r a t i o n . Pure s i l v e r n i t r a t e c r y s t a l s are d i s s o l v e d i n d i s t i l l e d water and then acetylene gas i s bubbled through t h i s s o l u t i o n u n t i l a p r e c i p i t a t e forms. The l a t t e r i s then compressed and molded to the d e s i r e d shape. The chemical r e a c t i o n which occurs when s i l v e r a c e t y l i d e i s detonated i s (Stettb a c k e r , 1940) AggCg.AgNO^ - 3Ag(vapor) + C0 2 + CO + 0.5Ng + 185 c a l (*-2) The detonation v e l o c i t y as measured by S t a d l e r (1939) was 3460 meters/sec. 30 4.3 The S t r a i n Gauge The type of s i g n a l d e t e c t o r r e q u i r e d was one wi t h the f o l l o w i n g p r o p e r t i e s : (1) n e g l i g i b l e d i s t u r b i n g i n f l u e n c e (such as s c a t t e r i n g e f f e c t s ) upon the passing wave (2) s u f f i c i e n t s e n s i t i v i t y to detect small-amplitude s i g -n a l s ( i n the m i c r o s t r a i n region) (3) c a p a b i l i t y of d e t e c t i n g wavelengths as short as 5 mm., which corresponds to a frequency of approximately 500 kc/s. This r e s t r i c t s the len g t h of the gauge to l e s s than 2.5 mm. The most advantageous p o s i t i o n to a t t a c h the s t r a i n gauge i s one where, f o r a given wave, the s t r a i n i s a maximum and the noise a minimum. By " n o i s e n i s meant unwanted waves such as f l e x u r a l and p waves. Consider a Rayleigh wave propagating along the surface of an i s o t r o p i c , l o s s l e s s medium. Let the d i r e c t i o n of propagation be along the x - a x i s , w i t h the z - a x i s normal to the surface. Let the corresponding p a r t i c l e d i s p l a c e -ments be u and w r e s p e c t i v e l y . Then u « A ( e " r z - 0 . 5 7 7 3 e " s z ) s i n k ( x - c t ) (4-3) 31 w = A(0.8475e" r z - 1.4679e~ s z) cos k(x-ct) ( 4 - 4 ) wh@r© k s wave number c 9 phase v e l o c i t y of Rayleigh wave v • 0,1 f o r Poisson 1 s r a t i o ?= 1 / 4 A = constant The d i f f e r e n t i a t i o n of (4-3) with respect to x and (4-4) with respect to z r e s u l t s i n the following expressions f o r the s t r a i n : = Ak(e" r z - 0 . 5 7 7 3 e ) cos k(x-ct) 3 X d w = A(^0.8475re" r z + 0.5773se" s z) cos k(x-ct) (4-5) 3z (4-6) Prom equation (4-5) i t can be seen that the s t r a i n amplitude i s equal to the product of the displacement am-p l i t u d e and the wave number, or e q u i v a l e n t l y the s t r a i n amplitude i s p r o p o r t i o n a l to the displacement amplitude d i v i d e d by the wavelength. From P i g . 4.2 i t can be seen that the s t r a i n has a maximum value ( ••= 0.42) at kz = 0. For t h i s 3x 32 reason the s t r a i n gauge was positioned on the truncated edge of the cylinder and aligned i n a tangential d i r e c t i o n . To minimize the influence of f l e x u r a l waves, the gauge was located midway between the inner and outer faces; i n beam bending theory t h i s would be referred to as the "neutral" axis. f o r a harmonic wave, the wavelength must be much greater than the gauge length. The reason f o r t h i s i s because the s t r a i n gauge measures the average s t r a i n occurring over i t s entire length and f o r wavelengths comparable to the gauge length, part of the gauge would be under com-pression while simultaneously another part would be under tension. Thus the length of the gauge puts a upper l i m i t upon the frequency of the wave that i s accurately detectable. For a Rayleigh wave propagating along the edge of a brass sheet t h i s cut-off frequency would be close to 500 kc/s fo r a gauge of length 2 mm. An important property of a s t r a i n gauge i s i t s "Gauge Factor", GF, which i s defined as A R / For a s t r a i n gauge to be an accurate transducer GF A L (t-7) 33 where L i s the l e n g t h of the unstr a i n e d gauge, A L i s the change i n length of the gauge, R i s the e l e c t r i c a l r e s i s t a n c e of the unstr a i n e d gauge, and A R i s the change i n r e s i s t a n c e of the gauge a s s o c i a t e d w i t h l e n g t h change A L. Semiconductor s t r a i n gauges w i t h gauge f a c t o r s v a r y i n g from 1 1 0 to 1 2 5 were used to detect the Rayleigh waves i n t h i s experiment; Table 4 . 1 shows the dimensions and composition of these gauges. M e t a l l i c s t r a i n gauges, i n comparison, have gauge f a c t o r s which are g e n e r a l l y l e s s than f i v e . The two important p r o p e r t i e s of these semi-conductor s t r a i n gauges are t h e i r l a r g e gauge f a c t o r and t h e i r i n s i g n i f i c a n t mass; the former property enables small s t r a i n s of the order of a m i c r o s t r a i n to be detected and the l a t t e r property ensures that s c a t t e r i n g of the Rayleigh wave w i l l be i n s i g n i f i c a n t . 4 . 4 P i e z o e l e c t r i c Transducers P i e z o e l e c t r i c transducers produce a mechanical displacement which v a r i e s d i r e c t l y as the a p p l i e d e l e c t r i c f i e l d . In a d d i t i o n the e f f i c i e n c y with which these c r y s t a l s can convert energy from e l e c t r i c a l to mechanical form i s q u i t e high i n the k i l o c y c l e range. Hence they provide a convenient means f o r producing time-harmonic waves i n metals. 34 4.4.1 E q u i v a l e n t C i r c u i t s of Transducers Recent i n v e s t i g a t i o n s i n t o the v i b r a t i o n p a t t e r n s of the ends of s o l i d c y l i n d r i c a l barium t i t a n a t e (BaTiO^) ceramics have shown a departure from plane p i s t o n - l i k e motion (Shaw [ 1 9 5 6 ] , Arnold and Martner [1959J , and Shaw and S u j i r DL960J ). Nevertheless, when a n a l y z i n g the performance of t h i s type of transducer under v a r i o u s ap-p l i c a t i o n s , i t i s convenient to assume plane p i s t o n - l i k e motion f o r the ends. Consider such a c r y s t a l w i t h a f i e l d E^, a p p l i e d as shown i n F i g . 4 . 3(a). The p i e z o e l e c t r i c r e l a t i o n s are T = c£ s - e E (4-8) D x = e M S , + e t E . (4-9) where T-^  and S-^  are the e x t e n s i o n a l s t r e s s and s t r a i n r e s p e c t i v e l y , i s Young's modulus, 6 n i s the p i e z o e l e c t r i c constant r e l a t i n g s t r e s s w i t h the a p p l i e d f i e l d , £^ i s the d i e l e c t r i c constant at constant s t r a i n , and D-^  i s the e l e c t r i c displacement. An important f a c t o r i n the s e l e c t i o n of a t r a n s -ducer i s i t s electromechanical c o u p l i n g f a c t o r , which i s 35 a measure of the e f f e c t i v e n e s s of the transducer i n energy-conversion and i s analogous to the c o u p l i n g c o e f f i c i e n t of a voltage transformer. With the exception of Rochelle s a l t (45° X-cut) f o r which k = O.78, c r y s t a l l i n e substances have values of k which are much lower than those f o r com-m e r c i a l f e r r o e l e c t r i c ceramics (k — 0.45 - 0 . 7 0 ) . Be-cause of t h e i r high l i n e a r c o u p l i n g c o e f f i c i e n t (0.70), ceramic transducers of the type US500 (developed by Sonus Corporation) were used f o r the generation of time-harmonic waves; Table 4.2 shows the dimensions and p r o p e r t i e s of these transducers. A convenient method f o r a n a l y z i n g the performance of p i e z o e l e c t r i c transducers i s through the use of e q u i -v a l e n t e l e c t r i c a l c i r c u i t s . The " d i r e c t " or "impedance" type of analogy i s used here; that i s , f o r c e —»- v o l t a g e , v e l o c i t y — - c u r r e n t , e t c . By s u b s t i t u t i n g the p i e z o -e l e c t r i c r e l a t i o n s i n t o Newton's Second Law of Motion, and by comparing the s o l u t i o n w i t h e l e c t r i c a l network theory, Mason (1942) has d e r i v e d the electromechanical equivalent c i r c u i t shown i n P i g . 4 . 3(b). For the case of " i n e r t i a d r i v e " where one end of the c r y s t a l d r i v e s a load and the other end i s f r e e , the l a t t e r r e a c t s against a i r which has a comparatively low mechanical impedance. Hence F, = 0 and t h i s i s 36 e q u i v a l e n t to short c i r c u i t i n g the ends on the l e f t of the c i r c u i t shown i n P i g . 4 . 3(b). When the c r y s t a l i s v i b r a t i n g at or c l o s e to i t s mechanical resonant frequency ( i . e . when the wavelength of the v i b r a t i o n i s one h a l f the c r y s t a l length) the c i r c u i t shown i n P i g . 4.3([c) i s a p p l i c a b l e . The combined reactance of the shunting elements, C 2 and M 2 , decreases as the frequency d e v i a t e s from the mechanical resonant frequency w i t h the r e s u l t that the output s i g n a l or response decreases. When a p i e z o e l e c t r i c c r y s t a l i s attached to a metal, the reac-tance of t h i s shunt element i s approximately twenty times as l a r g e as the mechanical impedance of the metal i n the frequency range < -f -^-f0 where f Q i s the resonant frequency; hence t h i s element can be neglected u> and the r e s u l t i n g c i r c u i t i s shown i n P i g . 4 . 3(d). Mechanical tuning of the transducer has already been in c o r p o r a t e d i n t o the above-mentioned c i r c u i t s by s e l e c t i n g a frequency range which i s near the mechanical resonant frequency. Another method of mechanical t u n i n g which r e s u l t s i n supe r i o r performance i s discussed by McSklmin (1959). The c i r c u i t may a l s o be tuned e l e c t r i -c a l l y . T h i s i s accomplished by i n s e r t i n g e i t h e r a s e r i e s or a shunt c o i l as shown i n F i g . 4 . 3(e); the s e r i e s c o i l r e s u l t s i n a low input impedance and the shunt c o i l i n a high input impedance. Since the RF pulse generator 3 7 (Model 6 0 0 , Madison I n d u s t r i e s ) used to d r i v e the t r a n s -ducer has a high output impedance ( — 1 0 K ) , the shunt c o i l set up was used. The i n s e r t i o n of the shunt c o i l n e u t r a l i z e s the e f f e c t s of the s t a t i c capacitance, w i t h the r e s u l t that (a) the e f f i c i e n c y of conversion of energy i s increased and (b) the ceramic can be pulsed over a wider range of frequencies by the pulse o s c i l l a t o r . For f r e q u e n c i e s s u f f i c i e n t l y removed from the mechanical resonant frequency, the shunting e f f e c t s of L — C „ and M 0 — C 0 become pronounced ( i . e . the impe-0 0 2 2 x dances across these branches decrease) w i t h the r e s u l t that the performance of the c r y s t a l becomes very poor. A convenient method f o r improving the performance under these c o n d i t i o n s i s to vary L Q ; as the frequency decreases, L Q i s increased and v i c e versa. F i g . 4 . 3 ( f ) i s then a p p l i c a b l e . 4.4.2 Importance of Geometry of Transducer For c i r c u l a r c y l i n d r i c a l s o l i d s , the aspect r a t i o , d e f i n e d as the r a t i o of the r a d i u s to the le n g t h , i s very i n f l u e n t i a l i n determining which modes of v i b r a t i o n pre-dominate at a given frequency. Shaw ( 1 9 5 6 ) has i n v e s t i -gated the a x i a l p a r t i c l e motion of the ends of BaTiO, d i s k s 38 and has observed s i g n i f i c a n t departures from a uniform p i s t o n - l i k e motion. In the t h i c k n e s s resonance regi o n , the electromechanical c o u p l i n g f a c t o r k f o r a p a r t i c u l a r mode was found to depend on the aspect r a t i o . I n general, the l a r g e r the aspect r a t i o , the greater the number of modes present. For higher frequencies such t h a t the wavelength i s l e s s than 1.25 times the le n g t h of the c r y s t a l , the motion of the d i s k i s confined to i t s p e r i -meter. Such a pronounced departure from uniform p i s t o n -l i k e motion could s e r i o u s l y d i s t o r t the waveform i n the medium to which the c r y s t a l i s attached. R a d i a l p a r t i c l e motions of the ends of short s o l i d c y l i n d e r s of BaTiO^ have been e x p e r i m e n t a l l y deter-mined by Arnold and Martner (1959). T h e i r i n v e s t i g a t i o n shows that as the aspect r a t i o of the ceramic i n c r e a s e s , the r a d i a l motions tend to predominate over the a x i a l motions. When the aspect r a t i o e q u a l l e d one, the maximum p a r t i c l e displacements i n the two orthogonal d i r e c t i o n s were equal, and when the aspect r a t i o was one h a l f , the r a d i a l motion was i n s i g n i f i c a n t . I n t h i s experiment the aspect r a t i o v a r i e d from 0.2 to 1.0. The f a c t t h a t one end of the c r y s t a l was glued onto a metal would tend to dampen h i g h l y any r a d i a l motion and c o n s i d e r i n g the r e l a -t i v e l y small aspect r a t i o s i n v o l v e d , i t would appear safe to conclude that r a d i a l p a r t i c l e motion i s I n s i g n i f i c a n t 39 i n t h i s experiment. Even i f the r a d i a l modes of a t r a n s -ducer are e x c i t e d i t does not i n v a l i d a t e the experiment. 4.4.3 C r y s t a l S c a t t e r i n g E f f e c t s A t r a i n of Rayleigh waves, propagating along the edge of a truncated c y l i n d r i c a l s h e l l (see P i g . 4.11(b)), encounters the transducer used to e x c i t e the waves at the t e r m i n a t i o n of each complete c i r c u i t . The r e s u l t i n g s c a t t e r i n g of the Rayleigh wave can be thought of as being due to (1) s c a t t e r i n g from a r i g i d o b s t a c l e (the transducer) and (2) a b s o r p t i o n of energy by an e l a s t i c object (the transducer) and the subsequent r e -r a d i a t i o n of energy. I n both cases mode conversion should occur w i t h the generation of the e x t e n s i o n a l and shear waves. The question of how much energy was converted by c r y s t a l s c a t t e r i n g i n t o the var i o u s modes was not of primary concern i n t h i s experiment: the important p o i n t was the shape of the tr a n s m i t t e d R ayleigh wave. I f , through s c a t t e r i n g and i n t e r f e r e n c e e f f e c t s , the tr a n s m i t t e d Ray-l e i g h waveform were s e r i o u s l y a f f e c t e d , then t h i s method of measuring the a t t e n u a t i o n of a time-harmonic Rayleigh wavetrain would be very l i m i t e d . P r e d r i c k s and Knopoff (i960) have i n v e s t i g a t e d 40 t h e o r e t i c a l l y the r e f l e c t i o n of a time-harmonic Rayleigh wave which i s obstructed by a h i g h impedance ob s t a c l e i n s hearless contact w i t h an e l a s t i c half-space of lower impedance. For the r e f l e c t i o n c o e f f i c i e n t of the v e r t i -c a l displacement component of the Rayleigh wave, they o b t a i n a value of O.265 when P o i s s o n 1 s r a t i o equals 0.25. The r e f l e c t e d wave i s found to be 90° out of phase w i t h the i n c i d e n t wave. In t h i s experiment the c r y s t a l i s glued onto the metal, and hence the c o n d i t i o n of s h e a r l e s s contact i s not a p p l i c a b l e . Secondly, the c r y s t a l i s e l a s t i c , w i t h an impedance c l o s e to that f o r the metals used. T h i r d l y , the dimensions of the c r y s t a l s (see Table 4.2) are g e n e r a l l y much l e s s than the wavelength of the Ray-l e i g h wave. Thus the theory of F r e d r i c k s and Khopoff i s not a p p l i c a b l e to t h i s experiment. In F i g . 4.11(b) i s shown an experimental a r -rangement f o r determining the a t t e n u a t i o n c o e f f i c i e n t of a time-harmonic Rayleigh wave. The p i e z o e l e c t r i c c r y s t a l generates two wavetrains, one t r a v e l l i n g clockwise and the other counterclockwise around the edge of the c y l i n -der. A f t e r making a complete c i r c u i t , both wavetrains a r r i v e at the c r y s t a l l o c a t i o n simultaneously and i n phase. A f t e r p assing by the c r y s t a l s i t e each wavetrain 41 should be a s u p e r p o s i t i o n of two waves; the t r a n s m i t t e d component of a wave t r a v e l l i n g i n one d i r e c t i o n (say counterclockwise) p l u s a small r e f l e c t e d component from a wave t r a v e l l i n g i n the opposite d i r e c t i o n . P i g . 4.11(a) shows from top to bottom the f i r s t , t h i r d and f i f t h wave detected by the same s t r a i n gauge. The corresponding t r a v e l path lengths are l/4c, 1 l/4c and 2 l/4c, where c i s the circumference of the c y l i n d e r . Since there i s no n o t i c e a b l e d i s t o r t i o n of the waveforms ( i i i ) and ( v ) , t h i s would i n d i c a t e e i t h e r that the amount r e f l e c t e d by the c r y s t a l i s i n s i g n i f i c a n t ( f o r t h i s p a r t i c u l a r case at l e a s t ) or the r e f l e c t e d wave i s e i t h e r i n phase or l80° out of phase wi t h the t r a n s m i t t e d wave. In some cases the t r a n s m i t t e d wave has s u f f e r e d a r e d u c t i o n i n amplitude as l a r g e as 30$ and the envelope of the wave-t r a i n has been modified. But i n no observed case was the wave d i s t o r t e d i n such a manner as to suggest the presence of a r e f l e c t e d wave 90° out of phase superim-posed on the t r a n s m i t t e d wave. 4.4.4 Benders or Bimorphs A bender i s a transducer constructed by cement-i n g together the f l a t f a c e s of two p i e z o e l e c t r i c c r y s t a l s i n such a way that when the combined u n i t i s f l e x e d , a voltage i s generated across the i n d i v i d u a l c r y s t a l s . 42 Because of i t s high s e n s i t i v i t y and ease of p o s i t i o n i n g , a bimorph i s w e l l s u i t e d as a d e t e c t o r of weak s i g n a l s i n a t r i g g e r c i r c u i t . The bimorph i s d i r e c t i o n s e n s i t i v e and f o r the d e t e c t i o n of a p wave f o r example, one of i t s f l a t f a c e s must be f a c i n g the oncoming p wave f o r maximum s e n s i t i v i t y . For a bender clamped at one end and a f o r c e a p p l i e d at the other end, the e q u i v a l e n t electromechanical c i r c u i t shown i n F i g . 4.4(b) i s a p p l i c a b l e near i t s mecha-n i c a l resonant frequency. Because of the s e r i e s L-C combination on the mechanical side of the e q u i v a l e n t c i r c u i t , the response i s a maximum at the mechanical resonant frequency and decreases on e i t h e r side of t h i s frequency. Since the dominant frequency of the t r i g g e r pulse and the mechani-c a l resonant frequency of the bender were s u f f i c i e n t l y c l o s e to each other i n t h i s experiment, the s e n s i t i v i t y of the bender was high ( i n f a c t , much higher than the s e n s i t i v i t y of the s t r a i n gauge c i r c u i t ) . 4.4.5 E f f e c t s of Bond between the Transducer and Metal For maximum t r a n s f e r of energy with minimum d i s t o r t i o n of waveform a t h i n l a y e r of bonding m a t e r i a l w i t h a low compliance i s r e q u i r e d . I n the equivalent 43 c i r c u i t shown i n P i g . 4 . 5(b), the bond appears i n the nature of a shunt compliance across the mechanical t e r -minals; the reason f o r t h i s i s th a t the s t r e t c h i n g of the bond i s governed by the d i f f e r e n c e i n p a r t i c l e v elo-c i t i e s of the c r y s t a l and the metal. Thus the e f f e c t i v e compliance of the bond l a y e r decreases as both the t h i c k -ness and the compliance of the bond decrease. As the c r y s t a l r i n g s , twice each c y c l e a l l the energy of the c r y s t a l i s stored i n k i n e t i c form i n IV^  and twice each c y c l e i n p o t e n t i a l form i n C^. Since Cg i s a r e a c t i v e element and does not absorb energy from the system, the smaller Cg can be made the l a r g e r i t s reactance ( f o r a given frequency) and t h i s r e s u l t s i n a greater percentage of the curren t f l o w i n g through the load r e s i s t a n c e R^. Consequently decreasing the bond t h i c k n e s s r e s u l t s i n a more r a p i d r a t e of abstrac-t i o n of energy from the system. Thus c r y s t a l r i n g i n g i s damped out more r a p i d l y . 4.5 E l e c t r o n i c C i r c u i t s and Experimental Procedure The various e l e c t r o n i c c i r c u i t s and i n s t r u -ments used f o r the d e t e c t i o n , a m p l i f i c a t i o n and record-i n g of Rayleigh pulses and s i n u s o i d a l R ayleigh waves are described and i l l u s t r a t e d . Two sets of Rayleigh pulses were detected, one set of the order of f o u r 44 m i c r o s t r a i n i n amplitude and another set approximately ten times as l a r g e . The s t r a i n amplitude of the s i n u -s o i d a l Rayleigh waves v a r i e d from approximately two to ten m i c r o s t r a i n . 4.5.1 Recording of Transient Rayleigh Pulses A schematic diagram of the d e t e c t i o n and reco r d i n g of a Rayleigh pulse (generated by an explosion) as i t makes m u l t i p l e t r i p s around the truncated edge of a c y l i n d r i c a l s h e l l i s shown i n F i g . 4 . 6 . This composite c i r c u i t may co n v e n i e n t l y be d i v i d e d i n t o three separate c i r c u i t s : the s i g n a l , the t r i g g e r and the voltage b i a s c i r c u i t s . (a) S i g n a l C i r c u i t The c i r c u i t used f o r the d e t e c t i o n , a m p l i f i c a -t i o n , and r e c o r d i n g of a s i g n a l i s shown i n F i g . 4 .7 . Semiconductor s t r a i n gauges (see Table 4.1) were attached w i t h a t h i n f i l m of Eastman 910 adhesive. E l e c t r i c a l i n s u l a t i o n between the s t r a i n gauge and the metal was obtained by g l u i n g a t h i n i n s u l a t i n g t i s s u e onto the metal and then a t t a c h i n g the gauge to t h i s t i s s u e . T h i s cement, which c o n s i s t s l a r g e l y of cyanoacrylate monomer, has the s p e c i a l property of hardening upon the a p p l i c a t i o n of pressure. I n a d d i t i o n , cured Eastman 910 adhesive Q has a value of Young's modulus of approximately K r 45 n e w t o n s s which i s only an order of magnitude or so meter1 l e s s than that f o r the metals used i n t h i s experiment. The time taken f o r a complete record (6 t r a c e s of the same Rayleigh pulse) was approximately one m i l l i -second. Since v a r i a t i o n s of the voltage and r e s i s t a n c e of the b a t t e r y and the temperature are n e g l i g i b l e i n such a short d u r a t i o n , the simplest type of s t r a i n gauge c i r -c u i t was employed. The s o - c a l l e d " s e r i e s " or "potentio-meter" c i r c u i t was used. The voltage output A e g. from the potentiometer c i r c u i t i s given by A e 9 = ( + A Rg) ( Rpot •+ R, -l-ARg R p o t + R 9 s ince A R 5 <; <: Rc, + R o o t (a) A e g = v R < l + a R * Rpot+ Rg A R„ Rp*+ Rg + (-IM-2) / A R V RpoT + R, / ~ Rg_V (b) V R,„\ + R 9  P P ° V R 9 ( ! + R » % ) V R 9 n o n l i n e o r t e r m s -ZJ -(***f + . . (4-10) Thus the output voltage v a r i e s l i n e a r l y w i t h the change i n r e s i s t a n c e only when the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : A R g < < R g and R p e r R . The g maximum value f o r A R g/R g i s approximately 1/100. A 4 6 value f o r R p o t was chosen such t h a t R p or /R was S approximately 15. Next a b a t t e r y was s e l e c t e d so that the steady s t a t e current i n the potentiometer c i r c u i t was the maximum value that the s t r a i n gauge could s a f e l y handle (about 30 m i l l i a m p s ) . To the output from the s t r a i n gauge c i r c u i t a high-pass f i l t e r ( f 1 = 4 0 kc/s) was connected to i s o l a t e the dc current i n the potentiometer c i r c u i t from the r e s t of the s i g n a l c i r c u i t and to f i l t e r out the low frequency n o i s e . I t a l s o d i s c r i m i n a t e s against the d i s p e r s i v e p a r t of the Ra y l e i g h pulse ( P i g . 3 . 4 shows that d i s p e r s i o n of the R a y l e i g h pulse becomes important when the wavelength becomes greater than one s i x t h the c y l i n d e r circumference; t h i s corresponds to a frequency of approximately 4 0 k c / s ) . A Burr-Brown (model 110) s o l i d s t a t e p r e a m p l i f i e r was se l e c t e d because i t has a wide pass band (-3db at 500 kc/s) and a high gain ( v a r i a b l e g a i n up to 300x). The s i g n a l from the p r e a m p l i f i e r was then f e d i n t o the "A" (upper beam) s i g n a l input of a Tekt r o n i x o s c i l l o s c o p e (type 502). Small amplitude pulses (around 5 /><-€. ) were recorded w i t h the preamp set at 300x and the v e r t i c a l sen-s i t i v i t y of the scope at 100 mv/cm. For l a r g e r s i g n a l s (around 50 ) , the corresponding s e t t i n g s were 60x and 200 mv/cm. 47 (b) T r i g g e r i n g C i r c u i t For a t r i g g e r i n g p u l s e , the p l a t e d i l a t a t i o n wave r a d i a t i n g a x i a l l y from the e x p l o s i v e source was used. A bender, w i t h one of i t s f l a t surfaces f a c i n g the charge, was l o c a t e d at a d i s t a n c e from the charge such that the d i l a t a t i o n wave a r r i v e d at the bender j u s t before the Rayleigh pulse reached the s t r a i n gauge. To minimize r i n g i n g i n the bender, the l a t t e r was clamped w i t h foam rubber, and to d i m i n i s h r e f l e c t i o n s , the bender was mounted i n a low v e l o c i t y f i b r e . A high-pass f i l t e r ( f ^ = 7.25 kc/s) was con-nected to the bender to r e j e c t low frequency waves (e.g. f l e x u r a l waves due to asymmetric placement of charge on the edge of the c y l i n d e r ) which are p o t e n t i a l but undesirable sources of t r i g g e r i n g p u l s e s . For small s i g -n a l s (a few m i c r o s t r a i n ) the p r e a m p l i f i e r , w i t h maximum gain s e t t i n g (300x), was found to be i n s u f f i c i e n t to t r i g -ger the 535 scope and so a s o l i d s t a t e a m p l i f i e r of var-i a b l e gain (3 - l8x) was i n s e r t e d f o l l o w i n g the preamp. I n s e r t i o n of a cathode f o l l o w e r between the preamp and the a m p l i f i e r r e s u l t e d i n more r e l i a b l e t r i g g e r i n g . F i g . 4 . 8(a) shows the f i r s t p o r t i o n of the t r i g g e r c i r c u i t . The t r i g g e r s i g n a l was then f e d i n t o the t r i g g e r input of a type 535A T e k t r o n i x o s c i l l o s c o p e , which possesses a single-sweep g a t i n g device. Once the single-sweep has 48 been t r i g g e r e d , the va r i o u s attachments to output gates A and B of scope (shown i n P i g . 4 . 8(b) and (c)) were designed to produce s i x t r i g g e r i n g pulses, w i t h a d j u s t a b l e delays between successive p u l s e s . The widths of the r e c t -angular pulses obtainable at gates A and B of the type 535 scope are c o n t r o l l e d by the r e s p e c t i v e h o r i z o n t a l time/cm knobs. The time i n t e r v a l between the commencement of the wave from gate A and that from gate B i s determined by the product of the s e t t i n g s on the delay-time m u l t i p l i e r and the A time/cm knob. Pour p o s i t i v e t r i g g e r i n g p u lses (1,3*5 and 6 ) are obtainable at the output of the c i r c u i t i n P i g . 4 . 8 ( b ) ; the numbers represent the sequence of a r -r i v a l of the pu l s e s . These pulses were f e d i n t o the A (lower beam) t e r m i n a l of the type 502 T e k t r o n i x scope. Two more t r i g g e r p u lses (2 and 4) are obtainable from the c i r c u i t shown i n F i g . 4 . 8 ( c ) . D i f f e r e n t i a t e d p u l ses are f e d i n t o a monostable v i b r a t o r which i s designed to t r i g g e r on p o s i t i v e peaks only; the l a t t e r correspond to the commencement of the s i g n a l s from gates A and B. The output from the monostable v i b r a t o r i s then d i f f e r e n t i a t e d , sent through a g a t i n g c i r c u i t and then f e d i n t o B (lower beam) of the 502 scope. The lower input s e l e c t o r i s set at A-B so that s i x p o s i t i v e peaks are a v a i l a b l e f o r t r i g -g e r i n g the 502, which i s set f o r i n t e r n a l t r i g g e r i n g on the lower beam. 49 (c) Voltage B i a s C i r c u i t Since s i x t r a c e s of the same Rayleigh wave (as i t makes successive t r i p s around the circumference of a c y l i n d e r ) were to be superimposed on a p o l a r o i d f i l m , i t was necessary to apply a d i f f e r e n t dc voltage b i a s to each t r a c e i n order to separate the waveforms. P i g . 4.9 i l l u s t r a t e s how t h i s was accomplished. The b i a s v o l t -ages from gates A and B and from the monostable v i b r a t o r were added and then f e d i n t o the B upper beam t e r m i n a l of the 502 scope. Care was taken to ensure that the t r a c e s (as seen on the 502 screen) from these b i a s v o l t a g e s were s t r a i g h t and h o r i z o n t a l . This was necessary because the input s e l e c t o r f o r the upper beam was set at A-B, and hence any disturbance from the Input t e r m i n a l B would be super-imposed on the Rayleigh pulse s i g n a l . The voltage b i a s e s were a p p l i e d i n such a man-ner as to p o s i t i o n the l a r g e r t r a c e s near the centre of the p o l a r o i d f i l m (see P i g . 6 . 3 ) . 4.5.2 Recording of Tims-Harmonic Rayleigh Waves A schematic diagram of the c i r c u i t used to gen-er a t e , d e t e c t , a m p l i f y and record time-harmonic R a y l e i g h waves i s shown i n P i g . 4.10. A p i e z o e l e c t r i c c r y s t a l (the generator) and two semiconductor s t r a i n gauges (the de t e c t o r s ) were glued onto the edge of the tube. The 50 R a y l e i g h wavetrains r a d i a t i n g i n both d i r e c t i o n s from the source were detected and photographed. Table 4 . 2 gives a d e s c r i p t i o n of the p r o p e r t i e s of the three f e r r o e l e c t r i c ceramics used. The mechanical resonant f r e q u e n c i e s l i s t e d i n t h i s t a b l e apply to a c r y s t a l w i t h both ends f r e e . For a c r y s t a l attached to a s o l i d t h i s resonant frequency i s lowered, the amount depending both on the nature of the bond and on the mechanical impedance of the s o l i d . S p e c i a l precautions were taken when a t t a c h i n g the p i e z o e l e c t r i c transducer to the metal. A very t h i n m e t a l l i c s t r i p . ( 3 X 0.2 X 0 . 0 0 4 cm) was soldered to each end of the c r y s t a l , w i t h any excess s o l d e r being squeezed out. These f l e x i b l e e l e c t r i c a l leads served to minimize the r e a c t i o n on the c r y s t a l as a consequence of s t i f f e l e c t r i c a l t e r m i n a l s . The c r y s t a l assembly was then glued to the edge of the metal w i t h Eastman 910 adhesive, w i t h care being taken to o b t a i n as t h i n a l a y e r of t h i s glue as p o s s i b l e . Such a setup was found to transmit s a t i s f a c t -o r i l y waveforms of f a i r l y uniform amplitude. Upon t r a -v e r s i n g the region where the c r y s t a l i s attached to the metal, the Rayleigh wavetrain experiences a d i m i n u t i o n of energy ( s c a t t e r i n g by a r i g i d body p l u s a b s o r p t i o n and r e - r a d i a t i o n by an e l a s t i c body). F i g . 4.11(a) shows the t r a n s m i t t e d waveforms to be u n d i s t o r t e d when compared 51 w i t h the i n c i d e n t waveform. Except f o r a l o s s of energy, such t r a n s m i t t e d waveforms have a uniform envelope and hence are s t i l l usable f o r a t t e n u a t i o n measurements. When d i s t o r t i o n of the waveform of the t r a n s m i t t e d Ray-l e i g h wave d i d occur, such cases were not used f o r a t t e n -u a t i o n measurements. The e f f e c t of c r y s t a l r i n g i n g on a t t e n u a t i o n measurements of time-harmonic R a y l e i g h waves can be ex-p l a i n e d w i t h the a i d of P i g . 4.11(b). Consider a s i n u -s o i d a l emf being a p p l i e d to a p i e z o e l e c t r i c c r y s t a l f o r a time T. Two Rayleigh wavetrains are generated, one t r a v e l l i n g i n the clockwise d i r e c t i o n towards and the other towards Sg. The l e n g t h L of each of these waves i s L = VT where V i s the v e l o c i t y of the Rayl e i g h wave along the edge of the tube. (L i s gener-a l l y about one quarter of the tube circumference.) To the t e r m i n a l end of each of these wavetrains w i l l be ap-pended an e x p o n e n t i a l l y damped wave due to c r y s t a l r i n g -i n g . Let the time constant f o r the l a t t e r wave be T D . The presence of t h i s t a i l w i l l have no d i s t u r b i n g i n f l u -ence upon the f i r s t R a y l e i g h wave s i g n a l detected by each s t r a i n gauge. However, the second Rayleigh wave s i g n a l detected b y each gauge can be contaminated by the presence of the t a i l from the f i r s t s i g n a l . Hence c r y s t a l r i n g i n g must be minimized so as not to i n t e r f e r e w i t h 52 the main wavetrains. In addition to the above-mentioned means f o r reducing c r y s t a l ringing, the addition of a r e s i s t o r across terminals C - D (Pig. 4.10) was found to be h e l p f u l . The i n s e r t i o n of a r e s i s t o r across A - C served to i s o l a t e the c r y s t a l from the pulse o s c i l l a t o r and i n ce r t a i n instances resulted i n a more uniform (although diminished i n amplitude) output waveform. A complete set of steady-state records would constitute (a) one group of records of time-harmonic Rayleigh waves of i d e n t i c a l s t r a i n l e v e l of approximately 4 /^ -fe over the frequency range from 100 to 500 kc/s and (b) another group at a s t r a i n l e v e l of approximately 40//€ over the same frequency range. Unfortunately a complete set was not obtainable f o r the following reasons: ( i ) the e f f i c i e n c y of conversion of e l e c t r i c a l to mechanical energy varied with frequency, ( i i ) i n some instances the shape of the envelope of the wavetrain varied with the applied voltage, ( i i i ) in-most cases i t was not possible to generate sinu-soidal waves with s t r a i n amplitudes greater than f i f t e e n microstrain. 53 CHAPTER V ANALYSIS OP RAYLEIGH-WAVE RECORDS 5.1 General The methods used to determine the a t t e n u a t i o n c o e f f i c i e n t of (a) the F o u r i e r components of a Ray-l e i g h pulse and (b) s i n u s o i d a l R a y l e i g h waves are des-c r i b e d . Schematic o u t l i n e s of these methods are shown i n F i g s . 5.1 and 5.2 r e s p e c t i v e l y . V a r i o u s sources of e r r o r i n the a n a l y s i s of a pulse spectrum are des c r i b e d ; the precautions taken t o minimize t h i s noise are presented. 5.2 A n a l y s i s of a Rayleigh Pulse from a D i s t a n t Impulse The Rayleigh wave generated by exploding a small charge i n contact w i t h the truncated edge of a cy-l i n d r i c a l s h e l l i s a t r a n s i e n t s i g n a l which experiences d i s p e r s i o n as w e l l as a t t e n u a t i o n . Hence the phenomenon i s a p e r i o d i c ; there i s no f i x e d fundamental p e r i o d and the waveform seen now does not recu r . Hence a F o u r i e r i n t e g r a l method was used to analyze t h i s type of waveform. A power s p e c t r a l technique may a l s o be used to analyze a t r a n s i e n t wave. However, a l l i n f o r m a t i o n r e l a t i n g to phase angles are l o s t when a power s p e c t r a l a n a l y s i s i s c a r r i e d out and because i n f o r m a t i o n concerning phase angles was 54 d e s i r e d , the l a t t e r technique was not used. Even though the phenomenon we wish to analyze i s a p e r i o d i c i t w i l l , n e v e r t h e l e s s , be shown that the Rayleigh pulse can, as an approximation, be t r e a t e d as a p e r i o d i c phenomenon. A phenomenon which i s a p e r i o d i c can under cer-t a i n c o n d i t i o n s be analyzed by a F o u r i e r s e r i e s technique. Consider, f o r example, a Rayleigh pulse (see F i g . 4.1(a)) propagating along the edge of a brass tube of 15.1 cm. diameter. The time taken f o r the pulse to make one com-p l e t e c i r c u i t around the c y l i n d e r (when d i s p e r s i o n i s n e g l i g i b l e ) i s 2 6 0 jxs and i n f o r m a t i o n i s o b t a i n a b l e at submultiples of t h i s fundamental p e r i o d . Then, i n the absence of d i s p e r s i o n , the F o u r i e r r e p r e s e n t a t i o n i s f i t ) = where n , . . . n , . . . -T*4 ( 5 - D 55 The F o u r i e r components are obtainable only at the d i s c r e t e set of frequencies 10* = ntOo =. n -2.TT n = 1 , 2 , 3 * . . . To ( 5 - 2 ) Information at c l o s e r i n t e r v a l s are simply i n t e r p o l a t e s and are obtainable by d i r e c t i n t e g r a t i o n ( J e f f r e y s [ l 9 6 4 J ) . Because the waveform i s of f i n i t e l e n g t h and has no d i s -c o n t i n u i t i e s , a necessary c o n d i t i o n f o r eq. ( 5 - 1 ) t o be t r u e , namely that j |-f(t)|dt be f i n i t e , i s s a t i s f i e d . However, owing to d i s p e r s i o n a s s o c i a t e d w i t h curvature, the a n a l y s i s becomes s l i g h t l y more complicated, d Let C n be the phase v e l o c i t y of a s i n u s o i d a l R a y l e i g h wave (wavelength ^ h ) propagating i n a d i s p e r s i v e medium; l e t C n be the corresponding v e l o c i t y i n a nondispersive medium. In the case under c o n s i d e r a t i o n , C i s a con-' n stant and i s equal to the v e l o c i t y of a Rayleigh wave propagating along the edge of a t h i n brass sheet (see eq. ( 3 - 2 ) ) . The wavelength /^ n equals where a i s the r a d i u s of the c y l i n d e r and n i s a p o s i t i v e i n t e g e r . From F i g . 3 . 4 i t can be seen that Cn = Cn + ^ Cn n = l , 2 , 3 * . . . ( 5 - 3 ) 56 where 8 C n represents the increase i n phase v e l o c i t y due to d i s p e r s i o n . Let o)J and u) n be the frequencies associated with the Rayleigh waves whose velocities are Cn and C n r e s p e c t i v e l y . Then eq. (5-3) may be written as a ( J r , = u)« + o<On n = 1,2 , 3 , . . . (5-4) where 8k)n represents the increase i n frequency due to d i s p e r s i o n . From Fig. 3 .4 i t can be seen that as the wavelength decreases, i . e . as n in c r e a s e s , Cn de-creases monotonically and approaches C . Hence § (Oo > > ••• 8 0 l n - . ==• 8 0 ) n (5-5) Consequently f o r l a r g e n, (5-6) Thus the pulse propagating i n the d i s p e r s i v e medium under c o n s i d e r a t i o n must be represented by an an-harmonic F o u r i e r s e r i e s . (Sommerfeld (1949) d i s c u s s e s 57 anharmonic F o u r i e r a n a l y s i s . ) However, since our i n t e r e s t i s confined to cases where n i s l a r g e , the frequencies ( u)n ) are approximately i n t e g r a l overtones of a fun-damental tone ( oJ 6 ) with the r e s u l t that the F o u r i e r s e r i e s r e p r e s e n t a t i o n of eg. (5-1) becomes a good approxi-mation to the anharmonic F o u r i e r s e r i e s , 5.3 F o u r i e r I n t e g r a l Method Which Incorporates F i l o n ' s equispaced time i n t e r v a l s and then analyzed by a F o u r i e r i n t e g r a l technique. Let f ( t ) represent the waveform and r iw) , i t s F o u r i e r transform; then Method The t r a c e of a Rayleigh pulse was d i g i t i z e d at f ( t ) and -aa f ( t l c o s c o t d t - f(t)sin«otdt 00 -00 - 0 0 C - i S (5-7) 58 where C i s the r e a l p a r t of F ( U J ) and -S i s the imaginary p a r t of F (CO) The amplitude d e n s i t y spectrum I F a n d the phase 0 are then given by I F ( w ) l = V c 2 + S* 0 = ar c t a n (-S/C) (5-8) Phase data can be analyzed by u s i n g the f o l l o w i n g theory (when a t t e n u a t i o n i s n e g l e c t e d ) ; l e t -f (x,t) r e -present the s i g n a l at a di s t a n c e x from the source at time t and l e t g (co) represent the F o u r i e r transform of f ( 0, t ) • Then » ( u > t - K x ) , f ( x , t ) = — — I g ( w ) e d w g ( u > ) e d c o : (5-9) where t = T + T and T i s the time o r i g i n f o r d i g i -t i z i n g the records. The F o u r i e r transform of the s i g n a l i s 5 9 F = g Cw) e ( 5 - 1 0 ) I n general 9 = | cj j e «1 F = |9je'* where tp = u ) ( T - x / c ) + e ( 5 - 1 1 ) Such a transform I s obtainable Tor every c i r c u i t . The transforms w i l l be denoted by s u b s c r i p t s . Let A = ) circumference of c y l i n d e r . set) ^ T J M - r. 5(9) = Then co ( S T — wt a T - A / c J + O J A C -p- - -7- ) ( 5 - 1 2 ) where C<s, i s the p l a t e Rayleigh v e l o c i t y at i n f i n -i t s frequency and C u> i s the p l a t e Rayleigh v e l o c i t y at angular f r equency 0 0 The v a r i a t i o n of S <p w i t h frequency i s i n -fluenced by trie manner i n which c<u v a r i e s w i t h f r e -quency. D i s p e r s i o n a s s o c i a t e d w i t h curvature has been 60 shown to be n e g l i g i b l e f o r the frequency range of i n t e r e s t . Consequently any anomalous behaviour of 8 0 wit h f r e -quency i s not due to t h i s source. However, f o r l i n e a r m a t e r i a l s d i s p e r s i o n i s a l s o a s s o c i a t e d w i t h a t t e n u a t i o n (Futterman [1962] ). Since t h i s type of d i s p e r s i o n depends upon the manner l n which 1/Q v a r i e s w i t h frequency, a d i s c u s s i o n of the dependence of 8 0 upon t h i s type of d i s p e r s i o n w i l l be postponed u n t i l the measurements of 1/Q have been obtained. The numerical e v a l u a t i o n of i n t e g r a l s such as are shown i n eq. (5-7) present no d i f f i c u l t y at low f r e -quencies. But as the frequency i n c r e a s e s the t r i g o n o m e t r i c f u n c t i o n o s c i l l a t e s more r a p i d l y ; t h i s n e c e s s i t a t e s d i g i -t i z a t i o n at smaller i n t e r v a l s . T h i s d i f f i c u l t y may be p a r t l y overcome by e n l a r g i n g the record. But there are se v e r a l disadvantages to e n l a r g i n g the record. D i s t o r t i o n s of the waveform may be introduced i n the process. I n a d d i t i o n e n l a r g i n g the record a l s o t h i c k e n s the beam tra c e and because measurements are made from a base l i n e to the middle of the t r a c e , t h i s i s another source of e r r o r . A method which circumvents t h i s d i f f i c u l t y at high f r e q u e n c i e s has been developed by F i l o n (Tranter [1959]). The essence of t h i s method l i e s i n the conversion of a d i g i t i z e d record i n t o a s e r i e s of p a r a b o l i c segments, i . e . i n t o a continuous record. The p a r a b o l i c equation f o r each arc i s m u l t i p l i e d 61 by the appropriate trigonometric function (a sine or cosine function) and then integrated by parts over the segment i n t e r v a l . These Integrals are then summed over the entire length of the d i g i t i z e d record. The f i n a l r e s u l t i s f ( t ) c o s w t d t = h f (t)sincjt d t - h OL[f (b) s i ncob - f(a)si nooaj -rx[f(b)coscob - f(a)cosojaj J (5-13) where C 0_ i s the sum of a l l the even ordinates of the Co curve y = f ( t ) cos cot between a and b Inclu-sive l e s s h a l f the f i r s t and l a s t ordinates, i s the sum of a l l the odd ordinates, s 0_ i s the sum of a l l the even ordinates of the curve y - f (t) s i n co"t between a and b in c l u s i v e l e s s h a l f the f i r s t and l a s t ordinates, S~„ n i s the sum of a l l the odd ordinates, cS—X and - 0 - co h (h i s the Interval), The expressions f o r oc and are as follows: 9 3 a e3/s 6* + 0 sin GcosG - 2 si rfQ 9 U + : o s £ G ) - 2 sm GcosG e3v = 4 s i n G Gcos e (5-14) 6 2 Asymptotic expressions f o r oc s {3 and Y f o r small 0 are given i n the Appendix. When P i l o n ' s method i s not used, the d i g i t i z a -t i o n i n t e r v a l i s governed by (a) the shape of the curve being d i g i t i z e d and (b) the freque n c i e s at which an ana-l y s i s i s d e s i r e d . However, by i n c o r p o r a t i n g P i l o n ' s method i n t o the F o u r i e r i n t e g r a l method, the d i g i t i z a t i o n i n t e r v a l i s governed predominantly by the shape of the curve being d i g i t i z e d . Care must be e x e r c i s e d when s e l e c t -i n g the d i g i t i z a t i o n i n t e r v a l i n order to ensure t h a t any sharp peaks which appear l n the o r i g i n a l t r a c e are not s e v e r e l y d i s t o r t e d during the a n a l y s i s . I n f a c t the accuracy of P i l o n ' s method i s governed by how c l o s e l y the shapes of the p a r a b o l i c segments approach the o r i g i n a l waveform. The s t r a i n records of Rayleigh pulses were d i g i t i z e d at equispaced i n t e r v a l s of approximately 0.3 us . The F o r t r a n program which i n c o r p o r a t e s F i l o n ' s method and which was used f o r the a n a l y s i s of the s t r a i n records of the Rayleigh pulses i s shown i n the Appendix. 5.4 Amplitude of S i g n a l The maximum amplitude of the f i r s t of the s i x s i g n a l s detected by the s t r a i n gauge was taken as a mea-sure of the s i z e of the Rayl e i g h p u l s e . By usi n g eq. (4-7) and by r e f e r r i n g to F i g . 4 .7 , the maximum s i g n a l 63 amplitude (MSA) i s e x p r e s s i b l e i n u n i t s of m i c r o s t r a i n as f o l l o w s : MSA( / ue ) = i o 6 (Rpot + R g + Rfl) (vss) ( f ( t ) ) (E) (GP) (R g) ( A p A ) (Enlgmt) (5-15) where Rpot , R g, Rg are the r e s i s t a n c e s of the b a l l a s t r e s i s t o r , the s t r a i n gauge and the b a t t e r y r e s p e c t i v e l y and this combination represents the combined r e s i s t a n c e of the potentiometer c i r c u i t ( i n ohms), E i s the e.m.f. of the b a t t e r y i n the potentiometer c i r c u i t ( v o l t s ) , VSS i s the v e r t i c a l s e n s i t i v i t y of the o s c i l l o s c o p e ( i n v o l t s / cm.), f ( t ) i s the ordinate ( i n cm.), GP i s the gauge f a c t o r of the s t r a i n gauge, Ap^ i s the gain of the pre-a m p l i f i e r , and Enlgmt i s the enlargement f a c t o r which r e l a t e s the s i z e of the photo of the o s c i l l o s c o p e screen to the s i z e of the screen i t s e l f . The small s i g n a l group had a maximum amplitude l n the range 2 to 7 /u.e and the lar g e s i g n a l group i n the 30 to 50 fxe range. The a n a l y s i s of two groups of s i g n a l s , one having a s t r a i n of approximately ten times the other, a f f o r d s a means of determining whether or not the a t t e n u a t i o n mechanism ( f o r the metals t e s t e d , and f o r the m i c r o s t r a i n r e g i o n and frequency range i n v e s t i g a t e d ) i s amplitude independent. 64 5 . 5 Noise Level of Record Typical noise records and t h e i r Fourier spectra are shown i n F i g . 5 . 3 . In the signal c i r c u i t shown i n F i g . 4.7, the p r i n c i p a l noise source i s at the input to the preamplifier p r i o r to the gain control and therefore the apparent contribution to the s t r a i n gauge i s indepen-dent of gain setting. Consequently traces (3) and ( 4 ) i n F i g . 5 . 3 (b) should have density spectra which are comparable to those shown i n (c). An inspection of the amplitude density spectra shown i n the next chapter shows that, f o r most of the records, the amplitude density i s well above the noise l e v e l i n the frequency range 100 kc/s to 400 kc/s; but for frequencies of 450 kc/s and 500 kc/s, the amplitude density of the 4 t h , 5 t h and 6 t h traces quite often becomes comparable to those of the noise spectra. Hence a cut-off was placed upon the amplitude density to ensure that only those values which were appreciably above the noise l e v e l were used i n determining the attenuation c o e f f i c i e n t . A value of 0,2 /xe /Mc/s was selected f o r the small-ampli-tude pulse records; t h i s value i s approximately l / 1 0 t h the average maximum value and corresponds roughly to l / 1 0 0 t h Che energy of the maximum value. At t h i s cut-off the signal-to-noise r a t i o i s approximately 3 : 1 f o r these 65 high f r e q u e n c i e s . For the large-amplitude s i g n a l s a c u t - o f f of approximately 0.5 fxt /Mc/s was s e l e c t e d . Although the noise spectra a s s o c i a t e d w i t h the small and the l a r g e s i g n a l s are of' the same order of magnitude, the p o s s i b i l i t y of i n t r o d u c i n g noise through the gener-a t i o n of waves other than R a y l e i g h waves i s much greater when generating the large-amplitude p u l s e s . For example, e i t h e r through c a r e l e s s p l a c i n g of the charge on the edge of the c y l i n d r i c a l s h e l l or the asymmetrical shape of the charge, f l e x u r a l waves could be generated. Hence the cut-o f f was increased from 0.2 to 0.5 M e /Mc/s to allow f o r the p o s s i b l e i n t r o d u c t i o n of noise from t h i s source. 5.6 D i s t o r t i o n s of a Pulse Spectrum During Harmonic A n a l y s i s spectrum of a pulse when only p a r t of the waveform i s ana-l y z e d and when the zero l i n e of the t r a c e i s d i s p l a c e d have been described by Grat s i n s k y (1962). The d i s t o r t i o n s introduced i n t o the amplitude A pulse of the form f(t) = (5-16) was s e l e c t e d and the le n g t h of the pulse analyzed was 6 6 g r a d u a l l y decreased. The main e f f e c t s were a di m i n u t i o n and broadening of the p r i n c i p a l peak i n the spectrum and the appearance of small peaks i n the high frequency r e g i o n . Since the beginning and the end of a t r a n s i e n t s i g n a l are masked by the everpresent noise, care was e x e r c i s e d when s e l e c t i n g the t e r m i n a l p o i n t s of a pulse i n order to minimize d i s t o r t i o n of the spectrum from t h i s source. The s e l e c t i o n of the approximate l o c a t i o n s of the t e r m i n a l p o i n t s of a pulse was not d i f f i c u l t f o r the records as-so c i a t e d w i t h the two brass c y l i n d e r s and the aluminum tube. However, the t r a c e s of both the small and the l a r g e amplitude pulses propagating around the edge of the copper c y l i n d e r was i r r e g u l a r over the e n t i r e l e n g t h of the t r a c e s , w i t h the i r r e g u l a r i t y becoming more pronounced wi t h each succeeding t r a c e . I n order to determine the e f f e c t on the spectrum when d i f f e r e n t t e r m i n a l p o i n t s were s e l e c t e d , a t y p i c a l t r a c e (see P i g . 5.4) was s e l e c t e d and the p a r t analyzed was g r a d u a l l y i n c r e a s e d . The v a r i a t i o n s i n the amplitude d e n s i t y at high f r e q u e n c i e s (400 kc/s to 500 kc/s) were of such magnitudes th a t r e l i a b l e values f o r the a t t e n -u a t i o n c o e f f i c i e n t were not obtainable at these f r e q u e n c i e s . Consequently the records a s s o c i a t e d with the copper tube were analyzed from 100 to 350 kc/s. The displacement of the zero l i n e of the tra c e tends to d i s t o r t the spectrum predominantly i n the low-67 frequency range. Consider the f u n c t i o n y i t ) = f ( t ) + </>CL) ( 5 -17 ) where f (t ) i s the a c t u a l or c o r r e c t f u n c t i o n and <p ( t ) i s the d i s t o r t i n g f u n c t i o n . ,The corresponding spectrum i s lp ( < j ) = Few) 4 <J)(oo) (5 -18) Three types of d i s t o r t i n g f u n c t i o n s with t h e i r r e s p e c t i v e amplitude d e n s i t y spectra w i l l now be given. TYPE I . The zero l i n e i s t r a n s l a t e d p a r a l l e l to the c o r r e c t base l i n e by a dist a n c e h^. cf.Ct) = o < t < x t < 0 , t > T | $ ( M Zk s m CO £ ( 5 -19 ) TYPE I I . The zero l i n e i s i n c l i n e d w i t h respect to the c o r r e c t base l i n e M l - -£ - t ) 0 < t < T t < 0 , t > T AJ( I - C05U)T ) 2 + ( COT - s in tOT )* (5 -20) 68 TYPE I I I . The zero l i n e i s r o t a t e d about i t s midpoint, (f>3 C t ) h s ( l - - f t ) 0 o < t < r t < o , t ^ T <L(GJ) ^ L C 0 S (5-21) Since each of the three amplitude d e n s i t i e s v a r i e s as p 1/oj or 1/cj , the d i s t o r t i o n i s most pronounced at low f r e q u e n c i e s as mentioned p r e v i o u s l y . Hand d i g i t i z a t i o n a l s o introduces n o i s e i n t o the amplitude spectrum. Bogert e t a l . (1962) i n v e s t i g a t e d the e f f e c t s of d i g i t i z i n g an earthquake record by two d i f -f e r e n t techniques; by an e l e c t r o n i c means and by hand d i g i t i z a t i o n . The e f f e c t s o n the analyses of d i g i t i z i n g by d i f f e r e n t techniques w e r e i n v e s t i g a t e d by comparing the coherence and the amplitude d e n s i t y spectra a s s o c i a t e d w i t h each technique. The. coherence i s an extremely s e n s i -t i v e measure of the agreement between t w o s e r i e s and i s defined as t h e r a t i o o f the absolute value of the cr o s s -p o w e r spectrum to the square r o o t o f t h e product of the auto-power spectra ( G a l b r a i t h [1963] ) . A coherence of 1 i n d i c a t e s that there i s a l i n e a r r e l a t i o n between the two s e r i e s ; a coherence of zero i n d i c a t e s that the two s e r i e s 69 are u n c o r r e l a t e d . ^ P i g . 5.5 shows the coherence and the amplitude spectra of an e l e c t r o n i c a l l y d i g i t i z e d e a r t h -quake record ( l a b e l e d CHP14) and a h a n d - d i g i t i z e d v e r s i o n (CHP14H) of the same record. The sample spacing f o r both the e l e c t r o n i c a l l y d i g i t i z e d and the h a n d - d i g i t i z e d records was 1/10 s e c ; hence the f o l d i n g frequency ( f n ) i n both cases i s 5 cps. The coherence becomes poor above 1.5 cps; the energy spectra tend to diverge at the high-frequency end. These r e s u l t s , which were obtained u s i n g a power s p e c t r a l method of a n a l y s i s , i n d i c a t e the inadequacy of hand d i g i t i z a t i o n at high f r e q u e n c i e s . But, since t h i s high-frequency noise i s introduced d u r i n g the d i g i t i z a -t i o n process and not during the a n a l y s i s , the same s i t u a -t i o n a p p l i e s to the F o u r i e r i n t e g r a l method of a n a l y s i s which i n c o r p o r a t e s F i l o n ' s method. Consequently the amplitude d e n s i t y spectrum i s l e s s r e l i a b l e at the lower and higher l i m i t s of the f r e -quency range f o r two reasons; poorer s i g n a l - t o - n o i s e r a t i o and d i s t o r t i o n s introduced d u r i n g a n a l y s i s . 5.7 A t t e n u a t i o n of Rayleigh waves c l o s e to the Source For a s i n u s o i d a l wavetrain the a t t e n u a t i o n i s given by the f i r s t term i n the expression -ocr- i ( k r - tot) e e (5-22) 70 where OC i s the attenuation coefficient, k i s the wave number, r i s the distance of travel from the source and OJ i s the angular frequency. For a pulse, eq. ( 5 -22 ) i s a p p l i c a b l e t o the F o u r i e r components of the pul s e . Close t o the source, however, the a p p l i c a t i o n of (5 -22 ) to determine the a t t e n u a t i o n c o e f f i c i e n t of a Rayle i g h wave w i l l l e a d t o erroneous r e s u l t s because of the presence qf both compressional and shear waves of appreci a b l e amplitudes. Consider the re g i o n c l o s e to a time-harmonic l i n e source a c t i n g normally on the surface of a semi-i n f i n i t e l o s s l e s s medium. Lamb (1904) has shown th a t the displacement of a p a r t i c l e on the surface i s due t o the presence of three types of waves: a Ra y l e i g h wave of constant amplitude, a compressional wave w i t h ampli--% tude p r o p o r t i o n a l to ( K*, V ) , and a shear wave w i t h amplitude p r o p o r t i o n a l tp ( T ) . The wave numbers f o r the r e s p e c t i v e waves are K R , and ; l e t the corresponding wavelengths be AR , and A# . E x p l i c i t expressions f o r the p a r t i c l e d i s -placements are given i n the Appendix. The amplitude of the compressional wave decreases by a f a c t o r of almost 200 as the wave propagates from V = ^Vin to r = 5 A.* • For the m a t e r i a l s of i n t e r e s t X«. =^  2. XR, ( f o r a given frequency). Consequently about 10 X * from the source, 7 1 the amplitudes of the compressional and shear waves w i l l have been reduced by geometric a t t e n u a t i o n to an i n s i g n i -f i c a n t l e v e l . Garvin ( 1 9 5 6 ) has i n v e s t i g a t e d the waveform of a Rayleigh pulse generated by a sudden displacement of a l i n e source l o c a t e d p a r a l l e l to and below the surface of a s e m i - i n f i n i t e l o s s l e s s medium. By t a k i n g the width of the pulse to be the time i n t e r v a l between half-maximum p o i n t s ( f o r p a r t i c l e motion normal to the s u r f a c e ) , he has shown th a t the Rayleigh pulse does not assume i t s f i n a l shape u n t i l a h o r i z o n t a l distance from the source of roughly s i x wavelengths i s reached. The i n t e r f e r e n c e of other types of waves wi t h the R a y l e i g h pulse i s thus manifest up to about s i x wavelengths from the source. I n order to o b t a i n r e l i a b l e values f o r the a t t e n u a t i o n c o e f f i c i e n t of the F o u r i e r components of a Rayleigh pulse and of a s i n u s o i d a l R a y l e i g h wavetrain, only those R a y l e i g h waves which have t r a v e l l e d at l e a s t 1 0 wavelengths from the source before being detected by the s t r a i n gauge were used to c a l c u l a t e the a t t e n u a t i o n c o e f f i c i e n t . This c r i t e r i o n was a p p l i e d to the amplitude d e n s i t y spectra of the Rayleigh pulse w i t h the r e s u l t t h a t , at f r e q u e n c i e s 1 0 0 kc/s, 1 5 0 kc/s and i n some cases 2 0 0 kc/s, values of the amplitude d e n s i t y were not used i n the f i n a l a n a l y s i s . 72 5 . 8 C a l c u l a t i o n of the Atte n u a t i o n of the F o u r i e r Components of a Pulse The method of l e a s t squares i s used to c a l c u -l a t e the a t t e n u a t i o n c o e f f i c i e n t of the F o u r i e r components of a pu l s e . Consider the amplitude d e n s i t y spectrum F (to) as defined i n eq. ( 5 - 8 ) . Let ^ r represent t h i s amplitude d e n s i t y at a dis t a n c e X r from the source. Since we are c o n s i d e r i n g the case where the pulse does not experience geometric a t t e n u a t i o n , may be represented by i ^ r fe= A c 6 & r = 1 , 2 , 3 , ...n ( 5 - 2 3 ) where oL i s the a t t e n u a t i o n c o e f f i c i e n t , LO i s the angular frequency, k i s the phase f a c t o r and Ao i s the amplitude d e n s i t y near the o r i g i n . The value of n i s g e n e r a l l y equal to s i x ; the e x c e p t i o n a l cases where n i s l e s s than s i x have been described. Where the enve-lope of the waveform i s considered the lo g a r i t h m of eq. ( 5 - 2 3 ) may be w r i t t e n as Z r = In ^ r = /S> - oLXr r = 1 , 2 , 3 , .. .n ( 5 - 2 4 ) where f3 i s a constant. The method of l e a s t squares 7 3 can now be used to o b t a i n expressions f o r @ and cx which are as s o c i a t e d w i t h the b e s t - f i t s t r a i g h t l i n e through the p o i n t s ( X r,Zr )> r = 1 , 2 , ...n . The expressions f o r @ and cx are as f o l l o w s : - $ Z r ^  * r + 5 X r Z r 5 X r |g _ r = | r r _ _ r - n £ x* + ( 5 Xr) r r n n 5 x rz r — 5 J r '"V r ( 5 - 2 5 ) The standard d e v i a t i o n , S Da. , of oC i s S D " | U ' - Z ) n - 2 V n ^ x ^ - ( I x r ) ' where v — 2. L R Z = n ( 5 - 2 6 ) The weighted average f o r oc and S D < * . are as f o l l o w s : N CL = 5 to; S D * = 1/-^ — n -1 ) ^ u)j j = 1 , 2 , .. .N ( 5 - 2 7 ) 74 where ojj = NPj / variance j NPj = number of p o i n t s used i n determining the slope of the j r h record d j = ocj - 5L where ~& = (I, a y u N = number of records over which oc i s averaged. The F o r t r a n programs used i n the a n a l y s i s are presented i n the Appendix, 5,9 C a l c u l a t i o n of A t t e n u a t i o n of Time-Harmonic Rayleigh Waves An average value of the a t t e n u a t i o n c o e f f i c i e n t was obtained by c o n s i d e r i n g the time-harmonic Rayleigh waves t r a v e l l i n g i n opposite d i r e c t i o n s along the semi-c i r c u l a r path (S.^ M — S 2) between the two s t r a i n gauges; t h i s s e m i c i r c u l a r path does not i n c l u d e the source c r y s t a l (see F i g , 4.11(b)), The c o e f f i c i e n t of a t t e n u a t i o n oc p i s given by $^ Ret Rcci M^- - 5 — 1 ~ a p = ^ Hcct-C (5-28) where C i s the mean circumference of the c y l i n d e r , R, c ^  i s the amplitude of the clockwise t r a v e l l i n g wave at S^, R cg i s the amplitude of the clockwise t r a v e l l i n g 75 wave at Sg, R c c i i s the amplitude of the counterclock-wise t r a v e l l i n g waves at S 0, and R i s the amplitude c cc2 of the counterclockwise t r a v e l l i n g wave at S-^ . Several values of OCp are obtainable from a s i n g l e record when d i s t o r t i o n of the waveform by the source c r y s t a l i s i n s i g n i f i c a n t . Let oTP represent the average of (cXp)j over n values; then OCp = n (5-29) Let d j = (oCp)j - OLp . Then the standard d e v i a t i o n of cxTp i s (5-30) At the low frequency end (100 kc/s to 150 kc/s) the a t t e n u a t i o n f o r a path l e n g t h of C/2 i s so small that r e l i a b l e values are not obtainable i f the method u s i n g eq. (5-28) i s employed. However, an upper l i m i t can be placed upon the value of the a t t e n u a t i o n c o e f f i c i e n t by c o n s i d e r i n g waves which have t r a v e l l e d a f u l l c i r c l e and 7 6 consequently have s u f f e r e d a di m i n u t i o n of energy not only from l o s s a s s o c i a t e d w i t h propagation along the edge of the c y l i n d e r but a l s o from c r y s t a l scattering,, Let ocs be the a t t e n u a t i o n due to the presence of the c r y s t a l . Then, since & p S o(. p + s , an upper l i m i t can be placed upon otp by t a k i n g an average value of o(p*s over many complete c i r c u i t s . From F i g . 4.11(b) i t can be seen t h a t the le n g t h of a wavetrain must be l e s s than one-half the ci r c u m f e r -ence of the tube. Let |_ wt a n d c represent the length of the wavetrain and the circumference of the tube respec-t i v e l y . I n general C/ < i _ t < 3 % * T h e a m P 1 1 _ tude d e n s i t y spectrum f o r such a wavetrain i s represented by a d i f f r a c t i o n curve. As the frequency decreases the maximum number of wavelengths i n a wavetrain a l s o decreases, and the d i f f r a c t i o n curve tends to spread out h o r i z o n t a l l y . Consequently at low freque n c i e s the d i s t o r t i o n of the wave-form due to d i s p e r s i o n and a t t e n u a t i o n become more pro-nounced. 5.10 I n t e r n a l F r i c t i o n The i n t e r n a l f r i c t i o n , 1/Q, may be c a l c u l a t e d from the f o l l o w i n g equation: OL C (2-2) 7 7 where C i s the phase v e l o c i t y and - f i s the ( c i r -c u l a r ) frequency. Both the phase v e l o c i t y and the f r e -quency of a time-harmonic wave are determined from the record ( p o l a r o i d f i l m ) of the wave. The top t r a c e of the record was w r i t t e n at a higher sweep rat e so t h a t the frequency of the wave could be measured to s u f f i c i e n t accuracy. The phase v e l o c i t y was determined by measuring the time taken f o r a p a r t i c u l a r phase t o make a complete c i r c u i t around the edge of the tube. 78 CHAPTER VI RESULTS 6.1 General The r e s u l t s of the study of the a t t e n u a t i o n of Rayleigh waves i n metals are presented as f o l l o w s : f i r s t the g r a i n s t r u c t u r e of the metals i s discussed, next the manner i n which 1/Q v a r i e s w i t h frequency f o r the metals stu d i e d i s compared w i t h the t h e o r e t i c a l ex-pr e s s i o n s f o r 1/Q which are described i n Chapter I I , and f i n a l l y the r e s u l t s f o r a l l the tubes are summarized and the i m p l i c a t i o n s discussed. 6.2 G r a i n S t r u c t u r e Microphotographs of the g r a i n s t r u c t u r e of a specimen from each of the f o u r tubes are shown i n P i g s . 6.1 and 6.2. Both specimens shown i n P i g . 6.1 c o n s i s t of y e l l o w (or c a r t r i d g e ) brass of approximate composition 70$ Cu - 30$ Zn. The g r a i n s vary markedly i n s i z e and shape but are a l i g n e d predominantly i n the a x i a l d i r e c -t i o n . The s i z e s of the g r a i n s i n the aluminum specimen are much l a r g e r than t h a t u s u a l l y found i n aluminum sam-p l e s . The g r a i n s do not vary markedly i n s i z e or shape. The copper specimen i s pure to the extent t h a t no second 7 9 phase appears. The g r a i n s vary c o n s i d e r a b l y i n s i z e and shape but are a l i g n e d predominantly i n the a x i a l d i r e c t i o n . The d i s t r i b u t i o n of g r a i n s i z e s i n annealed metals has been shown by Andrade e t . a l . ( 1 9 6 5 ) to f o l -low the e x p e r i m e n t a l l y confirmed law which i s represented by: n = n*«* e x p OL ( 6 - 1 ) where n i s the number of g r a i n s f o r which the diameter D l i e s w i t h i n a s p e c i f i e d narrow range, D Q i s the value of D f o r which n has the maximum value n „ and o<-max i s a dimensionless constant (which equals T t f o r the metals they have s t u d i e d ) . An accurate determination of D Q i s d i f f i c u l t because n does not vary a p p r e c i a b l y from n m a x f o r values of D which range from approxi-mately 2 0 $ below to 2 0 $ above D Q. For t h i s reason i n Table 6 . 1 a lower and an upper l i m i t are t a b u l a t e d f o r the average g r a i n diameter. 6 . 3 Sound S c a t t e r i n g E f f e c t s . The s c a t t e r i n g of a c o u s t i c waves i n granular m a t e r i a l s becomes appreciable when the a c o u s t i c wavelength 8 0 becomes comparable to the g r a i n s i z e . T h i s s c a t t e r i n g i s due to the anisotropy of the e l a s t i c constants i n the gr a i n s (Mason ( 1 9 5 8 ) ) . In t h i s study the sh o r t e s t wave-le n g t h of i n t e r e s t i s 4 mm, which i s s e v e r a l orders of magnitude l a r g e r than the g r a i n s i z e s shown i n Table 6 . 1 . Consequently the l o s s of a c o u s t i c energy from t h i s source i s n e g l i g i b l e . 6 . 4 Records of Rayleigh Waves A p o l a r o i d f i l m of a Rayleigh pulse which has made m u l t i p l e t r i p s around the edge of a metal tube i s shown i n P i g . 6 . 3 . The records of Rayleigh pulses shown i n subsequent f i g u r e s are t r a c i n g s from the o r i g i n a l p o l a r -o i d f i l m . The propagation c h a r a c t e r i s t i c s of the Rayleigh pulses vary from tube to tube and th e r e f o r e a t y p i c a l record i s presented f o r each tube. A p o l a r o i d f i l m of s i n u s o i d a l R a y l e i g h waves i s shown i n P i g . 6 . 4 . Since the propagation c h a r a c t e r i s t i c s of these s i n u s o i d a l waves do not vary a p p r e c i a b l y from tube to tube, P i g . 6 . 4 i s a good r e p r e s e n t a t i o n f o r a l l the tubes. 6 . 5 R e s u l t s f o r Brass Tube of 1 0 cm. Diameter Rayleigh pulses of 4 and 3 4 /xe amplitude res-p e c t i v e l y are shown i n P i g . 6 . 5 . The corresponding ampli-81 tude d e n s i t y spectra are shown i n P i g . 6 . 6 . The curves of phase s h i f t per two r e v o l u t i o n s ( of eq. (5-12)) are shown i n P i g . 6.7. (Note that §c/> i s not a true phase s h i f t because the phase s h i f t r e s u l t i n g from the d i f f e r e n c e i n zero times chosen i n d i g i t i z i n g the records i s in c l u d e d ; i f the l a t t e r were not i n c l u d e d then a l l the experimental records would be the same.) The observed manner i n which 8<f> v a r i e s w i t h frequency i s not the r e s u l t of d i s p e r s i o n a s s o c i a t e d w i t h c urvature. The s u b s t i t u t i o n of values of the phase v e l o c i t y which are shown i n Table 3.1 i n t o eq. (5-12) r e s u l t s i n values of 8c/> which vary approximately l i n e a r l y w i t h frequency over the range from 100 to 500 kc/s. The curvature of the &(/> versus frequency curves may be the r e s u l t of d i s p e r s i o n a s s o c i a t e d w i t h attenua-t i o n . For a medium f o r which the wave motion i s l i n e a r and the a t t e n u a t i o n c o e f f i c i e n t v a r i e s l i n e a r l y w i t h frequency, the phase v e l o c i t y i s given by (Putterman L1962J ) (6-2) where x i s the r a t i o of the frequency f to an a r b i -t r a r y small frequency f„ ( l e s s than 1 c / s ) , Q, i s a 82 constant, C V i s approximately equal to .1.8, and C i s a constant v e l o c i t y . The s u b s t i t u t i o n of eq. (6-2) i n t o eq. (5-12) r e s u l t s i n $</> having a n o n l i n e a r " frequency dependence of the form 80 oc  l/af Im. %o (6-3) When 1/Q i s independent of frequency, the n o n l i n e a r be-haviour of i s too small to account f o r the appre-c i a b l e n o n l i n e a r trend shown i n P i g . 6.7. However, i f 1/Q i n c r e a s e s with i n c r e a s i n g frequency, then under the assumption that eq. (6-3) i s s t i l l v a l i d the n o n l i n e a r trend i n S</> becomes grea t e r . The 1/Q values f o r the brass tube of 10 cm. diameter (shown i n Table 6.16) increase w i t h i n c r e a s i n g frequency by about the r i g h t amount to account f o r the observed n o n l i n e a r negative trend i n 8 f . A t t e n u a t i o n measurements are p l o t t e d i n P i g s . 6.8, 6.9 and 6.10. The dependence of a t t e n u a t i o n on s t r a i n amplitude can be determined from P i g . 6.8. In view of the f a c t that the r a t i o of the s t r a i n amplitude of the two pulses i s approximately l / 9 , the a t t e n u a t i o n depends very l i t t l e , i f at a l l , upon the s t r a i n amplitude i n the s t r a i n r egion from 4 to 34 />L6 and i n the f r e -quency range from 100 to 500 kc/s. The s t r a i n l e v e l s 8 3 a s s o c i a t e d w i t h the a t t e n u a t i o n measurements f o r the s i n u s o i d a l waves are shown i n Table 6 .2. For any a r b i t r a r y small frequency i n t e r v a l , i t can be seen from F i g . 6.10 that values of the a t t e n -u a t i o n l i e w i t h i n or close to the standard d e v i a t i o n of ea£h other. In view of t h i s o v e r a l l agreement the a t t e n -u a t i o n mechanism f o r the metal te s t e d i s considered to be l i n e a r i n the s t r a i n r e g i o n from 4 to 34 A*- € and l n the frequency range from 100 to 500 kc/s. 6.6 R e s u l t s f o r Brass Tube of 15 cm. Diameter T y p i c a l s t r a i n records of Rayleigh pulses of d i f f e r e n t s t r a i n amplitudes are shown i n F i g . 6.11. The t r a v e l paths f o r the f i r s t and the l a s t t r a c e s are 10 and 250 cm. r e s p e c t i v e l y . The corresponding amplitude d e n s i t y spectra are shown i n the next f i g u r e . The a t t e n u a t i o n measurements p l o t t e d i n F i g . 6.13 are weighted averages over three records; the com-ponent values are presented i n Table 6 . 3 . For the s i n -u s o i d a l waves the a t t e n u a t i o n measurements are shown i n the bottom graph of F i g . 6 . 9 . The three sets of a t t e n -u a t i o n measurements are superposed i n F i g . 6.14. Agreement between the a t t e n u a t i o n measurements v a r i e s from good to f a i r . The a t t e n u a t i o n measurements 84 shown i n P i g . 6.13 i n d i c a t e very l i t t l e amplitude depen-dence. The three sets of a t t e n u a t i o n measurements shown i n P i g . 6.14 are, f o r the most p a r t , c o n s i s t e n t w i t h each other. However, there i s one exception; at 450 kc/s there i s a discrepancy i n the a t t e n u a t i o n measurements. I t seems most l i k e l y t hat the standard e r r o r of the a t t e n -u a t i o n of the 3 3 y U £ wave has been underestimated and thus the d e v i a t i o n of t h i s s i n g l e p o i n t i s not regarded as s i g n i f i c a n t i n view of the o v e r a l l c o n s i s t e n c y . Hence the r e s u l t s i n d i c a t e that the a t t e n u a t i o n mechanism f o r t h i s metal (brass) i s a l s o l i n e a r i n the s t r a i n region and the frequency range i n v e s t i g a t e d . 6.7 R e s u l t s f o r Aluminum Tube of 13.7 cm. Diameter T y p i c a l s t r a i n records of Rayleigh pulses of d i f f e r e n t s t r a i n amplitudes are shown i n P i g . 6.15. For each record the t r a v e l paths f o r the f i r s t and l a s t t r a c e s are 10 and 240 cm. r e s p e c t i v e l y . In F i g . 6.15(a) the t r a c e s become more i r r e g u l a r (on e i t h e r side of the main s i g n a l ) w i t h i n c r e a s i n g d i s t a n c e ; t h i s may be an i n d i c a -t i o n of d i s p e r s i o n . The consequence of t h i s (explanations given i n the previous chapter) i s that d i s t o r t i o n s are introduced i n t o the pulse s p e c t r a ; the pulse spectra shown i n F i g . 6.16 show signs of i r r e g u l a r i t y . 85 Prom P i g . 6.17 i t can be seen that the ampli-tude dependence of the a t t e n u a t i o n i s very s l i g h t or none at a l l . Prom P i g . 6.19, the a t t e n u a t i o n a s s o c i a t e d with the pulses and the time-harmonic waves are seen to be i n f a i r agreement. The standard d e v i a t i o n s are, how-ever, q u i t e l a r g e . Much of t h i s can be a t t r i b u t e d to the d i s t o r t i o n of the pulse spectrum. However, there i s suf-f i c i e n t agreement to suggest t h a t , i f the techniques f o r r e c o r d i n g and a n a l y z i n g the s i g n a l s were improved, then the agreement would be much c l o s e r . (For example, the use of a wider o s c i l l o s c o p e screen would enable the record-i n g of a pulse to be taken at a f a s t e r h o r i z o n t a l sweep-speed. Consequently the F o u r i e r a n a l y s i s of the pulse would become more r e l i a b l e at f r e q u e n c i e s i n the neighbor-hood of 500 kc/s.) Nevertheless the o v e r a l l evidence i n d i c a t e s t h a t the a t t e n u a t i o n mechanism f o r the metal t e s t e d (aluminum) i s a l s o l i n e a r i n the s t r a i n r e g i o n and i n the frequency range i n v e s t i g a t e d . 6.8 R e s u l t s f o r Copper Tube of 12.5 cm. Diameter The s t r a i n records which are shown i n F i g . 6.20 have a pronounced o s c i l l a t o r y waveform both preceding and f o l l o w i n g the main s i g n a l . From Table 6.15, i t can be seen t h a t the phase v e l o c i t y of a Rayleigh wave propagat-i n g along the edge of the copper tube v a r i e s markedly 86 w i t h frequency. Consequently the spreading out of the s i g n a l i s probably the r e s u l t of d i s p e r s i o n . In order to o b t a i n t r a c e s with l e s s o s c i l l a t o r y waveform, the lower 3 db frequency ( f 1 i n P i g . 4.7) was increased from 40 kc/s to 80 kc/s, and the records thus obtained were the only ones presented i n the f i n a l r e s u l t s . The v a r i a t i o n s i n the amplitude d e n s i t y of t r a c e s 3 to 6 i n (a) and (b) of P i g . 6.20, as a r e s u l t of i n c r e a s i n g the number of sample p o i n t s , are presented i n Tables 6.9 to 6.12; the i n t e r v a l between sample p o i n t s i s 0.3 JULS The l a s t three columns were averaged at each frequency and p l o t t e d i n P i g . 6.21. The percentage v a r i a t i o n i n the amplitude d e n s i t i e s i n the l a s t three columns was used as a measure of the r e l i a b i l i t y of the r e s p e c t i v e d e n s i t y . For freq u e n c i e s l e s s than 300 kc/s the v a r i a -t i o n i s g e n e r a l l y l e s s than 5$; f o r f r e q u e n c i e s 300 to 500 kc/s the v a r i a t i o n i s g e n e r a l l y much gr e a t e r than 5$. The r e s u l t s are presented i n F i g s . 6.22, 6.23 a n d 6.18 and Tables 6.13 and 6.14 f o r the frequency i n t e r v a l 100 to 350 kc/s. However, even i n t h i s frequency i n t e r v a l the r e l i a b i l i t y of the r e s u l t s i s l e s s than that f o r the other tubes. Although the s c a t t e r i n the r e s u l t s at the high frequency end i s such that no d e f i n i t e d e c i s i o n can be made as to whether the a t t e n u a t i o n mechanism i s l i n e a r , 87 the measurements at the low frequency end i n d i c a t e that the a t t e n u a t i o n i s l i n e a r f o r the metal t e s t e d (copper). 6.9 Curve F i t t i n g A curve having an equation of the form OL = a f " (6-4) where OL i s the a t t e n u a t i o n c o e f f i c i e n t , f i s the frequency and a and n are constants, was f i t t e d to each set of a t t e n u a t i o n measurements by the method of l e a s t squares. Since our primary i n t e r e s t i s i n the general trend of the OL vs. f curve, any anomalous p o i n t s were f i r s t smoothed out and then the a n a l y s i s was c a r r i e d out. The r e s u l t s of t h i s a n a l y s i s are shown i n Table 6.17. The f i t between the t h e o r e t i c a l curves and the experimental data v a r i e s from good ( f o r the two brass tubes) to poor ( f o r the aluminum tube). I n a d d i t i o n n v a r i e s from tube to tube. Therefore i t i s u n l i k e l y t h a t any important s i g n i f i c a n c e can be attached to these r e s u l t s . Nevertheless the a n a l y s i s does show t h a t , f o r the metals st u d i e d , n i s c l o s e r to two than to one. I t i s t h i s l a t t e r o bservation that w i l l be used i n the i n t e r p r e t a t i o n of the r e s u l t s . 88 6.10 The General Trend of 1/Q An approximate expression f o r the frequency-dependence of 1/Q f o r the metals stu d i e d i s where n i s c l o s e r to one than to zero. This observa-t i o n that 1/Q f o r the metals stu d i e d does increase w i t h i n c r e a s i n g frequency i s not l i k e l y to be due to the cur-vature of the seismic model because s i m i l a r r e s u l t s were obtained f o r Rayleigh waves propagating along the s t r a i g h t edge of a brass sheet. I t could be that t h i s increase of 1/Q wi t h i n c r e a s i n g frequency may be the product of the set up and that the two-dimensional model i s at f a u l t . But t h i s i s r a t h e r d o u b t f u l . of a c o u s t i c energy i n s o l i d s i n Chapter I I , the theory of d i s l o c a t i o n damping, as f i r s t proposed by Koehler (1952) and l a t e r g e n e r a l i z e d by Granato and Lucke (1956) can best account f o r the observed manner i n which 1/Q v a r i e s w i t h frequency f o r the metals s t u d i e d . Stern and Granato (1962) have used the theory of d i s l o c a t i o n damping t o e x t r a p o l a t e Mc/s measurements of 1/Q ( i n cop-per s i n g l e c r y s t a l s ) to 10 kc/s and have obtained reason-(6-5) Of the f o u r t h e o r i e s d e s c r i b i n g the d i s s i p a t i o n 89 able agreement between theory and experiment. Thus t h e i r i n v e s t i g a t i o n tends to support the idea that d i s -l o c a t i o n damping i s operative over a wide frequency range (kc/s - Mc/s). I n a d d i t i o n these e x t r a p o l a t e d values of 1/Q f o r copper s i n g l e c r y s t a l s are of the same order of magnitude as the measurements of 1/Q that were obtained f o r the copper tube. 6.11 Anomalous Peaks Superimposed on the general trend of the a t -tenuation curves are s e v e r a l anomalous peaks which can be seen i n P i g s . 6.13 and 6.17. According to the t h e r -moelastic theory as proposed i n Zener (1938), r e l a x a t i o n peaks occur l n cubic metals or metals e x h i b i t i n g e l a s t i c a n i sotropy at the frequency f 0 which i s d e f i n e d i n eq. ( 2 - 6 ) . For the brass tube of 15 cm. diameter, f 0 i s approximately 450 kc/s, which agrees w i t h the peak shown i n the bottom graph of F i g . 6.13. For the alumi-num tube, f„ i s approximately 120 kc/s; the peak i n the top graph of F i g . 6.17 appears at t h i s frequency. For the copper tube f 0 i s approximately 240 kc/s; the peak i n F i g . 6.22 occurs at 300 kc/s. In s e c t i o n 6.2 i t was described why the accuracy of D Q i s not c l o s e r than 20$. Hence the discrepancy between measure-90 ment and theory i s l i k e l y to be due to t h i s source. Thus a l l the anomalous peaks are probably manifesta-t i o n s of the Zener e f f e c t . Values of 1/Q f o r the metals t e s t e d are d i f f i c u l t to c a l c u l a t e from equation (2-5) because of the u n c e r t a i n t y i n R, This i s because the g r a i n s i n the tubes are o r i e n t e d predominantly i n the a x i a l d i r e c t i o n whereas i n a t y p i c a l sample the o r i e n t a -t i o n i s at random, 6,12 Conclusion W i t h i n the l i m i t s of the e r r o r i n these e x p e r i -ments, there i s a strong i n d i c a t i o n t h a t the a t t e n u a t i o n process i n metals i s a l i n e a r process i n the s m a l l - s t r a i n r e g i o n and i n the frequency range i n v e s t i g a t e d . I n c e r -t a i n cases the standard d e v i a t i o n s are so l a r g e , probably due to experimental technique, that no d e f i n i t e c o n c l u -sions can be drawn; e,g. high frequency range i n P i g s . 6.19 and 6.23, I n s p i t e of these shortcomings we f e e l t hat there i s s u f f i c i e n t o v e r a l l evidence to i n d i c a t e that the a t t e n u a t i o n i n metals i s a l i n e a r process i n the 2 /xe to 40jU.e r e g i o n and i n the 100 kc/s to 500 kc/s frequency range. 91 P i g . 3»1. F i r s t symmetric mode ( M M ) of a f r e e e l a s t i c p l a t e of t h i c k n e s s 2. H and Poisson's constant of 1/4 ( a f t e r T o l s t o y and Usd i n ) . F i g . 3 , 2 . Phase v e l o c i t y of Ray l e i g h wave propagating around circumference of homogeneous c y l i n d e r w i t h Poisson's constant of 0 . 2 8 ( a f t e r O l i v e r ) . 92 Z /i z = o P i g . 3.3. C y l i n d r i c a l s h e l l showing co-ordinate system used. 93 A AMPLITUDE k T * (cm) 1 ! , ( , [- i | ' " -~^ J ^ 0 0.1 Q2 0.3 0.4 0.5 FREQUENCY (megacycles) (b) F i g . 4.1. T y p i c a l s t r a i n record of Rayl e i g h wave from a d i s t a n t impulse and a sine wave of one c y c l e are shown i n ( a ) . The corresponding amplitude s p e c t r a l d e n s i t i e s are shown i n (b) . The super-p o s i t i o n of the d i f f r a c t i o n curves 1 and 2 i n (c) r e s u l t i n the curve shown i n ( b ) f o r the sine wave. 94 0.50 STRAIN 0 4 0 -0 .30 0 . 2 0 --0 .10 F i g . 4 . 2 . S t r a i n p a r a l l e l ('—^7—) and perpendicular ( dw ) to the surface, as a f u n c t i o n of the product of the wave number ( K ) of the Rayleigh wave and the depth ( z ) below the surface. 9 5 Conducting Surfaces Direction of j Polarization (a) T—K L E LEGEND Note: MKS Units used. A = P = s = V = E = area of end of c r y s t a l d e n s i t y of c r y s t a l d i e l e c t r i c constant a t constant s t r a i n e l a s t i c compliance o l constant a p p l i e d f i e l d v e l o c i t y of compressional wave i n c r y s t a l a p p l i e d f i e l d ( i n a x i a l d i r e c -t i o n ) J - — w v j Z o T A N ^ u uj V Z o C o 0 e p a r t i c l e v e l o c i t y of end of c y l i n d e r i n a x i a l d i r e c t i o n angular frequency of v i b r a t i o n of c y l i n d e r e A / L = capacitance of clamped c r y s t a l - 8 A / L p i e z o e l e c t r i c constant r e l a t i n g s t r e s s w i t h a p p l i e d f i e l d V G = c c = 0 . c, = M, = r e s i s t a n c e of pulse generator voltage of pulse generator - e A A 2 S E L A p A L '2 -S E L 8 A 8 p A L F i g . 4 „ 3 ( . " E q u i v a l e n t c i r c u i t f o r p i e z o e l e c t r i c transducers, (a) P i e z o e l e c t r i c transducer. (b) Dynamic eq u i v a l e n t c i r c u i t , (c) Eq u i v a l e n t c i r c u i t f o r c r y s t a l f r e e at one end and operati n g i n the region of i t s mechanical resonant frequency ( a f t e r Mason [ 1 9 4 2 ] ) . 96 V G Co L E G E N D = resistance of pulse generator = voltage of pulse generator = £ S A / L = capacitance of clamped c r y s t a l C, M, - e 2 5 Ti Z A P A L / 2 = mechanical resistance of oad Lo = L o = t J M R = inductance to tune e l e c t r i -c a l l y c i r c u i t (d) near resonant frequency ' — where mechanical resonant f r e -quency of transducer v a r i a b l e inductance to tune e l e c t r i c a l l y a piezoelec-t r i c transducer at f r e -quency about 50$ below and 50% above the mechanical resonant frequency C = SE L F i g . 4.3. E q u i v a l e n t c i r c u i t s f o r p i e z o e l e c t r i c transducer. (d) E q u i v a l e n t c i r c u i t f o r c r y s t a l f r e e on one end and r a d i a t i n g energy i n t o a metal (e.g. b r a s s ) . (e) C i r c u i t (d) tuned e l e c t r i c a l l y f o r improved performance. ( f ) E l e c t r i c a l l y tuned c i r c u i t f o r frequen c i e s separated from the mechanical resonant frequency of the order of 50$ below and 50$ above the l a t t e r frequency ( a f t e r Mason [1942] ). 97 Eout Low Velocity Fibre 1 Fbamrubber (a) L Bender L = L W = L» = length of bender width of bender thickness of bender (b) Eout e Y o K M ±_hJH_ where e = 4 L piezoelectric constant relat-ing stress with applied field 4 e3 L Lw Lt dielectric constant at constant strain 4 L S ; where and Y 0 ( I - | K Z ) Young's modulus electromechanical coupling coeff. L» L P i g . 4 . 4 . (a) Cross s e c t i o n of p i e z o e l e c t r i c bender and support. (b) Electromechanical e q u i v a l e n t c i r c u i t f o r bender clamped on one end and d r i v e n by an a p p l i e d f o r c e (p) on the other end. ( a f t e r Mason [1942] ) . Piezoelectric Crystal Metal Pulse Oscillator M i (b) ( 2 0) 2Z T = P i g . 4 . 5 . open circuit impedance of (e) Fig. 4.5 as seen from 1-1. effective impedance of (e) Fig. 4.5 as seen from 2-2; effective compliance of bond, mechanical impedance of load (a) P i e z o e l e c t r i c c r y s t a l f r e e attached to load on other end. electromechanical c i r c u i t bondine Mason [ 1 9 on one end and (b) Eq u i v a l e n t showing e f f e c t of ^ . t e r i a l on performance of c r y s t a l ( a f t e r SIGNAL SOURCE^, TRIGGER DETECTOR SIGNAL DETECTOR M E T A L TUBE HIGH-FIL -PASS FER i PREAMF >LIFIER LARGE \ SIGNAL SMALL SIGNAL AMPLIFIER" 3 - l 8 x TYPE 535A "A" single sweep TEKTRONIX AC-GATE Bo 9 8 POTENTIOMETER CIRCUIT HIGH- PASS FILTER PREAMPLIFIER r 5 6 6 TRIGGER PULSES WAVE SHAPING CIRCUIT POTENTIAL DIVIDER UPPER BEAM A.B OB LOWER BEAM - O A —OB s Ul VOLTAGE DIVIDER (provides 6 different, voltage biases) DELAY CIRCUIT (monostable vibrator ) DIFFERENTIATING and GATING CIRCUIT P i g . 4.6. Schematic diagram of composite c i r c u i t f o r the d e t e c t i o n and re c o r d i n g of a t r a n s i e n t Rayleigh pulse (generated by an explosion) as i t makes m u l t i p l e t r i p s around the truncated edge of a c i r c u l a r c y l i n d r i c a l s h e l l . 99 POTENTIOMETER CIRCUIT Cylindrical s h e l l TYPE 502 SCOPE (Tektronix) ODUMONT POLAROID CAMERA TYPE 450 UPPER BEAM — O A OGd. F i g . 4.7. C i r c u i t f o r the d e t e c t i o n , a m p l i f i c a t i o n and p h o t o g r a p h i n g o f a t r a n s i e n t R a y l e i g h p u l s e produced by an e x p l o s i o n . PREAMP 0 - 3 0 0 x 100 Large Signal • H . 5 V + 3.0 V 100 k 10 k< 2N1S06 2 5 - 5 0 k > 1^ :37 k Ik 2.7 k * —TRIGGER INPUT 535 Scope "A" single BMORPH 'WALL of CYLINDER CATHODE FOLLOWER ( a ) SMALL SIGNAL s w e e P AMPLIFIER 3-I8X + 6 V S C 0 P E 4 4 k <l5k 1.5 V - i - 3 5 k lOOpf |,5 k i i i - i 4 Trigger pulses L5V" 5.5 k: B Gate Gd. LGate (b) MONOSTABLE VIBRATOR + 22.5 V 502 Scope o A Lower -o Beam A - B B I0k> -I.5V 30V < I R k o k L k X>k OUTPUT PULSE WIDTH = RC In 2 ,5V 2 Trigger pulses P i g . 4 . 8 . T r i g g e r c i r c u i t . (a) C i r c u i t t o t r i p 535 s c o p e % : s e t a t ' s i n g l e - s w e e p ) b y means o f t r i g g e r p u l s e f r o m v B i m o r p h . (b) C i r c u i t t o p r o d u c e 4 t r i g -g e r p u l s e s once 535 s c o p e h a s b e e n t r i g g e r e d , ( c ) C i r c u i t t o p r o d u c e 2 t r i g g e r p u l s e s once 101 A G A T E 5 [ 3 5 S C 0 P E r o—>— B G A T E :200 k :i00k 5.5 k< 5k ! Gd. Voltage Bias 502 Scope MONO MONO B G A T E A G A T E ->T.ime OUPUT WAVE FORM l b ) Voftao,e + 22.5 V 5 0 0 k OUTPUT from MONOSTABLE VIBRATOR (a) Time VOLTAGE BIAS AS SEEN ON 502 SCOPE SCREEN (AT SLOWl-SWEEP SPEED) (O P i g . 4.9. (a) Voltage b i a s c i r c u i t . (b) Components (plus o r i g i n of each component) of voltage b i a s versus time. (c) Voltage b i a s as seen on 502 screen. I n v e r s i o n of wave form occurs because input i s at B(upper beam) and s e l e c t o r switch i s at A-B s e t t i n g . 102 R.F. PULSE GENERATOR o MODEL 600 High MADISON Voltage INDUSTRIES Output Low (0-3000V) ?VSS0p«et 9 POTENTIOMETER CIRCUIT ATTENUATOR — . Triggeri / \ Input , P^olaroid V.—y Camera -oA Lower Beam TEKTRONIX B02 P i g . 4.10. C i r c u i t f o r the production, d e t e c t i o n and re c o r d i n g of a time-harmonic Ra y l e i g h wave t r a i n . 103 F i g . 4.11. The e f f e c t o f s c a t t e r i n g o f the R a y l e i g h wave by the source c r y s t a l . i 104 Polaroid film of six traces of the same transient Rayleigh pulse as it makes multiple trips around truncated edge of cylindrical shell Small-signal group 2-7 microstrain Large-signal group 30-50 microstrain Digitization of records at equispaced intervals (opprox. 03 //.sec intervals) Digital computer I.B.M. 1620-earlier work LB.M. 7040-later work Fortran program Fourier integral analysis which incorporates Ron's method Amplitude and Phase spectra Plot of Phase differences Digital computer Fortran program Least-squares fit determination of a Attenuation coefficient a and standard deviation of a Weighted average of a and its standard deviation Averaging over several records. Comparison of atten. coeff. Attenuation coefficients of time-harmonic waves FIG. 5.1 DIAGRAMMATIC REPRESENTATION OF THE METHOD OF ANALYSIS OF A TRANSIENT PULSE. Polaroid film of the two wavetrains (generated by a time - harmonic source) travelling in opposite directions along truncated edge of cylindrical shell. Amplitudes in 2—10//.strain range. Measurement of average amplitude of envelope of wavetrain Attenuation coefficient a calculated from eg. 5-28 In some instances upper limit on a calculated Average value of a standard deviation a Averaging over several values Comparison of attenuation coefficients t small large Attenuation coeff. from transient pulse analysis amplitude FIG. 5.2 DIAGRAMMATIC REPRESENTATION OF METHOD OF ANALYSIS OF TIME-HARMONIC WAVETRAIN. 106 (I) (a) Typical traces of the (electronic) noise present when recording small-amplitude pulses (2~7 micro-strain) . (3) ( 4 ) (b) Typical traces of the (electronic) noise present when recording large-amplitude pulses (30 - 50 microstrain)„ Noise spectrum ' microstrain ) 1 cycle/microsec.) 0.15 0.10 0.05-0.00 100 400 200 300 (C) Noise spectra of traces (1) and (2) 500 Freq. (KC/s) F i g . 5«3» Noise records and t h e i r Fourier spectra. 1 0 7 MICROSTRAIN 2 0 10 0 - 1 0 •20 d c bg a b c d 5/<sec. ( a ) Freq„ (kc/s) Ampli-tude Den-s i t y ( yU, strain-meg 1 0 0 2 3.91 2 4 . 8 7 2 4.92 2 4.32 1 5 0 2 8.69 2 8 . 9 5 2 8 . 5 4 2 8 . 9 0 2 0 0 2 8 . 5 3 2 8 . 2 3 2 7 . 6 1 2 6 . 5 6 2 5 0 2 4.44 2 4.86 2 5 . 7 3 2 6 . 3 3 3 0 0 1 2 . 1 6 1 2 . 3 1 1 4 . 0 3 1 4 . 5 5 3 5 0 9 . 5 3 9 o 2 1 9 . 9 7 9 . 8 5 400 3 . 4 9 3 » 1 5 3 . 1 4 3 . 8 1 4 5 0 1 . 9 1 1 . 9 2 2 . 5 2 2 . 2 3 5 0 0 0 . 9 1 0 . 9 7 O . 7 6 0 . 4 8 No. sample 1 6 3 p o i n t s 1 0 3 1 2 9 1 9 5 Terminal p o i n t s a-a b-b c - c d-d (b) F i g , 5 . 4 . In part (a) t r a c e 5 of F i g . 6 . 2 0 (b) ( f o r the copper tube) i s shown; the t e r m i n a l p o i n t s of d i g i t i z a t i o n are marked wi t h lower-case l e t t e r s . The corresponding amplitude d e n s i t i e s are tabu-l a t e d i n part ( b ) . The d i g i t i z a t i o n i n t e r v a l i s C 3 microsec. 108 db -10 -20 -30 -40 -50 i i i i ; _ i V V CHP I4H \ - ' - \ - ~ \ I I i \! CHP I 4 V - V V ^ *^ x >cs. ^> - y /"^  v v . — ' v v y / \ i i i i ,4 ! \ 4 F r e q . C P S . P i g . 5 .5. Log of energy spectra and coherence f o r the seismic record C H P 1 4 and i t s hand d i g i t i z e d v e r s i o n C H P 1 4 H . The dashed l i n e i n the coher-ence p l o t i s the expected coherence f o r wholly u n c o r r e l a t e d s i g n a l s ( a f t e r Bogert et a l . , 1962). 1 0 9 Microphoto of specimen from brass tube of d i a . 1 0 cm. Microphoto of specimen from brass tube of d i a . 1 5 cm. P i g . 6 . 1 0 Grain s t r u c t u r e at m a g n i f i c a t i o n 2 5 5 * . The top edge of each photo i s p a r a l l e l to the a x i a l d i r e c t i o n of the r e s p e c t i v e tube. 110 Microphoto of specimen from aluminum tube of d i a . 13.7 cm. Microphoto of specimen from copper tube of d i a . 12.5 cm. F i g . 6.2. Grain s t r u c t u r e at m a g n i f i c a t i o n 255x. The top edge of each photo i s p a r a l l e l to t h e ' a x i a l d i r e c t i o n of the r e s p e c t i v e tube. P i g . 6 . 3 . S t r a i n record of a t r a n s i e n t Rayleigh s i g n a l (from a d i s t a n t explosion) propagating along truncated edge of brass tube of 15 cm. d i a . The time sequence i n which the tr a c e s were taken i s , from top to bottom, 6, 2, 13 5, 4 and 3 . 112 IOyu.sec. 5yu.strain 5 0 ^ sec. > M lessee. • < — > F r e q u e n c y 500 K C / s 5 ^ . s t r a i n 50/isec. „ _ j j -• ; igaf fflrinf'-pv te? ,<!ljS|j||(i itiumi mjinr AMiift 111 pUlllr mW • F r e q u e n c y 150 K C / s P i g . 6 . 4 . S i n u s o i d a l Rayleigh wave t r a i n s (from a pie z o -e l e c t r i c transducer) propagating along edge of the brass tube of 10 cm. d i a . The frequency i s measured from the top tra c e of each record. 1 1 3 Rayleigh puise Brass tube —dia. 10 cm Distance 1/4 7TD 9/4 7TD 17/4 7TD 25/4 7TD 33/4 TT D 41/4 7TD (a) I 2 3 4 5 6 20yu.strain Rayleigh pulse Brass tube—dia. 10cm / Uvll J / \ L-v / \ f i — - - - ^ Distance I/47TD 9/4 7TD 17/4 7T D 25/4 7TD 33/4 7T D 41/4 7TD lOyxsec. (b) P i g . 6 . 5 . Each record shows 6 t r a c e s of the same Ray l e i g h pulse as i t makes m u l t i p l e t r i p s around the edge o f the BRASS tube of d i a . D = 1 0 cm. The d i s -tance from the e x p l o s i v e source that the pulse t r a v e l s before being detected by the s t r a i n gage i s shown f o r each t r a c e . 114 FREQ. (KC/s) P i g . 6.6. Amplitude d e n s i t y spectra of the records shown i n P i g . 6 ,5. The number a s s o c i a t e d w i t h each curve corresponds to the tra c e no. of the r e s -p e c t i v e record; t r a c e 1 i s at the top of each record, t r a c e 2 i s j u s t below i t and so on. P i g . 6 , 7 . Phase d i f f e r e n c e p l o t t e d a g a i n s t f r e q u e n c y f o r R e c . 6.5 (b) . The s u b s c r i p t s of r e f e r to the t r a c e n o . of the r e c o r d . 1 c i r c l e = 2 71 r a d i a n s . 1 1 6 .012 A t t e n u a t i o n c u r v e R a y l e i f h p u l s e of F i g . 6.5 ( a ) B r a s s t u b e of 10cm d i a . .008 a = 1.04 x I0"9 f 1 , 2 2 .004-100 200 300 400 500 F R E Q . ( K C / s ) .012 .008 A t t e n u a t i o n c u r v e R a y l e i g h p u l s e of F i g . 6.5 ( b ) B r a s s t u b e of 10 c m d i a . a =1.46 x 10"" f1-65 .004 100 200 300 400 500 F R E Q . ( K C / s ) F i g . 6 . 8 . A t t e n u a t i o n values f o r the t r a n s i e n t Rayleigh pulses shown i n F i g . 6 , 5 are f i t t e d by l e a s t -s q u a r e s - f i t curves. The standard d e v i a t i o n f o r each value i s represented by the r e s p e c t i v e v e r t i c a l l i n e segment. 1 1 7 A t t e n u a t i o n c u r v e FREQ. (KC/s) F i g . 6.9. L e a s t - s q u a r e s - f i t a t t e n u a t i o n curves f o r s i n u -s o i d a l R ayleigh waves. The experimental a t t e n -u a t i o n values are represented by c i r c l e s . 118 Attenuation c u r v e s , B r a s s t u b e of 10cm d i a . o 4 y a s t r a i n p u l s e F R E Q . ( K C / s ) P i g . 6 . 1 0 . S u p e r p o s i t i o n of a t t e n u a t i o n values shown i n P i g . 6 . 8 and the top of P i g . 6 . 9 . The s o l i d curve was f i t t e d t o the composite data p o i n t s by the method of l e a s t squares. 119 Rayleigh pulse Brass tube —dia. 15 cm Distance 1/4 7TD 5/4 7TD 9/4 7TD 13/4 7TD 17/4 7T D 21/4 7TD (a) Rayleigh pulse Brass tube-dia. 15 cm Distance I/47TD 5/47TD 9/4 7T D I3/47T 6 17/4 7TD 21/4 7TD F i g . 6.11. Each record shows 6 t r a c e s of the same Rayleigh pulse as i t makes m u l t i p l e t r i p s around the edge of the BRASS tube of d i a . D = 15 cm. The dis t a n c e from the e x p l o s i v e source that the pulse t r a v e l s before being detected by the s t r a i n gage i s shown f o r each t r a c e . 120 Rayleigh pulse spectra g» Fig. 6.11 j^a) Brass tube D sl5cm FREQ. (KC/s) F i g , 6„12„ Amplitude d e n s i t y spectra of the records shown i n F i g . 6.11.. The number a s s o c i a t e d with each curve corresponds to the t r a c e no. of the r e s -p e c t i v e record; t r a c e 1 i s at the top of each record, t r a c e 2 i s j u s t below i t and so on. 121 Weighted a v e r a g e a t t e n u a t i o n R a y l e i g h p u l s e « 4 / / . s t r a i n F R E Q . ( K C / s ) P l g o 6.13- Weighted-average a t t e n u a t i o n values are f i t t e d by a l e a s t - s q u a r e s - f i t curve. Values i n the top graph represent averages over records of 2.8, 3.0 and 6.3 m i c r o s t r a i n ; those i n the bottom graph are averages over 33* 4 8 and 35 m i c r o s t r a i n records. 122 Weighted—average attenuation Brass tube of 15 cm dia. ° -4yu.strain pulse A 33/xstrain pulse x sinusoidal waves (*= 4.73x10 1 4 f i - a e ; 1 _2E I 100 200 300 400 500 FREQ. (KC/s) P i g . 6.14. Superpositiftn of the a t t e n u a t i o n values shown i n ' F i g . 6.13 and i n the bottom-half of F i g . 6„9° The s o l i d l i n e represents the l e a s t -s q u a r e s - f i t curve through the composite data p o i n t s . 123 Rayleigh pulse Aluminum lube-dia. 13.7 cm Distance 1/4 7TD 5/4 7TD 9/4 7TD 13/4 7TD 17/4 7TD 21/4 7TD (a) I 2 3 4 5 6 20 jtxstrain Rayleigh pulse Aluminum tube-dia. 13.7 cm lO/tsec. < > I Distance I/47TD 5/4 7TD 9/4 TT D 13/4 7TD 17/4 7TD 21/4 7TD (b) Pig. 6,15. Each record shows 6 traces of the same Rayleigh pulse as i t makes multiple t r i p s around the edge of the ALUMINUM tube of d i a . D = 13.7 cm. The distance from the explosive source that the pulse t r a v e l s before being detected by the s t r a i n gage i s shown f o r each i-.race. 124 cn Q) E Rayleigh pulse spectra Fig. 6.15(a) Aluminum tube 400 500 FREQ. (KC/s) S" 30 '5 in 1 CO z Ul o Ul a _l a. < 2 0 -10 Rayleigh pulse spectra Fig- 6.15 (b) Aluminum tube 100 200 300 400 500 FREQ. (KC/sJ F i g . 6.1,6,, A m p l i t u d e d e n s i t y s p e c t r a o f t h e r e c o r d s s h o w n i n F i g . 6 . 1 5 . T h e n u m b e r a s s o c i a t e d w i t h e a c h t u b e c o r r e s p o n d s t o t h e t » r a c e n o . o f t h e r e s -p e c t i v e r e c o r d ; t r a c e 1 i s a t t h e t o p o f e a c h r e c o r d , t r a c e 2 i s j u s t b e l o w i t a n d s o o n . 125 .006 E u N » W or £ .004 • .2 , O 5 .002 •z I-Weighted - average attenuation Rayleigh pulse = 2.6 ytxstrain Aluminum tube a « 2.20 x I0~'8 f 2 , 6 100 200 300 - 400 500 FREQ. (KC/s) .006 Eu u oc Ul uj .004 z o 5 2 UJ I-Weighted -average attenuation Rayleigh pulse » 39yustrain Aluminum tube a » l . 4 2 x | 0 - ' 5 f 2 ' 2 0 .002 -I00 200 300 400 500 FREQ. (KC/s) P i g . 6.17. Weighted-average a t t e n u a t i o n values are f i t t e d by a l e a s t - s q u a r e s - f i t curve. Values i n the top graph represent averages over records of 2 .8 , 2.1 and 2»8 m i c r o s t r a i n ; those i n the bottom graph are averages over 33, 48 and 35 m i c r o s t r a i n records. 1 2 6 .006 i u • v . CO CC Ul Q. Ul 2 .004^ o 3 ui .002 h-I -< Attenuation curve Sinusoidal Rayleigh waves Aluminum tube -15 £. 2.20 t o o 200 300 400 500 FREQ. (KC/s) E o *s CO a: ui a . ui 2 o 3 Z Ul 5 Attenuation curve .004h Sinusoidal Rayleigh waves Copper tube .002 ii _L 100 200 300 400 500 FREQ. (KC/s) P i g . 6 „ l 8 „ A t t e n u a t i o n values f o r s i n u s o i d a l R ayleigh waves. The standard d e v i a t i o n i s shown f o r each value. The upper curve i s f i t t e d by the method of l e a s t squares. 127 .007h .006 .005 I .004 to oc Ul o_ ui z .003 z o z Ul .002 Weighted - a v e r a g e a t t e n u a t i o n A l u m i n u m t u b e o 2.6 y a s t r a i n p u l s e A 39/i.strain p u l s e x s i n u s o i d a l waves Ax A = 9-12x10 1 f .001 100 200 300 400 500 F R E Q . I K C / s ) P i g . 6.19. S u p e r p o s i t i o n of the a t t e n u a t i o n values shown i n P i g . 6.17 and the t o p - h a l f of P i g . 6.18. The s o l i d l i n e represents the le a s t - s q u a r e s -f i t curve through the composite data p o i n t s . 128 P i g . 6 .20. Each record shows 6 t r a c e s of the same Rayleigh pulse as i t makes m u l t i p l e t r i p s around the edge of the COPPER tube of d i a . D = 12.5 cm. The dist a n c e from the e x p l o s i v e source that the pulse t r a v e l s before being detected by the s t r a i n gage i s shown f o r each t r a c e . 129 Rayleigh pulse spectra FREQ. (KC/s) _ 30 Q> E v. c 2 +— in 20 >-\-to z UJ Q Ul a • a . S < 10 JRayleigh pulse spectra Fig. 6,20 (b) Copper tube 100 200 300 400 500 FREQ. (KC/s) P i g . 6. 21. Amplitude d e n s i t y spectra of the records shown i n P i g . 6.20. The number as s o c i a t e d with each tube corresponds to the t r a c e no. of the r e s -p e c t i v e record; t r a c e 1 i s at the top of each record, t r a c e 2 i s j u s t below i t , and so on. 130 Weighted average attenuation Rayleigh pulse - 3.0^.strain Copper tube .004 £ u >* CO a: ui a. ui z z .002 o 3 z Ul 100 200 300 400 500 FREQ. (KC/s) E co ox Ul a. ui z < z> z u> . 0 0 4 -Weighted average attenuation Rayleigh pulse « 37yu.strain Copper tube .002 100 200 300 400 FREQ. (KC/s) 500 Pig. 6.22. Weighted average attenuation over two records. The top curve i s averaged over records of 2.8 and 3„3 microstrain. The bottom curve i s aver-aged over records of 37 and 38 microstrain. 131 .004 E co or S .003h Ui z 3 .002 z Ui Si Weighted -average attenuation Copper tube o 3yu.strain pulse A 37 yastrain pulse x sinusoidal waves .001 A x 100 200 300 400 FREQ. (KC/s) 500 P i g , 6.23. S u p e r p o s i t i o n of the a t t e n u a t i o n c u r v e s shown i n P i g . 6.22 and i n the bot tom h a l f of P i g . 6 .18. 132 ' • ' i I i 1 1 L _ 0 100 200 300 400 500 FREQUENCY (KC/s) Pig. 6.24. Internal f r i c t i o n ( 1 / Q ) i n the metals i n v e s t i -gated as a function of frequency. Rayleigh waves i n the 2 - 7 microstrain region were used to obtain the above values. 133 TABLE 3 - 1 Numerical values computed f o r the case V = s % % V s V s V /A,U2 1 2 . 9 7 7 5 3 .9440 1.044 1 . 3 2 2 9 • . 2 7 5 1 - . 3 3 7 4 . 1 8 6 9 - . 5 3 9 5 4 . 9 3 5 6 . 9 6 9 1 . 1 0 1 8 • 3 1 2 5 - . 4 1 5 2 .2497 - . 6 0 1 6 5 . 9 2 9 9 . 9 5 8 1 . 0 0 9 8 . 3 3 7 1 - . 4 6 1 7 . 2 9 1 0 - . 6 3 0 2 6 . 9 2 6 7 .946 . 9 6 0 6 . 3 5 2 2 - .4914 . 3 1 7 3 - . 6 4 5 8 7 . 9 2 4 8 . 9 3 8 . 9 3 0 9 . 3 6 1 7 - . 5 1 1 2 .3346 ' - . 6 5 5 1 8 . 9 2 3 6 . 9 3 3 . 9 1 1 7 . 3 6 8 7 - . 5 2 5 2 . 3 4 7 2 - . 6 6 1 3 9 . 9 2 2 7 . 9 3 2 . 8 9 8 4 . 3 7 3 5 - . 5 3 5 1 . 3 5 6 1 - . 6 6 5 4 1 0 . 9 2 2 1 . 9 2 8 . 8 8 8 9 . 3 7 6 9 - . 5 4 2 5 . 3 6 2 7 - . 6 6 8 4 1 2 . 9 2 1 3 . 9 2 6 . 8 7 6 3 . 3 8 1 6 - . 5 5 2 6 . 3 7 1 5 - . 6 7 2 3 1 4 . 9 2 0 8 . 9 2 4 . 8 6 8 6 . 3 8 4 7 - . 5 5 8 9 . 3 7 7 1 - . 6 7 4 7 1 6 . 9 2 0 4 . 9 2 3 . 8 6 3 8 . 3 8 6 8 - . 5 6 3 1 . 3 8 0 9 - . 6 7 6 2 1 8 . 9 2 0 3 . 9 2 2 . 8 6 0 3 . 3 8 7 9 - . 5 6 5 9 . 3 8 3 2 - . 6 7 7 2 2 0 . 9 2 0 1 . 9 2 2 . 8 5 7 9 . 3 8 9 0 - . 5 6 8 1 . 3 8 5 2 - . 6 7 8 0 2 2 . 9 1 9 9 . 9 2 1 . 8 5 6 1 . 3 9 0 0 - . 5 6 9 7 . 3 8 6 7 - . 6 7 8 5 24 . 9 1 9 8 . 9 2 1 . 8 5 4 7 . 3 9 0 5 - . 5 7 1 0 . 3 8 7 8 - . 6 7 9 0 2 6 . 9 1 9 8 . 9 2 1 . 8 5 3 6 . 3 9 0 8 - . 5 7 1 8 . 3 8 8 5 - . 6 7 9 3 2 8 . 9 1 9 7 . 9 2 0 . 8 5 2 8 . 3 9 1 0 - . 5 7 2 5 . 3 8 9 0 - . 6 7 9 5 3 0 . 9 1 9 7 . 9 2 0 . 8 5 2 1 . 3 9 1 2 - . 5 7 3 1 . 3 8 9 5 - . 6 7 9 7 oo . 9 1 9 4 . 9 1 9 4 .8475 . 3 9 3 3 - . 5 7 7 3 . 3 9 3 3 - . 6 8 1 2 134 Table 4.1 Semiconductor s t r a i n gauge (type P01-05-120, manufactured by Microsystems, Inc). Composition S i l i c o n c r y s t a l Length 2 mm. Width 0.127 mm. Gauge factor 110 - 125 Table 4.2 Dimensions and properties of f e r r o e l e c t r i c ceramics (type US500 developed by Sonus Corporation). Length L (mm) 2.5 5,0 3,0 Radius r (mm) 1.0 1.0 3.0 Aspect r a t i o r/L 0.4 0.2 1.0 Mechanical resonance f r e q (kc/s) 720.0 360.0 600.0 Capacitance (pf) 150.0 75,0 125.0 Linear coupling c o e f f i c i e n t ( k ^ ) 0.70 0.70 0.70 135 Table 6.1 Composition and dimensions of metal tubes. Composition Thickness Diameter Length Av. G r a i n T D L Dia. Do (mm) (cm) (cm) (mm) Aluminum 1.95 13.75 120 0.055 - 0.065 Copper 1.65 12.55 120 0.040 - 0.060 Brass (10 cm. dia.) 1.70 10.0 120 0.020 - 0.030 Brass (15 cm. dia.) 1.65 15.1 120 0.015 - 0.025 1 3 6 Table 6 . 2 A t t e n u a t i o n c o e f f i c i e n t ( <X p ) f o r s i n u s o i d a l R a y l e i g h wave t r a i n s propagating along edge of BRASS tube of 10 cm. diameter. Preq. Ceramic No. of Average Atten. Standard_ L = le n g t h values s t r a i n c o e f f . dev. of cxP f R = r a d i u s n ( JJ. s t r a i n ) 6CP (kc/s) (mm) (nepersy ) SD*f 100 L5 . 0 R l 1 125 L5 . 0 R l 1 1 5 0 L5 . 0 R l 3 1 6 0 L5 . 0 R l 1 1 9 5 L5 . 0 R l 2 225 L2 . 5 R l 3 235 L5.0 R l q L2 . 5 R l 245 L5.0 R l 5 260 L5.0 R l 4 317 L5 . 0 R l 1. L2 . 5 R l 3 4 3 L5 . 0 R l 2 385 L2 .5 R l 6 400 L2 . 5 R l 7 4 3 3 L5 . 0 R l 1 442 L2 . 5 R l a. L5.0 R l 0 475 L5.0 R l 1 500 L2 .5 R l 6 5 5 0 L2 . 5 R l 2 272 L5.0 R l 3 0 . 0 0 1 0 0 . 0 0 0 7 3 16 7 7 1 9 5 6 17 7 6 29 5 6 3 2 7 8 36 6 .7 4 8 5 7 4 5 4 10 4 7 9 7 60 6 6 6 1 10 8 6 8 13 6 7 5 8 5 7 8 ' 6 3 1 0 0 10 7 1 0 6 6 6 0 . 0 1 2 0 0 . 0 0 0 9 CL p+s 8 0 . 0 0 5 0 137 Table 6 . 3 Attenuation c o e f f i c i e n t of Rayleigh pulse propagating along edge of BRASS tube of 15 cm. diameter. Records 1, 2 and 3 correspond to 2.8, 3.0 and 6.3 microstrain records respectively. Preq. (kc/s) f Record l a b e l # Atten. coeff. (cm"1) a Stand. dev. of a SDa. Weighted average CX Stand, dev. of 6c 1 0.0001 0.0004 100 2 1 4 0.0001 0.0001 3 0 6 1 0.0006 0.0005 150 2 9 2 0.0008 0.0001 3 9 5 1 0.0011 0.0002 200 2 10 2 0.0010 0.0001 3 7 5 1 0.0019 0.0002 250 2 15 3 0.0017 0.0001 3 • 17 10 1 0.0029 0.0002 300 2 20 2 0.0024 0.0003 3 24 2 1 0.0034 0.0004 350 2 27 2 0.0031 0.0004 3 42 5 l 0.0043 0.0002 400 2 41 3 0.0044 0.0004 3 56 4 1 O.OO65 0.0006 450 2 56 6 0.0062 0.0003 3 64 9 1 0.0086 0.0010 500 2 70 10 *0.0075 0.0007 3 61 20 138 Table 6.4 Attenuation c o e f f i c i e n t of Rayleigh pulse propagating along edge of BRASS tube of 15 cm. diameter. Records 1, 2 and 3 correspond to 35* 27 and 36 microstrain records respectively. Preq. Record Atten. Stand. Weighted Stand. (kc/s) l a b e l coeff. dev. average dev. of oc f (cm"1) of oc # oc 04 1 0.0001 0.0002 100 2 4 2 0.0003 0.0001 3 3 6 1 0.0007 0.0001 150 2 7 1 0.0007 0.0000 3 7 3 1 0.0012 0.0001 200 2 15 1 0.0012 0.0001 3 4 5 1 0.0017 0.0002 250 2 18 1 0.0017 0.0001 3 13 2 1 0.0026 0.0002 300 2 27 1 0.0027 0.0001 3 29 4 1 0.0038 0.0003 350 2 36 3 0.0037 0.0001 3 41 6 l 0.0060 0.0007 400 2 48 3 0.0050 0.0005 3 61 8 1 O.OO76 0.0015 450 2 73 7 0.0074 0.0002 3 78 26 1 0.0085 0.0020 500 2 72 14 0.0077 0.0004 3 77 20 139 Table 6 . 5 A t t e n u a t i o n c o e f f i c i e n t ( d P ) f o r s i n u s o i d a l R a y l e i g h wave t r a i n s propagating along edge of BRASS tube of 1 5 . 1 cm. diameter. Preq. Ceramic L = l e n g t h f R = r a d i u s (kc/s) (mm.) No. of Average values s t r a i n n (JJ. s t r a i n ) A t t e n . Stand, c o e f f . dev_^ _ OLP of otP (nepers/ c m) SDoC P 120 L3.0 R3 1 , 6 0 . 0 0 0 7 0.0001 135 L3.0 R 3 3 4 11 4 165 L2 . 5 R l 3 5 9 2 185 L2 . 5 R l 2 6 15 2 209 L2 . 5 R l 2 6 14 6 242 L2 . 5 R l 6 6 17 3 304 L2 . 5 R l 4 6 22 3 322 L2 . 5 R l 4 8 26 2 357 L2 . 5 R l 4 8 31 5 383 L3.0 R l 2 4 39 4 450 L2 . 5 R l 4 8 60 4 488 L2 . 5 R l 1 4 66 6 525 L3.0 R3 1 4 72 6 556 L2 . 5 R l 1 5 0.0100 0 . 0 0 0 6 OCpts 140 L2 . 5 R l 5 0.0014 205 L2 . 5 R l 4 0.0021 420 L 2 . 5 R l 5 0.0053 I 140 Table 6,6 Attenuation c o e f f i c i e n t of Rayleigh pulse propagating along edge of ALUMINUM tube of 13.7 cm. diameter. Records 1, 2 and 3 correspond to 2.8, 2.1 and 2.8 microstrain records respectively. Freq. Record Atten. Stand. Weighted Stand. (kc/s) l a b e l coeff. dev. average dev. f (cm"1) of oc of oc # Oc S D * CX S D * . 1 0.0003 0.0007 100 2 10 4 0.0008 0.0002 3 8 5 1 0.0008 0.0003 150 2 9 2 0.0009 0.0001 3 7 4 1 0.0008 0.0002 200 2 4 1 0.0005 0.0002 3 11 4 1 0.0006 0.0002 250 2 13 4 0.0007 0.0002 3 8 3 1 0.0011 0.0001 300 2 11 5 0.0011 0.0001 3 14 1 1 0.0015 0.0005 350 2 24 3 0.0015 0.0004 3 11 2 1 0.0026 0.0006 400 2 34 9 0.0032 0.0004 3 40 7 1 0.0044 0.0001 450 2 42 6 0.0045 0.0002 3 53 4 1 0.0052 0.0010 500 2 85 16 0.0057 0.0009 3 67 10 141 Table 6.7 A t t e n u a t i o n c o e f f i c i e n t of R a y l e i g h p u l s e p r o p a g a t i n g a l o n g edge of ALUMINUM tube of 13.7 cm. diameter. Records 1, 2 and 3 correspond to 33.» 48, and 35 m i c r o s t r a i n r e c o r d s r e s p e c t i v e l y . Preq. Record A t t e n . Stand. Weighted Stand. (kc/s) l a b e l c o e f f . dev. average dev. f (cm" 1) of oc of oc OC S D c OC SD«* 1 0.0001 0.0002 100 2 4 7 0.0002 0.0004 3 6 3 1 0.0005 0.0002 150 2 5 3 0.0003 0.0003 3 1 2 1 0.0009 0.0001 200 2 9 5 0.0008 • 0.0002 3 5 2 1 0.0006 0.0002 250 2 10 2 0.0008 0.0001 3 7- 2 1 0.0014 0.0001 300 2 14 1 0.0014 0.0001 3 12 2 1 0.0022 0.0002 350 2 16 2 0.0020 0.0002 3 21 3 1 0.0038 0.0002 400 2 27 4 0.0034 0.0005 3 26 4 1 0.0053 0.0004 450 2 44 5 0.0050 0.0003 3 51 5 1 0.0077 0.0009 500 2 51 8 0.0064 0.0007 3 67 7 142 Table 6 . 8 Attenuation c o e f f i c i e n t ( CX P ) f o r s i n u s o i d a l Rayleigh wave t r a i n s propagating along edge of ALUMINUM tube of 1 3 * 7 5 cm. diameter. Preq. Ceramic No. of Average Atten. Stand. L = leng t h values s t r a i n c o e f f . dev._ f R = r a d i u s n ^ of o( P (kc/s) (mm.) ( u s t r a i n ) p ' ( n e Pers / c m ) s D s P 9 5 L 3 . 0 R 3 1 3 - too small 1 0 5 L 3 . 0 R3 1 3 - to measure 1 6 3 L 2 . 5 R l 1 8 0 . 0 0 0 5 0 . 0 0 0 3 1 7 7 L 3 . 0 R3 1 4 7 3 2 1 5 L 2 . 5 R l 2 1 0 1 0 4 245 L 2 . 5 R l 3 2 0 1 2 2 2 6 0 L 2 . 5 R l 2 4 1 0 3 2 8 0 L 2 . 5 R l 1 1 1 1 3 4 2 9 0 L 2 . 5 R l 3 1 6 14 5 3 3 0 L 2 . 5 R l 2 1 0 2 0 2 3 5 3 L 2 . 5 R l 5 7 2 5 2 3 8 8 L 2 . 5 R l 8 6 3 5 7 4 0 5 L 2 . 5 R l 3 5 3 6 4 4 5 5 L 2 . 5 R l 2 5 48 4 4 6 3 L 2 . 5 R l 1 5 5 3 8 5 1 5 L 2 . 5 R l 2 5 0 . 0 0 6 4 0.0004 440 L 2 . 5 R l 4 0.0046 143 Table 6,9 The v a r i a t i o n i n the amplitude d e n s i t y (of the Rayleigh pulse propagating around the edge of the copper tube of 12.5 em, diameter) as a r e s u l t of i n c r e a s i n g the number of sample p o i n t s . The i n t e r v a l between sample p o i n t s i s 0.30 microseconds. Trace 3 F i g . 6.20(a) Preq. (kc/s) Ampli-tude Den-s i t y ) 100 1.76 1.79 1,76 1.81 1.83 150 1.82 1.77 1,76 1.72 1.68 200 2.32 2.24 2,29 2.28 2.33 250 2.12 2.12 2.17 2.16 2.14 300 1.73 1.69 1.68 1,68 1.69 350 1.57 1.48 1,42 1.45 1.46 400 1.01 0.95 O.96 0,92 0.93 450 0.84 O.85 0,88 0,91 0.90 500 0.45 0,42 0,45 0,44 0.45 No. sample 189 p o i n t s 105 125 141 169 Trace 4 P i g , 6,20(a) Preq, (kc/s) Ampli-tude Den-s i t y ( ^ M / s ) 1 100 1.61 1,56 1,58 1.58 1,64 150 1.78 I .83 1.84 I .78 1.79 200 2,22 2,15 2,16 2.17 2.09 250 2,08 2,02 2,04 2.05 2,06 300 1.40 1,40 1.37 1.35 1.31 350 1.41 1.33 1,25 1.26 1.35 400 0.79 0,77 0.75 0.75 0,75 450 0.60 0.58 O.63 O.67 0,72 500 0.32 0,32 0.32 0.27 0,23 No. sample 134 I65 183 p o i n t s 101 117 144 Table 6 . 1 0 The v a r i a t i o n i n the amplitude d e n s i t y (of the Rayleigh pulse propagating around the edge of the copper tube of 1 2 . 5 cm. diameter) as a r e s u l t of i n c r e a s i n g the number of sample p o i n t s . The I n t e r v a l between sample p o i n t s i s 0 . 3 0 microseconds. Trace 5 F i g . 6 . 2 0 ( a ) Freq. (kc/s) Ampli-tude Den-s i t y ( ^ / M c / s ) 1 0 0 1 . 4 4 1 . 4 5 1 . 5 9 1 5 0 1 . 7 5 1 . 7 1 1 . 8 0 2 0 0 1 . 8 1 1 . 7 9 1 . 8 9 2 5 0 I . 8 7 1 . 8 4 1 . 7 6 3 0 0 1 . 2 3 1 . 2 0 1 . 2 1 3 5 0 1 . 1 4 1 . 1 1 1 . 0 0 400 O . 6 7 0 . 6 1 0 . 5 9 4 5 0 0 . 5 9 0 . 5 0 0 . 5 1 5 0 0 0 . 2 7 0 . 1 9 0 . 1 6 No. sample p o i n t s 1 2 3 1 5 9 1 9 1 Trace 6 F i g . 6 . 2 0 ( a ) Freq. Ampli- Den- ) (kc/s) tude s i t y \ / 7s 1 0 0 1 . 6 2 l o 5 7 1 . 6 1 1 . 6 6 1 . 6 3 1 5 0 1 . 5 6 1 . 6 0 1 . 6 3 1 . 6 0 1 . 6 3 2 0 0 I . 8 5 I . 8 5 1 . 8 2 1 . 8 2 1 . 7 8 2 5 0 1 . 5 9 1 . 5 6 1 . 5 4 1 . 5 7 1 . 6 1 3 0 0 0 . 8 8 0 . 9 4 0 . 9 8 0 . 9 1 0 . 8 9 3 5 0 1 . 0 5 0 . 9 9 1 . 0 0 1 . 0 0 1 . 0 5 400 0 . 4 0 0 . 4 0 0 . 3 6 0 . 3 6 0 . 3 5 4 5 0 O . 5 8 O . 5 8 0 . 6 0 O . 6 5 0 . 7 1 5 0 0 0 . 2 5 0 . 2 9 0 . 2 8 0 . 2 4 0 . 2 2 No. sample 143 1 8 3 p o i n t s 1 0 3 1 2 7 1 5 9 145 Table 6.11 The v a r i a t i o n i n the amplitude d e n s i t y (of the Rayleigh pulse propagating around the edge of the copper tube of 12.5 cm. diameter) as a r e s u l t of i n c r e a s i n g the number of sample p o i n t s . The i n t e r v a l between sample p o i n t s i s 0.30 microseconds. Trace 3 P i g . 6.20(b) Freq. (kc/s) Ampli-tude Den-s i t y ( M c / S ) 100 25.94 26.56 26.31 26.17 26.78 150 32.07 32.40 31.61 32.32 31.72 200 32.59 32.40 31.64 31.84 31.69 250 28.12 28.28 28.29 27.92 27.57 300 18.08 18.49 19.26 19.81 20.31 350 13.03 12.80 13.60 13.81 13.41 400 6.58 6.04 6.31 6.09 6.40 450 4.49 4.38 4.22 4.41 4.01 500 3.38 3.48 3.28 3.48 3.67 No. sample p o i n t s 109 129 149 164 195 Trace : 4 F i g . 6.20(b) Freq. (kc/s) Ampli-tude Den-s i t y 100 24.43 24.86 24.77 24.22 24.53 150 29.97 29.84 29.19 29.37 28.85 200 30.24 30.40 29.70 30.26 30.51 250 26.27 26.79 26.99 26.58 26.31 300 15.39 15.67 16.73 17.56 17.93 350 11.53 11.75 12.46 12.78 12.61 400 4.07 3.99 3.68 3.97 4.18 450 2.05 1.96 1.45 1.75 1.47 500 1.44 1.78 1.35 1.37 1.52 No. sample 185 p o i n t s 109 127 149 173 146 Table 6.12 The v a r i a t i o n i n the amplitude d e n s i t y (of the Rayleigh pulse propagating around the edge of the copper tube of 12.5 cm. diameter) as a r e s u l t of i n c r e a s i n g the number of sample p o i n t s . The i n t e r v a l between sample p o i n t s i s 0.30 microseconds. Trace 5 P i g . 6.20(b) Freq. (kc/s) Ampli-tude Den-s i t y 100 23.91 2 4 . 8 7 2 4 . 9 2 2 4 . 3 2 150 28.69 28.95 28.54 28.90 200 28.53 28.23 27.61 26.56 250 2 4 . 4 4 2 4 . 8 6 25.73 26.33 300 12.16 12.31 1 4 . 0 3 14.55 350 9.53 9.21 9.97 9»85 4 0 0 3.49 3.15 3 . 1 4 3.81 450 1.91 1.92- 2.52 2.23 500 0.91 0.97 • 0.76 0 . 4 8 No. sample p o i n t s 105 129 163 193 Trace 6 F i g , 6.20(b) Freq. (kc/s) Ampli-tude Den-s i t y 100 2 3 . 0 0 23.09 23.22 22.77 23.26 150 28.54 28.05 27.99 27.82 27.03 200 26.27 25.97 25.43 26.50 26.96 250 21.55 21.98 2 1 . 4 6 20.97 2 0 . 4 6 300 10.71 11.51 11.69 12.50 12.96 350 9 . 4 2 9.70 10,59 10.76 10.74 4 0 0 5.01 4 . 8 l 4.83 5.06 5.17 450 3.25 3.26 2.58 2.28 2.59 500 1.38 1.51 1.08 1.16 1.18 No. sample p o i n t s ~ 117 131 1 4 9 171 185 147 Table 6,13 Attenuation c o e f f i c i e n t of Rayleigh pulse propagating along edge of COPPER tube of 12.5 cm. diameter. Records 1, 2, 3 and 4 correspond to 2.8, 3,3, 37 and 38 micro-s t r a i n records respectively. Preq. Record Atten, Stand, Weighted Stand. (kc/s) l a b e l coeff, dev. ofoc average dev. f (cm"1 ) of oc # oc SDoc OC S DSL 100 1 0,0009 0,0004 0.0009 0.0000 2 9 4 150 1 0.0004 0.0004 0.0007 0.0003 2 9 4 200 1 0,0014 0.0002 0.0012 0.0002 2 10 2 250 1 0.0020 0,0003 0.0021 0.0001 2 23 6 300 1 0.0043 0.0004 0.0041 0.0003 2 38 5 350 1 0,0032 0.0003 0.0036 0.0003 2 48 6 100 3 4 0.0011 8 0.0001 1 0,0008 0.0003 150 3 4 0.0013 11 0.0002 2 0.0012 0.0001 200 I 0,0016 9 0,0001 1 0,0014 0.0002 250 3 4 0.0017 21 0.0004 2 0.0020 0.0001 300 I 0.0031 42 0,0003 2 0.0039 0.0004 350 0.0025 0,0004 0.0032 0.0005 39 4 1 4 8 T a b l e 6 „ l 4 A t t e n u a t i o n c o e f f i c i e n t ( 0C P ) f o r s i n u s o i d a l R a y l e i g h wave t r a i n s p r o p a g a t i n g a l o n g edge of COPPER tube o f 12.5 em. d i a m e t e r . P r e q . Ceramic No. o f Average A t t e n . S t a n d . _ L = l e n g t h v a l u e s s t r a i n c o e f f . dev. ofotp f R = r a d i u s n ^ ( k c / s ) (mm.) ( a s t r a i n ) p ' (nepers/) S D « P 'cm 130 L 2 . 5 R l 4 7 0.0008 0.0005 155 L 2 . 5 R l 1 3 11 4 165 L 2 . 5 R l 8 7 15 3 215 L 2 . 5 R l 1 4 20 5 235 L 2 . 5 L5.0 R l R l 5 8 13 5 257 L 2 . 5 R l 5 8 17 3 280 L5.0 R l 1 6 20 5 300 L 2 . 5 R l 8 8 26 10 320 L 2 . 5 R l 5 6 32 12 340 L2 . 5 R l 7 7 31 7 362 L 2 . 5 L5.0 R l R l 8 8 33 7 375 L5.0 R l 8 L 3 . 0 R3 7 0.0034 0.0002 L 2 . 5 R l 150 L 3 . 0 R 3 5 0.0012 235 L 5 . 0 R l 8 0.0025 370 L 5 . 0 R3 7 0.0040 149 Table 6.15 Measured phase v e l o c i t i e s of Rayleigh waves. Brass Brass Aluminum Copper Frequency tube tube tube tube (kc/s) 10 cm d i a . 15 cm d i a . 13.75 cm 12.5 cm d i a . d i a . C C C C (km/s) (km/s) (km/s) (km/s) 100 1.89 2.78 125 135 1 4 5 150 200 210 235 245 260 272 300 320 330 3 4 0 350 360 375 385 395 4 0 0 410 4 4 0 455 500 550 1.88 1.86 2.05 2.02 2 . 0 4 1.88 1.86 2.03 I . 8 7 I . 8 5 1 . 8 4 2.78 2.05 2.05 1.87 1.83 2 . 0 4 1.86 2.78 2.73 2 . 0 4 I . 8 3 2.08 2.08 1.86 I . 8 3 2.73 2.03 2.03 I . 8 3 2.71 2.01 2.01 1.85 2.71 1.92 I .85 1.98 1.83 2.70 1.98 I . 8 5 2.70 1.96 I .85 I . 8 3 150 Table 6,16 I n t e r n a l f r i c t i o n 1/Q i n the metals l i s t e d below f o r Rayleigh waves i n the 2 - 7 m i c r o s t r a i n r e g i o n . Brass Brass Aluminum Copper tube tube tube tube 10 cm. 15 cm. 13.75 12.5 d i a . d i a , cm. d i a . cm, d i a , (icc7s) VQ VQ VQ VQ x I O 5 x I 0 5 x I 0 5 x I 0 5 100 90 6 71 59 150 80 32 53 30 200 89 29 22 39 250 88 40 25 55 300 91 47 32 89 350 98 52 37 68 400 105 64 69 450 117 80 86 500 121 87 98 151 Table 6.17 „n L e a s t - s q u a r e s - f i t expression of the form OC = cf f i t t e d to experimental a t t e n u a t i o n data. (X = a t t e n u a t i o n c o e f f i c i e n t ; c = constant; f = frequency i n cps; n = constant exponent. M a t e r i a l Comments BRASS TUBE 10 cm. d i a . BRASS TUBE 15 cm. d i a . ALUMINUM TUBE 13.7 cm. d i a . 4 /A € p u l se P i g . 6 .7 (top) 34 yU€ pulse P i g . 6.7 ( b o t . ) S i n u s o i d a l wave P i g . 6 .8 (top) Composite P i g . 6 . 9 . 4 /U£ pulse P i g . 6.12 (top) 33 ue pulse P i g . 6.12 (bot.) S i n u s o i d a l wave F i g . 6 . 8 (bot.) Composite P i g . 6.13 3 /Ue pulse P i g . 6.16 (top) 39 JU £ pulse F i g . 6.16 (bot.) S i n u s o i d a l wave F i g . 6.17 (top) Composite F i g . 6.18 1.04 x 10" 9 •11 1.46 x 10 1.47 x 10 1.36 x 1.0 1.37 x 10* 1.58 x 10 1.23 x 10 4.73 x 10 2 .20 x 10 1.42 x 10' 1,72 x 10' 9.12 x 10 -10 -10 •16 -14 -11 -14 •15 -15 -15 -16 n 1.22 1.55 1.37 1.38 2.42 2.05 1.52 1.96 2 .16 2.20 2.20 2.24 STDV. n 0 .06 0.05 0.06 0.04 0.20 0.05 0.10 0.10 0.20 0.16 0.10 0.11 152 APPENDIX General AI F o r t r a n program on F o u r i e r i n t e g r a l method which i n c o r p o r a t e s F i l o n ' s method„ This program was w r i t t e n to analyze records of seismic pulses which are d i g i t i z e d at equispaced i n t e r v a l s and f o r which the f i r s t and l a s t o r d i n a t e s must be zero. The number of sample p o i n t s must be odd. A l l F o r t r a n program f o r determining the a t t e n u a t i o n c o e f f i c i e n t alpha by the method of l e a s t squares. The data are f i t t e d to a l i n e whose equation i s y = beta + alpha x wi t h the r e s u l t that values of alpha are p r i n t e d out as negative numbers. The stan-dard d e v i a t i o n of alpha i s a l s o p r i n t e d out, A I I I F o r t r a n program on d i s p e r s i o n of Ray l e i g h waves. The d i s p e r s i o n of Rayleigh waves as a r e s u l t of the curvature of a tube i s i n v e s t i g a t e d . The waves considered are those surface waves which are confined to the truncated edge of the tube. 153 AIV Asymptotic expressions f o r o( , fi and V f o r small 6 These expressions are a s s o c i a t e d w i t h P i l o n ' s method. I n the l i m i t as 6 tends to zero, 0(. tends to zero, /3 tends to 2/3, If tends to 4/3 and P i l o n ' s method y i e l d s the same r e s u l t s as does Simpson's r u l e , AV Lamb's problem. The waves confined to the surface and which are generated by a time-harmonic l i n e source l y i n g on the surface and v i b r a t i n g normal to the surface of a semi-i n f i n i t e s o l i d are considered. The p a r t i c l e d i s p l a c e -ments p a r a l l e l ( U ) and normal ( ur ) to the surface are considered. 154 A I . Fortran program f o r determining the harmonic components of a transient signal using a Fourier Integral Analysis which incorporates F i l o n ' s Method. FORTRAN SOURCE LIST SOURCE STATEMENT K = 1 READ (5,956) IN, IM 956 FORMAT (212) 999 J = 1 PRINT 25 250F0RMAT (6lH N T FO DELFY FYMAX H SENST GNP 1ENLGMT/) 500READ 100, N, T, FO, DELFY, FYMAX, H, SENST, GAINP, ENLGT,RPOT, 1 RG, GR, E 1000P0RMAT ( 13, P8 . 3 , 3F8.5, P7 . 3 , F7 - 4 , F4.0, F5.2, 1 F6.0, P5-0, F5.0, F4.0 ) PRINT 110, N, T, FO, DELFY, FYMAX, H 110 FORMAT (1X13, Po .3 , 3 P 8 . 5 , P7-3 ) DIMENSION Y(300), TA(300) GO TO 175 150 READ 1500, N 1500 FORMAT ( 13) 175 READ 200, (Y(I), I = 1,N) 200 FORMAT ( 16F5.2) FY = FO PRINT 250 250 FORMAT ( 48H FREQUENCY C(MICSTN) PHASE(CIRCLE)//) 275 OMGPY = 2. x 3.1416 x FY THETA = OMGPY x H IF ( THETA - 0.10) 285, 285, 280 2800BETA = 2.x(THETAx(l.+(C0S(THETA))xx2) -2.x(SIN(THETA)) xCOS(THETA)) l/THETAxx3 GAMMA = 4.x(SIN(THETA) - THETAxCOS(THETA))/THETAxx3 GO TO 290 285 BETA=0.66666667+O:13333333xTHETAxx2-(THETAxx4)x4./105. GAMMA=1.3333333-0.13333333xTHETAxx2+(THETAxx4)/210. > 290 DO 300 I = 1, N AI = I 155 300 TA(I) = OMGPY x (AI - l o ) x H SUMPL =0.0 SUMEV =0.0 SUMOD =0.0 SUMBE =0.0 SUMDD =0.0 NM1 = N - l NM2 = N-2 DO 400 I = 1, N, NM1 400 SUMPL = SUMPL + Y ( l ) x S I N ( T A ( l ) ) DO 500 1 = 2, NM1, 2 500 SUMEV = SUMEV +GAMMAxY(l)xSIN(TA(l)) DO 600 1 = 3, NM2, 2 600 SUMOD = SUMOD + BETAxY(l)xSIN(TA(l)) SUMSI = SUMFL + SUMEV + SUMOD A = (H/(SQRT(2 . x 3 . l 4 l 6)))xSUMSI DO 700 I = 1, N, NM1 700 SUMBE = SUMBE + Y(l)xCOS(TA(l)) DO 800 1 = 2, NM1, 2 800 SUMEN = SUMEN +GAMMAxY(l)xCOS(TA(I)) DO 900 1 = 3, NM2, 2 900 SUMDD = SUMDD + BETAxY(l)xCOS(TA(i)) SUMCO = SUMBE + SUMEN + SUMDD B = (H/(SQRT(2 .x3ol4l6)))xSUMCO C = SQRT( AXX2 + Bxx2 ) C = (10.xx6)x(RP0T+RG)xSENSTxC/(ExGRxRGxGAINPxENLGT) Z = -A/B PHASE = ATAN(Z) IP ( B - 0. ) 19, 20, 20 19 PHASE = PHASE + 3.14 20 PHASE = PHASE/6.2832 PRINT 1000, FY, C, PHASE 1000 FORMAT (IXF8 . 5 , P13 . 3 , F19 - 5 ) FY = FY + DELPY IP ( FY - FYMAX ) 275, 275, 1100 1100 J = J + 1 IF ( J - IN) 150, 150, 1200 1200 K = K + 1 IF (K - IM) 999, 999, 1300 1300 STOP END 1 5 6 A I I . F o r t r a n program f o r determining the a t t e n u a t i o n c o e f f i c i e n t alpha by the Method of Least Squares. FORTRAN SOURCE LIST SOURCE STATEMENT DIMENSION C ( 5 0 0 0 ) , X(lOO) READ 1 5 0 , NCYL 1 5 0 FORMAT ( I I ) READ 200, FO, DELFY, FYMAX 200 FORMAT ( 3 F 8 . 5 ) L = 1 1 2 5 K = 1 READ 2 5 0 , NRDS, MNTRS 2 5 0 FORMAT ( 2 1 3 ) READ 2 7 5 , ( X(IR), IR = 1, MNTRS ) 275 FORMAT ( 9 F 8 . 3 ) 288 FREQ = FO PRINT 300 300 FORMAT ( 5 1 H FREQUENCY BETA ALPHA STD DEV ALPHA//) 3 1 0 READ 3 1 5 , NTRS 315 FORMAT ( 13) READ 320, ( C ( I ) , I = 1, NTRS) 320 FORMAT ( 16F5.2) 3 5 0 DO 400 1 = 1 , NTRS 400 C(I) = AL0G(C(I)) SUMX =0.0 DO 500 IT = 1/ NTRS 500 SUMX = SUMX + X(IT) SUMCT =0.0 DO 6 0 0 IT = 1, NTRS 6 0 0 SUMCT = SUMCT + C(IT) SUMXC =0.0 DO 7 0 0 IT = 1, NTRS 7 0 0 SUMXC = SUMXC 4- (X(IT)) x (C(IT)) SUMXX =0.0 DO 8 0 0 IT = 1, NTRS 8 0 0 SUMXX = SUMXX + (X(IT)) xx2 PNTRS = NTRS DENOM = PNTRS x SUMXX - SUMX xx 2 TEST = ABS ( DENOM ) ERROR = 0.0001 IF ( TEST - ERROR ) 1050, 1050, 8 5 0 157 850 ALPHA = (PNTRS x SUMXC - SUMX x SUMCT ) / DENOM BETA = ( SUMCT x SUMXX - SUMXC x SUMX ) / DENOM D = PNTRS x SUMXX - SUMX xx 2 SUMDS = 0 . 0 DO 900 IT = 1, NTRS 900 SUMDS = SUMDS + ( BETA + ALPHA x X ( l T ) - C(IT) ) xx 2 RE = SQRT( SUMDS / ( PNTRS - 2 . ) ) STDEV = RE x SQRT (PNTRS / D) PRINT 1000, PREQ, BETA, ALPHA, STDEV 1000 FORMAT ( F9»5, F13 . 4 , F 1 3 A F 1 3 » 6 ) 1050 FREQ = FREQ + DELFY IF ( FREQ - FYMAX) 310, 310, 1100 1100 K = K + 1 IF ( K - NRDS) 288, 288, 1200 1200 L = L + 1 IF ( L - NCYL ) 125, 125, 1300 1300 STOP END 158 A i n . F o r t r a n program on d i s p e r s i o n of Rayleigh waves. FORTRAN SOURCE LIST SOURCE STATEMENT EN = 0.500 S = 3 . PRINT 100, EN 100 FORMAT ( 5HEN = , P 6 . 3 / ) PRINT 200. S 200 FORMAT ( 4HS = , F4.0 ) PRINT 125 1250F0RMAT (58H ZETA1 ZETA2 XISQ U1/U2 X2/DEL2 X1U1/DEL1U2 F 1NXI/) XISQ =0.84 4 A = 2.x(EN+l.)xXISQ - 4.x(2.xEN+l.)/Sxx2 CHI = (XISQ - 2.x(EN+l»)x(l.+Sxx2)/Sxx2)xXISQ B = - 4.x(2.xEN+l.)/Sxx2 + CHI + (XISQ - l . ) x A C = (XISQ - l.)xCHI RAD = Bxx2 - 4.xAxC IF (RAD) 9,400,400 400 Z1SQ = (-B + SQRT(RAD))/(2.xA) IF (Z1SQ) 9 , 4 5 0 , 4 5 0 4 5 0 Z2SQ = (-B - SQRT(RAD))/(2.xA) IF (Z2SQ) 9, 5 0 0 , 5 0 0 5 0 0 ZETA1 = SQRT(ZISQ) ZETA2 = SQRT(Z2SQ) GAMA1 = 1. - ZETAlxx2 - XISQ GAMA2 = 1. - ZETA2xx2 - XISQ XI = (4.xENxZETAlxx2)/Sxx2 + GAMAlx(XISQ - 2.x(EN+l.) /Sxx2) X2 = (4„xENxZETA2xx2)/Sxx2 + GAMA2x(XISQ - 2.x(EN+l.) /Sxx2) DELI = ZETAlx ( - 4.x(EN+1.)/Sxx2 - GAMAlx(XISQ - 2.x (EN+1.)/Sxx2)) DEL2 = ZETA2x(-4.x(EN+1.)/Sxx2 - GAMA2x(XISQ - 2.x (EN+1.)/Sxx2)) Y l = 2.xGAMAlx(-ENxZETAlxx2 + EN+1.) Y2 = 2.xGAMA2x(-ENxZETA2xx2 + EN + 1.) 159 U1DU2 = ( -1. + ZETA2 x X2 / DEL2 ) / ( -1. + ZETAlxXl / DELI ) X2DD2 = X2 / DEL2 XU = ( XI / DELI ) x ( U1DU2) OFNXI = ((l.+EN)xZETAlxDELl/EN + XI + Yl/Sxx2)x (-DEL2 + ZETA2xX2) 1 - ((l.+EN)xZETA2xDEL2/EN + X2 + Y2/Sxx2)x (-DELI + ZETAlxXl) PRINT 600, ZETA1, ZETA2, XISQ, U1DU2, X2DD2, XU, FNXI 600 FORMAT ( 2F8.4, F8.3, 3F8.4, F12.6 ) 9 XISQ = XISQ +0.01 IF (XISQ - 0.90) 4, 4, 12 12 S = S + 1. PRINT 200, S IF ( S - 10.) 700, 700, 800 700 XISQ =0.84 GO TO 4 800 STOP END 160 A IV. Asymptotic expressions f o r o t , £ , and Y - expressions used i n P i l o n ' s method. For small 0 , 45 315 4 7 2 5 • t * Q _ 2 | 2 e 2 _ 4_e^ 2 e 6 ^ 3 15 1 0 5 567 Y = _5_ _ 1©! 4 _§L ®1 4 ' 3 15 210 11340 161 A V . Lamb's problem Consider a time-harmonic l i n e source a c t i n g normally to the surface of a s e m l = i n f i n i t e s o l i d and l y i n g on the s u r f a c e . Let the x and y axes l i e on the surface with the y a x i s p a r a l l e l to the l i n e source and the z a x i s p o i n t i n g down i n t o the s o l i d . Then the p a r t i c l e displacements u and w In the x and z d i r e c t -ions r e s p e c t i v e l y are (see Ewing et a l [1957] ) U = '- -y 1 exp[i(o)t- KXH + -^ -f{c (k . x ) " 1 expft^t.-k.xj] w = -+ D. ( kB x) 2 ex p[i (<M - k px w h e r e C Wl K( kg - k l f 2 ( k2 - 2 k2.)3 exp(-i \ ) D c 'l V 2 TT ki kP2 exp(-i 4 ) (k 2 - 2 v S D, 4 i V 1 6 2 K(2K 2- k 2 - 2 V K 2 - k2 V > - kB2) F ' ( K ) V *2 -F ' ( K ) F(K) = ( 2 K 2 - k , 2 ) 2 - 4 K 2 v / F'(K) = ^F(K) v =\A<2- k.2 V' = VK2- k,2 k, = Q = c o n s t a n t K , k* , a nd k0 are the wave numbers of the Rayleigh wave, the compression wave and the shear wave r e s pect ively . and H = -163 BIBLIOGRAPHY Andrade, E. N. da C., and D.A. Aboav, D i s t r i b u t i o n of grain size i n annealed metals, Nature, Vol. 207, 68-69, 1965. Arnold, J . S., and J. G. 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