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The reflection and transmission of Rayleigh waves Clement, Maurice James Young 1961

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- i -THE REFLECTION AND TRANSMISSION OP RAYLEIGH WAVES by MAURICE JAMES YOUNG CLEMENT B. Sc., University of B r i t i s h Columbia, i960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physics We accept t h i s thesis as conforming to the required standard . THE UNIVERSITY OF BRITISH COLUMBIA August, I 9 6 I In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be alloi\red without my written permission. Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. September 18, 1961 ABSTRACT The techniques of two-dimensional model seismology were employed to investigate the problem of Rayleigh waves incident upon the boundary between two s o l i d e l a s t i c media. The object of the investigation was the determination of the r e f l e c t i o n and transmission c o e f f i c i e n t s . As a preliminary, the s p e c i a l case of a quarter space was examined to test the hypothesis that the r e f l e c t i o n c o e f f i c i e n t could be successfully approximated by selecting the amplitude of the r e f l e c t e d Rayleigh wave so that the normal and tangential stresses imposed on the free surface are minimized i n the least square sense. Analogously for the half space, i t was proposed that the r e f l e c t i o n and transmission c o e f f i c i e n t s should be such as to minimize the differences (or residuals) between the respective stresses and displacements on either side of the discontinuity. This i s a reasonable assumption since the bound-ary conditions require continuity i n this case. I t was found that the agreement between the measured and calculated values of the c o e f f i c i e n t s was only q u a l i t a t i v e and i t had to be con-cluded that the proposed hypotheses were not s u f f i c i e n t to explain the r e f l e c t i o n and transmission of Rayleigh waves. For a half space consisting of an aluminium a l l o y and p l e x i g l a s s , the r e f l e c t i o n and transmission c o e f f i c i e n t s were measured as a function of the angle with which the plane i n t e r -face between the two media meets the free surface. The angle was varied i n steps of 10° from 0° to 180° and the observed co e f f i c i e n t s are presented with no attempt at a t h e o r e t i c a l derivation. - i l i -A CKNOWLEDGEMENTS I would l i k e to express my sincere thanks to Dr. J . C. Savage, who suggested the topic for this t h e s i s . His guidance throughout a l l stages of the work was greatly appreciated. For his help, thanks are also due to Mr. L. Mansinha, a fellow graduate student, who was responsible f or most of the apparatus used i n th i s research. Furthermore, the f i n a n c i a l assistance of the National Research Council of Canada i n the form of a Bursury i s grate-f u l l y acknowledged. The work was also supported by research grants from the American Petroleum I n s t i t u t e and the National Research Council. - i v -TABLE OP CONTENTS INTRODUCTION CHAPTER I CHAPTER I I CHAPTER I I I CHAPTER IV ELASTIC WAVE THEORY 1.1 Equations of Motion 1.2 Boundary Conditions 1.3 Rayleigh Waves 1.4 Plate Wave Ve l o c i t i e s 1.5 Attenuation REFLECTION AND TRANSMISSION OF RAYLEIGH WAVES 2.1 Previous Work 2.2 Quarter Space Hypothesis 2.3 Half Space Hypothesis APPARATUS 3.1 Introduction 3.2 Transducers and Holders 3.3 Experimental Arrangement MEASUREMENTS AND RESULTS 4.1 Wave V e l o c i t i e s 4.2 Reflection and Transmission Coefficients 4.3 Comparison with Theory 4.4 Variat i o n of Coefficients with Interface Angle 4.5 Conclusions page 1 4 6 6 9 10 BIBLIOGRAPHY 12 14 16 22 23 24 28 31 35 36 40 42 -v-ILLUSTRATIONS AND TABLES to follow page FIG. 1 Reflection and Transmission Coefficients for a Corner (After de Bremaecker) 12 FIG. 2 (a) Cylinder Transducer 23 (b) Transducer Holder 23 FIG. 3 Block Diagram of Apparatus 24 FIG. 4 An Example of a Multiple Trace Seismogram 28 FIG. 5 Travel-Time Curves f o r Plexiglass 30 FIG. 6 Reflection Data for Three Quarter Spaces 33 FIG. 7 Reflection and Transmission Data f or Plexiglass — Aluminium 90° Half Space 34 FIG. 8 Transmission Data f or Plexiglass - Aluminium (120°- 60°) Half Space 38 FIG. 9 Reflection and Transmission Coefficients for the "Al to P i " Half Space as a Function of the. Interface Angle 39 FIG. 10 Reflection and Transmission Coefficients for the "PI to A l " Half Space as a Function of the Interface Angle 39 TABLE I Measured Properties of Aluminium, Plexiglass and Glass 35 TABLE I I Comparison of Reflection and Transmission Coefficients with Theory 35 -1-INTRODUCTION For the case of a plane body wave incident on a plane interface between two e l a s t i c media, the c a l c u l a t i o n of the f r a c t i o n of energy that i s carried away i n each of the r e f l e c t e d and refracted waves has received extensive treatment over the past s i x t y years (1, 2, 3, 4)*. Results embodying mostly theory and computations may be found for numerous special cases (5, 6, 7). Suppose then, a longitudinal or transverse wave i s incident i n medium 1 against medium 2, the two media having d i f f e r e n t e l a s t i c properties. Continuity would make i t necessary f o r the normal and tangential stresses and displacements to be equal, respectively, on the two sides of the dis c o n t i n u i t y . For the general case, therefore, s i x boundary conditions must be s a t i s f -ied. Equations have been obtained describing these conditions i n terms of the amplitudes and angles of r e f l e c t i o n and r e f r a c t -ion of the various r e f l e c t e d and refracted waves produced at the discontinuity. The mathematical problem of the r e f l e c t i o n and transmission of surface waves i s , however, much less tractable than the analogous one for body waves. Imagine a corifiguration which, when referred to a Cartesian coordinate system, may be described as follows: / / 7 ' ' / / |\ \ ~ \ \ "x / / medium 1 / / \ \ medium 2 \ \ / y u , X , £ / / \ \ P " X ' 5 ' \ / / / / ///IX \ \ \ \ \ *The numbers i n parentheses indicate references given in.the bibliography. -2-The space z< 0 i s a vacuum; the space z> 0, x< 0 i s f i l l e d with a s o l i d medium 1 characterized by a density and Lame e l a s t i c parameters fx and \; the space z>0, x> 0 i s f i l l e d with a s o l i d medium 2' characterized by s i m i l a r constants 5 } yu' } and A . A harmonic Rayleigh wave propogating i n the p o s i t i v e x d i r e c t i o n i s incident upon the discontinuity. The problem i s to f i n d the amplitudes of the r e f l e c t e d and transmitted Rayleigh waves. The mathematical solution of the problem i s unknown at present, p r i n c i p a l l y because ,the boundary conditions, although simple, cannot be s a t i s f i e d by Rayleigh waves alone; there i s some conversion from surface waves to body waves and the ampli-tudes of the l a t t e r depend on t h e i r angles of emergence from the discontinuity. As a r e s u l t of the d i f f i c u l t i e s involved, very l i t t l e work has been done and only a few papers have been published on t h i s subject. De Bremaecker (8) studied experimentally the effect of a corner on.the r e f l e c t i o n and transmission of Ray-leig h waves. Later Lapwood (9) examined t h i s same problem from a t h e o r e t i c a l standpoint. I t appears that no one has published any work on the problem outlined above which i s of special interest to both pure and applied seismologists. The importance of t h i s problem i s concerned with the propogation of mantle Rayleigh'waves with periods of about 200 seconds. What has heretofore been interpreted as attenuation may be the r e s u l t of r e f l e c t i o n or transmission at the oceanic-continental bound-ar i e s . This p o s s i b i l i t y has only become plausible with the recent discovery that the upper mantle beneath continents d i f f e r s s i g n i f i c a n t l y from that beneath oceans. - 3 -It was thought desirable, therefore, to investigate experi-mentally the problem of Rayleigh waves incident upon a boundary between two elastic media and to attempt to predict the observed reflection and transmission coefficients by some suitable hypothesis. Also, i t was thought interesting to observe the variation of these coefficients for an appropriate "half space" as a function of the angle with which the plane interface meets the free surface. The object of the research reported in this thesis, then, was to examine these two problems using the methods of two-dimensional model seismology. c - 4 -CHAPTER I ELASTIC WAVE THEORY 1 .1 Equations of Motion We s h a l l now describe b r i e f l y some features of the trans-mission of a disturbance through a material substance which we s h a l l assume to be homogeneous, i s o t r o p i c , and per f e c t l y e l a s t i c . We take the undisturbed configuration as standard and s h a l l ignore the effects of possible flucuations i n the l o c a l external forces during the passage of the disturbance. In the circum-stances just described, the relevant equation of motion of the disturbance i s : (3) ?|D = ( \ ^ ) v ( V - b ) +y-V 2D ( i - i ) where D = (u,v,w) i s the vector displacement. I f we take the divergence of both sides, we obtain: ?|,(V-D) - ( V 2 / O V Y V - 7 ; ) ( 1 . a ) Also, i f we take the c u r l of both sides of ( l - l ) we have: .iL(\7*D) = K 7 2 (V*P) d-3) dt ' Equations ( 1 - 2 ) and ( 1 - 3 ) are both forms of the vector wave equation and may therefore be immediately interpreted. By ( 1 - 2 ) a d i l a t a t i o n a l (or i r r o t a t l o n a l ) disturbance, may be transmitted through the substance with v e l o c i t y =< , where r \ By ( 1 - 3 ) a r o t a t i o n a l (or equivoluminal) disturbance,^\7* D, may be transmitted with speed ^3 , where - 5 -(1 - 5 ) We notice o^^jS . In seismology the two types of waves are ca l l e d respectively the primary P and the secondary S waves. In the case we have taken, both v e l o c i t i e s depend only on the e l a s t i c parameters and the density of the material and therefore, the waves are nondispersive. For displacements i n a s o l i d body i t i s convenient to define a scalar p o t e n t i a l and a vector p o t e n t i a l "^0¥x, ,"Vj) as follows: (10) n -- ffl u. h9 (1-6) or i n vector form: In general, the equation of motion ( l - l ) represents, as we have seen, the propogation of a disturbance which involves both equivoluminal (6-0) and i r r o t a t i o n a l (X)=0) motion, where Q =V' 0 and JTL =-£-^y ^  • However, by introduction of the potentials *Y and separate wave equations are obtained f o r these two types of motion. We can write equation ( l - l ) i n the form: V ( V 2 ? ) + ^ 2 [ V * ( V 2 ^ ) ] ( i - 8 ) I t i s easy to see that t h i s equation w i l l be s a t i s f i e d i f -6-the functions ^ and '"V are solutions of the following equations: 2 C D J _ (1-9) 1.2 Boundary Conditions The stress-displacement relations for an isotropic medium may be written in the usual notation (10) as: I f the material to which the equations of motion are applied i s bounded, some special conditions must be added. At a free boundary of a s o l i d or l i q u i d , a l l stress components across the surface must vanish. In the problems concerned with i n the next chapter, i t w i l l be assumed that the s o l i d e l a s t i c media are welded together at the surface of contact, implying continuity of a l l stresses and displacement components across the boundary. I f these stress boundary conditions did not obtain, we would have i n f i n i t e accelerations at the free surfaces or interfaces, whichever the case may be. 1.3 Rayleigh Waves The f i r s t mathematical theory of e l a s t i c surface waves applicable to earthquakes was published by Lord Rayleigh i n 1885 (11) and further developed by Lamb i n 1904 (12). Consider a homogeneous, is o t r o p i c half space which has a (1-10) - 7 -plane free surface z - 0 and an interface x = 0 (see diagram page l ) . Assume a simple harmonic wave t r a i n t r a v e l l i n g i n the pos i t i v e x di r e c t i o n such that the disturbance i s ( i ) independent of the y coordinate and ( i i ) decreases rap i d l y with distance z from the free surface. Waves s a t i s f y i n g the second condition are ca l l e d surface waves. Since we are here concerned only with two dimensions (condition ( i ) ) , the displacements corresponding to the above waves may be written from equations ( 1 - 6 ) as: where subscript 2 has been dropped from As a solution of the wave equations ( 1 - 9 ) , Rayleigh chose the following set of potentials which express a simple harmonic wave propogating i n the posi t i v e x d i r e c t i o n : where f and g were undetermined functions of z and k^ =^/t , C) being the Rayleigh wave v e l o c i t y . I t can e a s i l y be shown that to s a t i s f y the boundary conditions at z = 0 , the exponent-' i a l terms i n ( 1 -12) must be i d e n t i c a l . Substituting ^ for example i n the wave equation ( 1 - 9 ) , we obtain: g -(V-fc«')f = 0 ( 1 . 1 3 ) where k^ = to/<*. . We also set k^ =• °->/fi . Furthermore i t i s convenient, at th i s point, to introduce r and s defined by: r 2 . k* - kS • s> • k,2 --8-The i n t e g r a l o f ( 1-13) i s s fe = A,-* + (1-15) From t h e s e r e s u l t s and c o n d i t i o n ( i i ) above t h a t r e q u i r e s t h e m o t i o n t o d e c r e a s e w i t h i n c r e a s i n g z, we c a n w r i t e t h e a p p r o p r i a t e s o l u t i o n s o f ( 1 - 9 ) f o r s u r f a c e waves a s : p r o v i d e d t h a t tY p ^ °^ , s o t h a t r a n d s a r e r e a l and n o n - n e g a t i v e . From t h e s e p o t e n t i a l s t h e d i s p l a c e m e n t s u and w may be d e r i v e d f r o m e q u a t i o n s ( l - l l ) . I n a s i m i l a r way t h e SH component ( m o t i o n p e r p e n d i c u l a r t o t h e x - z p l a n e i n t h e y d i r e c t i o n ) i s g i v e n by: The a r b i t r a r y c o n s t a n t s A, B, and C c a n be d e t e r m i n e d f r o m t h e b o u n d a r y c o n d i t i o n s . S i n c e we assume t h a t t h e p l a n e b o u n d a r y o f t h e h a l f s p a c e i s a f r e e s u r f a c e , t h e s t r e s s e s p £ p ? 1 , and p x i must v a n i s h a t z = 0 . I t f o l l o w s t h a t C = 0 ( i . e . t h e r e i s no m o t i o n i n t h e y d i r e c t i o n ) and ... F o r t h e r e t o e x i s t n o n z e r o v a l u e s f o r A and B, we must have t h e f o l l o w i n g r e l a t i o n w h i c h i s known "as R a y l e i g h ' s e q u a t i o n : ( 1 - 1 6 ) ( 1 - 1 8 ) -9-C 2 ^ 2 - ^ ) 2 " 4k*rs = 0 (1-19) I f we set y%= vp2an& (y©<a, then after some algebraic manipulation equation (1-19) becomes: y2[y6 -8y* +(Z4-\6f2)»2 - l6()-t2)] = 0 (1-20) The t r i v i a l root V-0 may be neglected and the factor i n brackets always has a root i f 0 ^  X ^°<. Therefore under these conditions surface waves can e x i s t . Note that V depends only on <f which i n turn depends only on the e l a s t i c parameters of the material. In other words, there i s no dispersion f o r the homogeneous half space. However, i t should be noted that the Rayleigh wave v e l o c i t y for a half space with a surface layer of a dif f e r e n t e l a s t i c material i s very much dependent on the frequency or wavelength. This effect i s observed l a t e r i n the experimental work. Because of the exponential decay with depth, the Rayleigh wave'motion becomes neg l i g i b l e at a distance of a few wavelengths from the free surface. I t can also be shown that the p a r t i c l e motion i s ' e l l i p t i c a l retrograde i n contrast to the c i r c u l a r d i rect o r b i t for surface waves on water. The v e r t i c a l displace-ment i s about one and a half times the horizontal displacement at the surface. 1.4 Plate Wave V e l o c i t i e s In two-dimensional model seismology, t h i n sheets of materials are used and therefore we wish to determine the v e l o c i t i e s of propogation of d i l a t a t i o n a l and shear waves i n -10-a t h i n plate and of Rayleigh waves on the edge of such a plate. We r e s t r i c t ourselves to waves i n which the p a r t i c l e motion i s symmetrical about the median plane of the plate, thus excluding any type of wave motion which involves bending of the plate. I t has been shown (13) that for t h i n sheets, the shear wave ve l o c i t y - i s the same as i n an i n f i n i t e s o l i d , namely ... ... while the d i l a t a t i o n a l wave v e l o c i t y must be replaced with a plate d i l a t a t i o n a l v e l o c i t y V., given by: It was also shown that the equation for Rayleigh waves on the surface of a s e m i - i n f i n i t e s o l i d (equation (1-20)) remains unchanged with the exception that the plate d i l a t a t i o n a l v e l o c i t y Vp replaces the i n f i n i t e s o l i d d i l a t a t i o n a l v e l o c i t y cx . Thus for wavelengths large compared to the thickness of the sheet, equations transform from half space to half sheet propogation simply by replacing <x. with Vp . Other than this change, Rayleigh wave propogation along the edge of the half sheet i s 'analogous to propogation along the surface of a half space. 1.5 Attenuation The damping of e l a s t i c waves results mainly from imper-fections i n e l a s t i c i t y and "i n t e r n a l f r i c t i o n " . The propogation of damped sinusoidal plane waves may be expressed by SL where the complex wave number i s given i n terms of an absorption -11-c o e f f i c i e n t g and a propogatlon v e l o c i t y : .. k* - " A - ' S - fe, ( 1_ 2 2 ) I t has been shown (14) that ... g ~ <v/Z*Qt (1-23) ... where 1/Qg i s the i n t e r n a l d i s s i p a t i v e parameter. I t was also demonstrated by Press and Healy (15) that f or a p a r t i c u l a r case - plexiglass - g i s not exactly proportional to the frequency. They found that ... <g =(.00/3Zf + .027)/tfl. (1-24) ... where f i s i n kil o c y c l e s / s e c . This empirical r e l a t i o n w i l l be used l a t e r f o r comparison with the experimental r e s u l t s . - 1 2 -CHAPTER I I THE REFLECTION AND TRANSMISSION OF RAYLEIGH WAVES 2 . 1 Previous Work Before proceeding to the proposed r e f l e c t i o n and trans-mission hypotheses, we w i l l consider b r i e f l y the recent work on t h i s subject of Rayleigh wave r e f l e c t i o n and transmission. (a) "Transmission and Reflection of Rayleigh Waves at Corners" - De Bremaecker Using a polystyrene sheet ( l / l 6 " thick with plate Poisson's r a t i o 0 . 1 7 ) , de Bremaecker investigated the phenomenon occur-ing when a Rayleigh wave i s incident upon a corner, the angle of which was varied between 0° and 1 8 0 ° . He found that part of the energy i n the form of a Rayleigh wave was transmitted around the corner and i t was the r a t i o of the amplitude of t h i s wave to that of the incident wave that he cal l e d the transmission c o e f f i c i e n t . This d e f i n i t i o n d i f f e r s from that used l a t e r , as w i l l be seen. He also found a r e f l e c t e d Rayleigh wave, the remainder of the energy (about 50$) going into P and S body waves. The surprising r e s u l t of de Bremaecker's work i s that for 0 less than 1 3 0 ° there are extremely strong and sharp variations i n the r e f l e c t i o n and transmission c o e f f i c i e n t s as shown i n f i g . 1. On the other hand, the f r a c t i o n of energy transformed into body waves (U 2) i s f a i r l y constant i n th i s region. Also there appears to be P i g . 1 Values of Coefficients of Transmission and Reflection and Energy Trans-formed into Body Waves as a Function of the Angle (after de Bremaecker) - 1 3 -nothing special about the r i g h t angle corner ( © = 9 0 ° ) . The uncertainties of de Bremaecker's measurements are generally 10$ and i n some of the worst cases are as high as 2 0 $ . This serious drawback to the experimental technique was also encountered i n the work described i n Chapter IV. (b) "The Transmission of a Rayleigh Pulse Round a Corner" - E. R. Lapwood Since t h i s topic has very l i t t l e i n common with the research outlined i n t h i s t h e s i s , i t i s mentioned only b r i e f l y and that mainly for the sake of completeness as the number of papers on the general subject of the r e f l e c t i o n and transmission of Rayleigh waves i s extremely small. The problem i s taken as two-dimensional and the incident Rayleigh pulse has the form found by Lamb i n his c l a s s i c a l paper of 1904 ( 1 2 ) . The d i f f e r e n t i a l equations and boundary conditions are reduced to operational form by use of a Fourier' transform with respect to time and Laplace transforms with respect to the two coordinates of the quarter space - x and z. The solution of the problem i s reduced to that of two simultan-eous i n t e g r a l equations. An i t e r a t i v e s o l u t i o n was found and from t h i s i t was possible to i s o l a t e the parts which describe the incident Rayleigh pulse and the pulse transmitted around the corner. It was found that the form of the transmitted pulse was greatly changed i n ways depending on the values of the v e l o c i t -ies of P, S, and Rayleigh waves. This i s substantiated by de Bremaecker's work which c l e a r l y showed that a Rayleigh pulse -14-i s transmitted around a corner and does change shape, 2.2 Quarter Space Hypothesis As explained earlier, the mathematical problem of the r e f l e c t i o n and transmission of Rayleigh waves i s much less tractable than the analogous one for body waves. With the aid of the diagram, consider a Rayleigh wave free boundary x= 0 . Using equation (1-16), we can express the incident and r e f l e c t e d Rayleigh waves as follows: where the subscript 1 refers to the incident wave and subscript 2 to the re f l e c t e d wave. I t should be noted that B ; i s deter-mined by A ( and B£ by A^ from equation (1-18); that i s , there are only two independent constants i n equation ( 2 - l ) . Now the boundary conditions require that the normal and tangential stresses, p x x and p x ^ respectively, should both vanish on the free surface x = 0. But no choice of A 2 (or B 2) w i l l s a t i s f y t h i s condition. Therefore, i t appears reasonable to propose the following working hypothesis: Hypothesis 1: The amplitude of the Rayleigh wave re f l e c t e d at a quarter space may be approximated by choosing that ampli-tude so that the stress imposed at the r e f l e c t i n g surface (x = 0) X t r a v e l l i n g i n the negative x d i r e c t i o n incident upon the plane (2-1) -15-by the i n c i d e n t and r e f l e c t e d R a y l e i g h waves w i l l be a minimum i n the l e a s t square sense. Note t h a t the body waves produced by such a r e f l e c t i o n are n e g l e c t e d . Thus the r e f l e c t i o n c oef-f i c i e n t R = A 2/A, w i l l be chosen so t h a t ... (2-2) ... w i l l be a minimum at x = 0. S u b s t i t u t i n g e q u a t i o n (2-1) i n t o e q u a t i o n (1-10) and s e t -t i n g x =0, we o b t a i n e x p r e s s i o n s f o r the two r e l e v a n t s t r e s s e s . /v* = ^ V ^ - A U " r * - ^ ^ k ; Y e ^ s a V " " | ( 2 ~ 3 ) Using the r e l a t i o n between the A's and the B's g i v e n by eq u a t i o n (1-18), the above two r e l a t i o n s reduce t o : (2-4) where i t i s convenient to i n t r o d u c e a. and b g i v e n by: E v a l u a t i n g the i n t e g r a l (2-2), which we wish t o minimize, we have . . . = A; I 2. (2-5) - 1 6 -... where M and N are f u n c t i o n s of the frequency and the e l a s t i c constants of the q u a r t e r space m a t e r i a l . Por a minimum, ^ 1 ^ 0 . S o l v i n g t h i s simple r e l a t i o n f o r R, we have ... A / - A 7 ( 2 - 6 ) A/ + A ? C l e a r l y R i s not a f u n c t i o n of the frequency to, s i n c e the l a t t e r w i l l c a n c e l out i n the r a t i o N-M/N + M. This, of course, should be expected as there i s no c h a r a c t e r i s t i c l e n g t h i n the geometry of the problem t o favour c e r t a i n wavelengths. 2 . 3 H a l f Space Hypothesis The h a l f space c o n f i g u r a t i o n i s again shown i n the sk e t c h below. Consider a R a y l e i g h wave pro p o g a t i n g i n the n e g a t i v e \\ \ \ w / / / t f \ medium 2 \ \ / / / medium 1 / / / / ? > A V / \ \ \ \ \ N / / / / -S2 <.(ust + Lx) ( 2 - 1 ) x d i r e c t i o n , i n c i d e n t upon the plane i n t e r f a c e x = 0 . Assuming r e f l e c t e d and t r a n s m i t t e d waves, the p o t e n t i a l s c h a r a c t e r i z i n g the R a y l e i g h wave motion i n medium 1 are i d e n t i c a l to those g i v e n i n s e c t i o n 2 . 2 , namely ... 9 = A , £ r V ^ t + M ) + A, k>x) V - B, In medium 2 c o n t a i n i n g the t r a n s m i t t e d wave (primed par-ameters), the corresponding p o t e n t i a l s are as f o l l o w s : >i) •) ( ( 2 - 7 ) - r ' E i JL -17-For the two s o l i d e l a s t i c media i n contact, the boundary-conditions require continuity f o r both the stresses and the displacements. In other words, the differences between the respective stresses and displacements on either side of the interface (also c a l l e d residuals) must vanish. However, no choice of A 2 and Ai w i l l make these residuals i d e n t i c a l l y zero; but i t is possible to minimize them. Hypothesis 2: The r e f l e c t i o n and transmission c o e f f i c i e n t s of Rayleigh waves may be determined by minimizing i n a least square sense, the residual stresses and displacements imposed at the boundary x= 0 by the incident, r e f l e c t e d and transmitted Rayleigh waves. That i s , the r e f l e c t i o n c o e f f i c i e n t R- A 2/A ( and the transmission c o e f f i c i e n t T = A3/A| w i l l be chosen so that the following i n t e g r a l i s a minimum at x=0: r 2 Jo where I, and I 2 remove the units from the expression, so that both stresses and displacements may be minimized together. They are simply the corresponding integrals f o r the incident wave alone and are given by ... . (2-8) (2-9) and o o I = [ I " . I * + l^o\2}dz (2-10) .. . where p x x o , p K*o > u Q and w0 are the stresses and displace-ments produced by the incident wave ... -18-Substituting the potentials (2-1) , (2-7) and (2-11) into equations (1-10) and ( l - l l ) , we obtain the following expressions: and /^xx = j=>x\ UL w - - r ( A , • A . Y * ' ' * w 1 a* v V 0 . - r A , U ' r 4 -After substituting these 12 equations into I 0 and evaluating -19-t h e I n t e g r a l , w e a r r i v e a t t h e f o l l o w i n g s i m p l i f i e d e x p r e s s i o n , w h e r e t h e c o e f f i c i e n t s A t o L a r e f u n c t i o n s o n l y o f t h e e l a s t i c c o n s t a n t s o f t h e t w o m e d i a ; i . e . y U . , A , ^ , JlA.' > \ > a n d ^ ' % + (C + £ *H * K ) T l + ( D + D T ( i + R ) + ( F - I ) T ( l - R ) ( 2 - 1 2 ) S e t t i n g £ » i i . = _L_ = VJMZ-Z-w e c a n w r i t e t h e 12 c o e f f i c i e n t s a s f o l l o w s ; A = / / L2 2q-b C L 2 Z r 2 CL n - -2 / A' v tt' fl-'i oJ>' 4(1-- 8 ( 1 s* + s' r ' + S * r * + s ' r*+r' J 4--20-1 - / 4 s r s V _ 2sr _ 2 s ' r ' + i / _ 0-<f'VO / L _ _ ± + _L i _ - 2 o - < ' v y v _ ± = = +  where c<0 and ^S0 are dimensionless quantities given by .« For a minimum, •=. 0. Solving these two equations for R and T, we have ... V - Y \ (2-13) T = X + Y R - 2 1 -. where X Y u V - ( D +L * F *1) 2(C *E *K) ( - 0 - L • • / ) 2{C * E * * ) •B- G ) ( F +1 - D ~ •L) z(A - J •< -B *• 6 ) (F + I -D-L) Note that R and T are not functions of the frequency Co . This fact w i l l permit the use of pulses, rather than harmonic wave t r a i n s , to prove or disprove experimentally the v a l i d i t y of equations ( 2 - 6 ) and ( 2 - 1 3 ) . Since any pulse may be resolved into Fourier components, each of which w i l l have r e f l e c t i o n and transmission c o e f f i c i e n t s independent of the frequency, the sum or integration of these components should also have R and T c o e f f i c i e n t s independent of co . - 2 2 -CHAPTER III APPARATUS 3• 1 Introduction Geophysicists have come to realize the meagerness of their knowledge of the various factors affecting the generation and transmission of seismic waves. Experiments dealing directly with the transmission of these waves within the earth are d i f f i c u l t to carry out because of unknown conditions. Conse-quently, a number of investigators, in recent years, have attacked problems of elastic wave propogation by studying waves' of ultrasonic frequencies travelling through small scale, two and three-dimensional models and many papers have been published on this subject ( 1 3 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 ) . The short wavelengths required in a seismic model to give wave-front patterns geomet-r i c a l l y similar to those in the earth necessitate the use of the high frequencies. As sources and detectors of such high frequency waves, piezoelectric crystals or transducers are used, primarily because under identical stimuli, they are capable of almost perfect duplication. Such duplication is made use of in " displaying on an oscilloscope stationary patterns which are characteristic of the transient particle motion at a point in the model.' These patterns are usually photographed, the result-ing picture being a miniature seismogram which can be studied at leisure. It appears that two-dimensional models in the form of thin sheets are the most convenient, mainly because of low cost and simplicity in obtaining desirable configurations. In -23-practice, these sheets are generally l / l 6 " thick and only wavelengths long compared to this thickness are employed. This is because dispersion becomes important for shorter wave-lengths . 3 . 2 Transducers and Holders Both the transmitting and receiving transducers used in this work were Barium Titanate cylinders - 1/10" thick and 1/4" in diameter. These are sensitive mainly to normal motion and in our case, motion normal to the edge of a plate. A change in the thickness of the crystal produces a charge on its two faces, the resulting voltage being amplified many thousands of times and displayed on the oscilloscope. Conversely a volt-age applied across the transducer causes a sudden change in dimension, a pulse being transmitted to any medium in contact. Both faces of the cylinders were metal plated to which leads were soldered. The small drops of solder had f l a t faces about a mm. square and i t was these that made contact with the model. The design of a suitable holder or support within which the crystal transducer w i l l perform i t s function is extremely important ( 2 1 ). In choosing a material on which to mount the crystal, i t must be remembered that energy may be transmitted in both directions perpendicular to the crystal faces and therefore energy may be radiated into the backing. This energy i s , of course, wasted and may cause reflections from the end of the holder which confuse the seismogram. This was avoided by backing the transducer with substances of very small ultra-sonic transmission - air and Bakelite. The Barium Titanate to follow page—2gn Barium Titanate Solder P i g . 2(P) H v U n r l r i ^ " ' 1 Tranflrtune-p Bakelite A i r Backing Transducer P i g . 2(b) Transducer Holder -24-cylinders were mounted in Bakelite "buttons" as shown in f i g . 2(b), and these were cemented to hollow Bakelite rods. The crystal f i t t e d snugly into the button and i t was not necessary to use cement, the pressure of the front face of the piezo-electric element against the model being sufficient damping to cause free vibrations to die out quickly. This is a very simple support but i t was found by t r i a l to be satisfactory and was used in a l l subsequent experiments. 3 . 3 Experimental Arrangement The three elastic media utili z e d in this research were thin sheets (l/l6") of (i) an aluminium alloy designated as 2024 - T 3 ; ( i i ) clear plexiglass; and ( i i i ) ordinary plate glass. Their maximum dimensions never exceeded three feet. The models were supported horizontally on a thick foam rubber sheet, the air spaces being sufficiently large so that the surface wave propogation was not affected appreciably. Both the transducer holders were arranged so that the front face of each crystal came in contact with the edge of the plate model. The Bakelite rods were held in clamps which could be moved parallel to the plate edge along a guide bar graduated in centi meters. It was possible to measure relative distances to ± 1 mm A block diagram of the apparatus is shown in f i g . 3 . E l e c t r i c a l pulses of approximately 15 microsecond duration are applied to the transmitting transducer in contact with the model. The resulting acoustical energy arriving at various points along the edge of the plate model is detected by an identical transducer, amplified and then displayed on a cathode display signal t r i g g e r input input pre-amplifier pass band f i t e r pre-amplifier audio i s c i l l a t o thyratron pulser receiver pulse shaper transmitter F i g , 3 Block Diagram of Apparatus - 2 5 -ray oscilloscope. The r e s u l t i n g pattern i s photographed with a Polaroid Land camera, and the miniature seismogram i s obtained immediately. An audio o s c i l l a t o r triggers the thyratron pulser 20 times a second. This low pulse r e p e t i t i o n rate insures that a l l e l a s t i c reverbations i n the model due to one pulse have ceased before the onset of the next pulse. A portion of the o s c i l l a t -or output also triggers the oscilloscope sweep, thus making i t : possible for a standing wave pattern to be observed on the scope screen. The trigger l e v e l of the oscilloscope can be adjusted' so that there i s no delay between the t r i g g e r i n g of the sweep and the application of the pulse to the transmitting transducer. This pulse i s merely the result of the d i f f e r e n t i a t i o n and attenuation of the output pulse of the thyratron which i s about Ov ^ 3000 v o l t s . The r e s u l t i n g pulse of about 700 v o l t s i s shown i n the small diagram to the l e f t . With the oscilloscope employed i n t h i s work, the time base was known accurately enough to forego the use of time pips. The t r a v e l times were measured d i r e c t l y from the seismogram with a r u l e r , Por the time base most often used, about 400 microsec-onds of sweep were shown at one time,, A variable pass band f i l t e r was placed i n the receiving c i r c u i t . The reason for t h i s i s that the attenuation varies with the frequency (see section 1 . 5 ) , so that a narrow frequency band i s desirable i f one i s interested i n determining the atten-uation. Also the f i l t e r helps remove unwanted low frequencies - 2 6 -such as the 6 0 cycle pickup from the lines. The voltage amplification of the receiving unit was usually 1 0 0 , 0 0 0 but at times only 1 0 , 0 0 0 was sufficient,, Since the preamplifiers which were available have maximum gains of only 1 0 0 0 , i t was necessary to connect two in series; but i t was found that they could not be connected directly. However, the pass band f i l t e r inserted between the amplifiers worked satis-factorily. This was probably because the output impedance of the type of amplifier used is too high, while that of the f i l t e r is relatively low, the output being across a cathode follower. To decrease pickup from the pulser, a l l lines were shielded and i f the aluminium alloy was being used as part of a model, i t was grounded. If the aluminium was in contact with the trans-mitting crystal i t was automatically grounded since the front face of the transducer was always kept at ground potential. On the other hand, i f the aluminium was in contact with the re-ceiving transducer, then a problem arose. The amplifiers are of the "push-pull" type and therefore neither of the two Input leads can be connected to ground. It was found after t r i a l and error, that a small piece of "Saranwrap" or cellophane inserted between the receiver and the aluminium plate would insulate the front face of the crystal from ground potential and yet be thin enough" to transmit the seismic pulse unchanged. Although i t was not needed for insulation, the cellophane was also used with the plexiglass when the latter was glued to the aluminium to form a "half space". This effort to keep the conditions as constant as possible for the two media was important, since the amplitudes of the Rayleigh waves on both sides of the interface -27-between the two elastic materials were to be compared to arrive at a transmission coefficient. -28-CHAPTER IV MEASUREMENT AND RESULTS 4.1 Wave V e l o c i t i e s Before proceeding to the r e f l e c t i o n and t r a n s m i s s i o n c o e f -f i c i e n t s , the d i l a t a t i o n a l , shear and R a y l e i g h wave v e l o c i t i e s f o r the e l a s t i c m a t e r i a l s employed i n t h i s experiment must he determined. This w i l l make i t p o s s i b l e to compare equations (2-6) and (2-13) w i t h experiment. To c a l c u l a t e wave v e l o c i t i e s , one g e n e r a l l y makes a p l o t of t r a v e l times versus d i s t a n c e t r a v e l l e d by the wave and f o r t h i s i t i s convenient t o make as many r e c o r d i n g s as p o s s i b l e of d i f f e r e n t t r a v e l times on one seismogram. The apparatus d e s c r i b e d i n the f o r e g o i n g chapter p r o v i d e s f o r r e c o r d i n g the output of only one r e c e i v i n g t r a n s d u c e r at a time. M u l t i p l e t r a c e r e c o r d s can be obtai n e d , however, by moving the P o l a r o i d Land camera t o a number of s u c c e s s i v e p o s i t i o n s . Each exposure i s made f o r one p o s i t i o n of the s i n g l e ; d e t e c t o r which can be moved between exposures to d i f f e r -ent p o s i t i o n s along the edge of the model as r e q u i r e d . In t h i s way, i t was p o s s i b l e by u t i l i z i n g the v e r t i c a l sweep c o n t r o l on the o s c i l l o s c o p e to take up to 32 exposures on one r e c o r d , thereby economizing on f i l m . P i g . 4 i s an example of a m u l t i p l e t r a c e t r a n s m i s s i o n r e c o r d of " p l e x i g l a s s to aluminium"f Both the d e t e c t o r and * From' now on t h i s terminology and i t s a b b r e v i a t e d form "PI to A l " w i l l be used to i n d i c a t e t h a t the wave source i s i n con t a c t w i t h the p l e x i g l a s s and t h a t the R a y l e i g h wave i s t h e r e f o r e i n c i d e n t from the p l e x i g l a s s upon the i n t e r f a c e between the two e l a s t i c media. to follow page 28 - 2 9 -transmitter were on the same edge of the plate model. The horizontal distance of the detector was successively increased i n steps of one centimetre between exposures. The f i r s t two lines of peaks correspond to the a r r i v a l of the di r e c t Rayleigh wave, the detector being on the same side of the interface as the wave source. Their slope.;gives the v e l o c i t y of Rayleigh waves'in p l e x i g l a s s . Note that although the amplitude of the Rayleigh pulse decreases s t e a d i l y due to the high attenuation of p l e x i g l a s s , the form and duration are e s s e n t i a l l y preserved. The l a s t l i n e of peaks i n f i g . 4 corresponds to the a r r i v a l of the transmitted Rayleigh wave, the detector now being on the opposite side of the plane in t e r f a c e . I t s slope gives the v e l o c i t y of the surface waves i n aluminium. The shape of the pulse has been s l i g h t l y altered even though the transmission c o e f f i c i e n t must be independent of frequency because there i s no reason for one frequency to be favoured. The mechanism which deforms the pulse could be the following. There i s a conversion of Rayleigh waves to P and S waves along the i n t e r -face x = 0 . Some of these P and S waves s t r i k e the surface z = 0. Being spherical waves, they are d i f f r a c t e d at t h i s plane bound-ary, creating Rayleigh waves. These secondary Rayleigh waves inte r f e r e with the d i r e c t l y transmitted wave to y i e l d the s l i g h t l y distorted transmitted Rayleigh wave. This process, however, i s not dependent upon the frequency. It i s quite evident that the record beyond the Rayleigh phase i s complicated by r e f l e c t i o n s from various boundaries. The primary portion of each trace i s also disturbed s l i g h t l y by noise from the electronics of the apparatus. Because of - 3 0 -t h i s and high attenuation, the P and S phases observed on the edge of the plate model are so poorly marked that they are indiscernible on t h i s type of record. To determine the d i l a t a t i o n a l and shear v e l o c i t i e s , i t was necessary to place the receiving transducer on the opposite edge of the sheet model where the P and S body phases could be detected. On some records i t was possible to observe the P and S waves refl e c t e d from the opposite edge of the plate with the receiver on the same side as the wave source. In t h i s case i t was usually the S-P r e f l e c t i o n which was detected more strongly than the S-S, since the Barium Titanate cylinders are '^sensitive mainly to normal motion. The S-P phase i s caused by a pulse making one traverse of the plate i n the form of a shear wave and then being transformed on r e f l e c t i o n into a longitud-i n a l wave. The calculations of the three wave v e l o c i t i e s f o r alumin-ium, plexiglass and glass are summarized i n Table I following page 3 5 . As an example, the travel-time plots for plexiglass are shown i n f i g . 5- In brackets are given the v e l o c i t i e s as determined by two other investigators. The l a t t e r values seem to be consistently lower than those calculated from the slopes of the graphs shown here. Perhaps the plexiglass used i n each case was of a s l i g h t l y d i f f e r e n t composition. The r e l a t i v e l y small experimental errors would c e r t a i n l y indicate t h i s . At t h i s point, a puzzling r e s u l t should be mentioned. Once the d i l a t a t i o n a l and shear wave v e l o c i t i e s are found, i t i s possible to calculate ^ , the Rayleigh wave v e l o c i t y from equation ( 1 - 2 0 ) and i t should be expected that t h i s would agree to follow page 30 -31-with the measured value. But for the aluminium and plate glass t h i s was found to be not true. Aluminium, for example, has P and S v e l o c i t i e s such that 6l = f 3/* 2- 0 „33. This was calculated from a plot s i m i l a r to f i g . 5. Substituting t h i s value of 6Z into equation (1-20) and selecting the only accept-able solution, we have y C t A ^ ^ ~ 0 .85 . The measured value i s the same as that f o r glass, namely V* enured = 0.91. An estimate of the errors involved w i l l not allow t h i s measured value to be compared favourably with the calculated one. No explanation i s offerred for this disagreement„ With the aluminium a l l o y used by Oliver et a l (13), the same discrepancy e x i s t s . 4.2 Reflection and Transmission Coefficients (a) Quarter Space; Consider the arrangement of apparatus shown i n the sketch below, where T i s the transmitter and R the receiver, x i s the distance of the receiver from the y y * / / > / / / / corner. Let T be at x = J L Then i f the attenuation i s g and i f A 0 i s the amplitude of the wave at x = JL , the amplitudes of the direct and re f l e c t e d waves as measured at x w i l l be A and Ay. respectively where ... (from equation (1-22), section 1,5) (4-1) R i s the r e f l e c t i o n c o e f f i c i e n t and B and B s are factors which take into account the d i r e c t i v i t y of the receivers. For some ' A = A. B J L A, - A„RBV-^ X ) -32-reason, these two factors d i f f e r for the two directions pos-s i b l e along the edge of the plate. B characterizes the res-ponse of the receiver to a pulse t r a v e l l i n g from l e f t to r i g h t ; B', the response to a pulse from r i g h t to l e f t . If we l e t 1*^ = Ar/A be the apparent r e f l e c t i o n c o e f f i c i e n t , then we have ... = R 2 l £ 2 * * (4-2) a or - R*. = - U - ( n ^ +- Iqx (4-3) B Hence a plot of -lnR^ vs. x should be a straight l i n e with slope 2g and intercept -InR-lnB'/B. I f the receiver i s rotated 180° about i t s own a x i s , then c l e a r l y B and B* should be i n t e r -changed and the same plot would have an intercept of -lnR+lnB'/B° A method e x i s t s , therefore, f or determining the magnitude of B'/B. This was done experimentally and i t was found that f o r a p a r t i c u l a r orientation B ?/ B = 1.1. I t would be very impract-i c a l to have' to reverse the receiver for every measurement, considering the number of records that had to be taken. I t was found that R could be estimated with s u f f i c i e n t accuracy by keeping the receiver f i x e d at the same orientation for a l l meas-urements and assuming that B'/B remained constant. I t i s r e a l -ized that t h i s i s not s t r i c t l y correct, since these d i r e c t i o n a l factors "depend not only on the orientation of the receiver about i t s own a x i s , but also on the angle which t h i s axis makes with the edge of the model. Por each recording, t h i s angle could be s l i g h t l y d i f f e r e n t . But because of the low accuracy of th i s type of measurement, t h i s method was accepted as s a t i s f a c t o r y . The accuracy of experiments of t h i s type i s severely l i m i t e d -33-because of wide scatter i n measurements and the interference of the r e f l e c t e d Rayleigh pulse with the wave t r a i n following the main incident pulse. Por some positions of the detector the peak-to-peak amplitude of the r e f l e c t e d wave simply could not be measured. The r e f l e c t i o n measurements for the three quarter spaces of aluminium, plexiglass and glass are shown i n f i g . 6 and the resul t s are summarized i n Table I I following page 3 5 . The attenuation c o e f f i c i e n t g was so small f o r aluminium and glass that i t could not be determined accurately; however, i t i s estimated to be less than 0.003/cm. for both cases. These re-s u l t s are for 1 0 0 k c / s e c , the frequency most often used i n the work described i n t h i s report. Por plexiglass the attenuation i s r e l a t i v e l y large and was calculated as 0.06/cm. The value found by Press (15) can be calculated from the empirical r e l a t -ion (1-24) and for lOOkc, the r e s u l t i s 0.063/cm., which i s i n very good agreement. ( b) Half Spaces The half space of aluminium and p l e x i -glass was chosen so as to put the minimizing hypothesis describ-ed i n Chapter I I to a severe t e s t . I f the e l a s t i c parameters of the two media were approximately equal, then i t would be expected that the hypothesis would predict the r e f l e c t i o n and transmission c o e f f i c i e n t s f a i r l y accurately, since f o r i d e n t i c a l e l a s t i c constants, c o e f f i c i e n t s of R = 0 and T = l would be given by equation (2 - 1 3 ) . The e l a s t i c constants of aluminium and p l e x i g l a s s , however, d i f f e r markedly. The r a t i o of the wave v e l o c i t i e s and densities i s approximately 2^, about the largest possible for materials r e a d i l y a v a i l a b l e . 2.00 -InR* 1.00-0.00 " Aluminium R- 0.32 n n " " IL a o o o Q - - 0 * 6 ^ -1.1 InR t , cm. 0 10 20 30 2.00 •lnR^ 1.00 0.00 Plexiglass 0 10 R = 0.38 -,- • cm. 15 2.00 Glass R'0.25 ° o • 1 C i ° lnR A 1.00 U U O o t 1 cm. a 10 20 30 F i g . 6 Reflection Data for Three Quarter Spaces n CD LA) J O - 3 4 -Por t h i s t e s t , the two sheets of aluminium and plexiglass were glued together so that the interface was at right angles to the edge along which the amplitude measurements were to be made. The "half space" was about 3 feet square, A f a i r l y strong bond between the two media was obtained with a r e s i n (fibreglass) glue - "Esterex 101". The aluminium and plexiglass did not come i n contact anywhere along the j o i n t , the separation being of the order of 0 .01 cm. The transmission of Rayleigh waves across t h i s t h i n layer of glue between two plates of aluminium was measured and found to be e s s e n t i a l l y perfect; i . e . R •= 0 and T - 1. On measuring the transmission c o e f f i c i e n t , B and B* may be forgotten, since the incident and transmitted waves are incident on the receiver R from the same d i r e c t i o n (see diagram). V \~i V V / T / / X \ \ \R \ \ \ / medium 1 > \ medium 2 \ \ / / / A l l that Is required i n t h i s case i s to plot as functions of x, the negative logarithm of the amplitude A of the incident pulse, measured on one side of the boundary and that of the amplitude At of the transmitted wave, measured on the other side. These plots give two straight lines as shown i n f i g . 7 which do not meet at the boundary. The negative logarithm of the transmission c o e f f i c i e n t T i s simply the difference i n the intercepts at the boundary. The r e f l e c t i o n results are also shown i n f i g . 7> the pos i t i v e intercept of the r e f l e c t i o n graph (b) being equal to -1.1 InR. The factor 1.1 i s due to the d i r e c t i o n a l terms B and B'„ Another method of determining R Is to simply p l o t -InA as 2.00 ' (a) A l to PI -lnA t 1.00 -lnA r O 0.00 p, C> ^ ii *\ n n ° ft. - Q -1.00 ° O 4 0 O O » -' « ~ ( J « - y •» -InA boundary R-0.25 T-0.58 t i i t . cm. 20 40 60 80 2,00 - (b) PI to A l R--0.46 T- 0.76 1,00 -n - n - a 0 0 0 o • A Q ° ° 0 O O 0 0.00 — -inA t 1.00 o i i i i boundary > cm. 20 40 6o 80 Pig . 7 Reflection and Transmission Data for PI - A l 90° Half Space -35-a function of x (a) and compare the intercept with that of the -InA p l o t . It should be mentioned that for the half space there e x i s t two cases. The wave may be incident on the interface from the plexiglass side (PI to Al) or from the aluminium side (Al to P i ) . The r e f l e c t i o n and transmission c o e f f i c i e n t s are quite d i f f e r e n t i n each of these cases „ 4.3 Comparison of Results with Theory The res u l t s given i n f i g . ' s 6 and 7 may now be compared with the co e f f i c i e n t s computed from the two formulae derived i n Chapter I I . Substituting the measured wave v e l o c i t i e s into equation (2-6), the r e f l e c t i o n c o e f f i c i e n t s for the three quart-er spaces consisting of aluminium, plexiglass and glass were calculated. Equation (2-13) requires the densities of alumin-ium and plexiglass and so these were determined and the r e f l e c t -ion and transmission c o e f f i c i e n t s f o r the 'Al to PI 1 1 and 'PI to A l ' half spaces computed, remembering that the primed variables characterize the medium containing the transmitted wave. Since a d i g i t a l computer was used to evaluate the 12 coe f f i c i e n t s A to L required by equation (2-13), i t was possible to vary the wave v e l o c i t i e s s l i g h t l y to obtain an idea of the errors involved. These re s u l t s are summarized i n tabular form i n Table I I and are presented along with the estimated errors which for the experimentally measured c o e f f i c i e n t s are about 10$, The subscripts m and c are used to indicate 'measured8 and 'calculated 8 values respectively. There seems to be reasonable agreement for the r e f l e c t i o n to follow page 35 Aluminium Plexiglass Glass 5 6 0 0 m/sec 2 4 0 0 m/sec 5620 m/sec is 3200 m/sec 1 4 0 0 m/sec 3500 m/sec 3050 m/sec 1260 m/sec 3 3 4 0 m/sec 0.955 0.900 0.955 ( W o 0.920 0.918 0.909 0.572 0.583 0.623 Poisson's Ratio 0.258 0 . 2 4 2 O.185 g < 0 .002/cm 0 . 0 6/cm < 0.003/cm 2.77 gm/cc 1.22 gm/cc -Table I Measured Properties of A l , PI and Gl R c T™ xm A l 0.32 t . 0 4 0.31± .06 - -PI O.38 i . 0 4 0.08 t . 0 4 - -Gl 0 . 2 5 t .03 0 . 3 1 i o 0 5 - -A l to PI 0.25 ± o 0 3 0 . 0 1 ± . 0 1 0 . 5 8 ± . 0 6 0.88 ± .03 PI to A l - 0 . 4 6 ± . 0 3 - 0 . 3 7 ± . 0 1 0.76± . 0 4 0 . 1 7 t . 0 1 Table I I Comparison of Reflection and Transmission Coefficients with Theory -36-coefficients for aluminium and glass. Also a change in phase is predicted for the reflected wave from the PI to Al bound-ary and this is indeed observed. The measured and calculated values for this last case are f a i r l y close, but the ranges of errors certainly have no point in common. Aside from these slight agreements there is a marked difference between theory and experiment and i t must be concluded that the minimizing hypotheses are not sufficient to explain the reflection and transmission of Rayleigh waves. As a further test, an attempt was made to minimize the stresses on the edge of the corner used in de Bremaecker's work as described in section 2.1 (see f i g . l ) . Again very poor agreement with experiment was found. Since the working hypotheses were a failure, i t was de-cided to determine empirically the variation of the reflection and transmission coefficients as a function of the interface angle - that i s , the angle with which the plane boundary bet-ween two elastic materials meets the free surface. This^work is outlined in the next section. 4.4 Variation of Coefficients with Interface Angle Plexiglass and aluminium were again chosen for two reasons, A strong variation in coefficients should be observed for two materials having considerably different elastic properties, i f such a variation exists. The second reason was simply a matter of f a c i l i t y , as the 'half space' model described in the last section was available. From this 3 foot square sheet of aluminium and plexiglass, - 3 7 -a c i r c l e of about 90 centimetres i n diameter was cut, one diameter being along the interface between the two materials. This c i r c l e was then sawed i n half with the cut at 10° to the interface. Thus two semicircles were obtained, one with a 10° wedge of aluminium glued to a 170° wedge of plexiglass and s i m i l a r l y with the other, the angles being interchanged. These two pieces represented four possible h a l f space models, designated as PI to A l 10°, PI to A l 170°, A l to PI 10°, and A l to PI 170° where the angle refers to the f i r s t w ritten medium. An example i s shown i n the sketch below where T i s the A l to PI 170 wave source as before. After r e f l e c t i o n and transmission data had been obtained for each of the models, the two semicircles were glued back together forming the o r i g i n a l c i r c l e which was again cut, t h i s time at 20°to the int e r f a c e . Thus four new configurations were provided and four r e f l e c t i o n and four transmission c o e f f i c i e n t s were measured with the seismic model-ing apparatus. This procedure was repeated i n steps of 10° up to 90° . The advantage of t h i s method i s that two angles are obtained at once; e.g. PI to A l 20° and PI to A l l60° . Also any problem of matching the edges f o r gluing i s removed, since they both belong to the same cut. Furthermore, the amount of aluminium and plexiglass consumed i s kept at a minimum, since the same pieces are used again and again. . -38-F i g . 8 i s an example i n graphical form of the transmission data for the 120°- 60° interface angle. One seismogram s i m i l a r to f i g . 2 was taken for each model. Of the 32 or so traces a v a i l a b l e , 18 are usually used for the incident and r e f l e c t e d waves, the detector being moved i n steps of 1 cm. between exposures. The remainder of the traces are used for measure-ments of the amplitude of the transmitted wave with the detector being advanced i n steps of 2 cm. Not a l l amplitude data i s shown i n f i g . 8 because of interference and obvious scatter. As the attenuation i s already known,only a few correct measure-ments are necessary. Dispersion of the Rayleigh waves was encountered for the small angled models such as A l to PI 10°. The sheet consisting of a t h i n wedge of aluminium on plexiglass approximates a lay-ered medium for which dispersion i s important as mentioned i n section 1.3. As the Rayleigh wave progresses farther and farther along the edge of the aluminium towards the apex of the wedge, the dispersion becomes more and more pronounced. When the perpendicular distance of the interface from the free edge i s more than one wavelength*, the dispersion becomes ne g l i g i b l e since at that depth the p a r t i c l e motion i s very small r e l a t i v e to that at the surface. Therefore a l l amplitude measurements were taken f a r enough from the plane interface so that disper-sion was not observed on the seismograms. Since the dispersion was rather minor i n the cases considered, there was never any problem i n deciding what the amplitude of the transmitted wave * The wavelengths i n aluminium and plexiglass at lOOkc are about 3 cm, and 1 cm. respectively. t o f o l l o w page 38 T- 0.56 2 . 0 0 h 0-00 6 ^ ~ - o - ^ o - l n A ^ A l t o P I 60° ^ ^ " ^ ^ ^ T- 0.57 > -InA » B „ " " u 0 • O-0  cm, 20 40 60 80 F i g . 8 T r a n s m i s s i o n Data f o r P I - A l (120°- 60°) H a l f Space -39-was. In f a c t , the transmitted wave tended to regain i t s o r i g i n a l pulse shape even though the incident wave would be dispersed when propogating along the edge of a small angled section of a model. Prom a l l the diagrams s i m i l a r to f i g . 8, i t i s possible to plot the co e f f i c i e n t s R and T as a function of the angle 0 o © where PI to A l <9 and A l to PI © have already been defined„ This i s shown i n f i g . ' s 9 and 10. The accuracy of the trans-mission c o e f f i c i e n t s i s estimated at about 10$ while that of the r e f l e c t i o n c o e f f i c i e n t s could i n some cases go as high as 20$. The accuracy of the l a t t e r i s lower, because the r e f l e c t -ed wave, which i s usually of a small amplitude, encounters interference with the incident wave while the transmitted wave does not. Prom these i l l u s t r a t i o n s , i t i s seen that the transmission c o e f f i c i e n t f or the A l to PI case remains f a i r l y constant from 30° to 110°, while that of the PI to A l half space o s c i l l a t e s i n t h i s region. In spite of the 10$ accuracy, i t i s f e l t that these o s c i l l a t i o n s are r e a l . For both cases, the r e f l e c t i o n c o e f f i c i e n t r i s e s from zero to a maximum and then decreases again to zero. For 'PI to A l 1 , R i s negative, i n d i c a t i n g a change i n phase. In f i g . 9 the r e f l e c t i o n curve i s dashed from 20° to 60° because the coefficients are uncertain i n t h i s range. This i s because f a i r l y strong r e f l e c t i o n s are detected from the opposite side of the aluminium wedge and these confuse the seismogram. This i s i l l u s t r a t e d i n the diagram below. In f i g . ' s 9 and 10 the ordinates of the curves chosen for 0° and 180° are ce r t a i n l y open to question. Furthermore, since F i g . 9 Reflection and Transmission Coefficients for "Al to PI" Half Space as a Function of the Interface Angle -40-measurements were taken at only 10 intervals, i t is impossible to be certain that some minor flucuations have not been missed. How much or in what way the shape of these curves w i l l change with a different choice of elastic media can not be said at present. 4.5 Conclusions It was the aim of this project to test by experiment two hypotheses which attempted to predict the reflection and trans-mission coefficients for Rayleigh waves. In the foregoing pages two formulae consistent with these hypotheses were derived and the experimental measurement of the reflection and transmission coefficients described in some detail. After comparison of the measured with'the calculated coefficients, i t was concluded that the proposed minimizing hypotheses were not satisfactory and that they must be rejected. This problem w i l l require much more work before a solution is forthcoming. Perhaps research along lines similar to those employed by Lapwood with the corner would prove f r u i t f u l . A discrepancy was observed between the.computed and meas-Oliver et a l (13) would indicate that this discrepancy is real, but i t w i l l not be certain u n t i l a detailed study, of the errors ured values of Calculations made from the results of -41-i s made. This should be done by using least squares to deter-mine the best f i t for travel-time curves; then the residuals and hence the standard errors may be computed. For an a r b i t r a r y choice of two e l a s t i c media joined to form a 'half space 8, the r e f l e c t i o n and transmission c o e f f i c -ients were measured as a function of the interface angle. The values of the co e f f i c i e n t s o s c i l l a t e d as expected, but i n such a manner that no pattern was recognizable. I t would appear that the r e f l e c t i o n and transmission of Rayleigh waves has a very complex dependence on t h i s angle between the plane Inter-face and the free surface. - 4 2 -BIBLIOfifiAPHY 1. Knott, C.G.: P h i l . Mag. 48, 90, 1899 2. MacElwane, J.B.s Physics of the Earth VI, B u l l e t i n of the National Research Council, no. 90, 116 3. Bullen, K.E.: "An Introduction to the Theory of Seismology" 2nd. ed., Cambridge University Press, 1953 4. Je f f r e y s , H.s Monthly Notices Roy. Astron. S o c , Geophys. Suppl. 1, 321, 1926 5. Dana, S.W.s B u l l . S e i s . Soc. Amer. 3j£» 1^ 9, 1944 6. Gutenberg, B.: B u l l . Seis. Soc. Amer. 3 j i , 85, 1944 7. Ergin, K.: B u l l . S eis. Soc. Amer. 4 2 , 349, 1952 8. De Bremaecker, J . : Geophysics 23_, 253, 1958 9. Lapwood, E.R.: Geophys. J . Roy. Astron. Soc. 4, 174, 1961 10. Ewing, W.M., Jardetzky, W.S., and Press, F.s " E l a s t i c Waves i n Layered Media", McGraw-Hill, 1957 11. Rayleigh, Lords Proc. London Math. Soc. 17, 4, 1885 12. Lamb, H . J P h i l . Trans. Roy. Soc. (London) A, 2 0 3 , 1, 1904 13. Oli v e r , J . , Press, P., and Ewing, W,M.s Geophysics 19, 202, 1954 1 4 . Newlands, M.s J . Acoust- Soc. Amer. 26, 434, 1954 15. Press, P. and Healy, J.s J . App. Phys. 28, 1323, 1957 16. Northwood, T.D. and Anderson,, D.V.s B u l l . S eis. Soc. Amer. 43_, 239, 1953 17. Evans, J.P., Hedley, CP,, E i s l e r , J.D. and Silverman, D.s Geophysics 19_, 220, 1954 18. Levin, F.K. and Hibbard, H.C.s Geophysics 20, 19, 1955 19. Obrien, P.N.s Geophysics 20, 2 2 7 , 1955 20. Clay, C S . and Mc N e i l , H.s Geophysics 20, 766, 1955 21. C a r l i n , B.: "Ultrasonics" McGraw-Hill, i960 

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