{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Physics and Astronomy, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Manning, Gavin N.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-07-16T00:36:37Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1987","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"The principle of operation of a novel rotating disk ultrasonic intensity meter is studied. Its characteristics are explained by a competition between acoustic radiation pressure and viscous drag on the disk. Acoustic streaming does not play a significant role in the operation of this meter as it is now configured.\r\nExperiments are described which were done to find the optimum dimensions and position for a nylon disk. In this optimum configuration, the rotation rate of the disk is related to the ultrasonic intensity by a power law. This relationship is theoretically predicted and found to hold as the ultrasonic intensity varies by a factor of at least ten.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/26496?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"A NEW ULTRASOUND INTENSITY METER: CHARACTERIZATION AND OPTIMIZATION by Gavin N. Manning B.Sc. Simon Fraser University, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard The University of British Columbia August 15, 1987 \u00a9 Gavin Manning 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P H Y S I C S The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A B S T R A C T The principle of operation of a novel rotating disk ultrasonic intensity meter is studied. Its characteristics are explained by a competition between acoustic radiation pressure and viscous drag on the disk. Acoustic streaming does not play a signif-icant role in the operation of this meter as it is now config-ured. Experiments are described which were done to find the optimum dimensions and position for a nylon disk. In this optimum configuration, the rotation rate of the disk is related to the ultrasonic intensity by a power law. This relationship is theoretically predicted and found to hold as the ultrasonic intensity varies by a factor of at least ten. ii T A B L E O F C O N T E N T S ABSTRACT ii. LIST OF TABLES vi. LIST OF FIGURES vii. ACKNOWLEDGEMENT x. 1. Introduction 1 1.1 What This Thesis is About, and Why it Matters 1 1.2 Thesis Organization 4 2. Radiation Pressure 5 2.1 What is Radiation Pressure? 5 2.2 Equations of Fluid Motion 6 2.3 Calculating Radiation Pressure in One Dimension 8 2.4 Range of Validity for Our Solution 10 2.5 Different Types of Radiation Pressure 12 2.6 The Relationship of Radiation Pressure to Momentum Flux . . . . 15 2.7 Torque on a Disk due to Radiation Pressure 18 2.8 Reflection and Refraction of Sound at a Liquid-Solid Interface . . . 20 2.9 Summary 24 3. Acoustic Streaming 26 3.1 What is Acoustic Streaming ? 26 iii 3.2 Equations of Motion for a Viscous Fluid 27 3.3 Calculation of Acoustic Streaming 28 3.4 Streaming Between two Parallel Walls 31 3.5 Acoustic Streaming Summary 34 4. The Drag on a Rotating Disk in a Still Fluid 36 4.1 The Drag on a Rotating Disk in a Still Fluid 36 5. Acoustic Cavitation 39 5.1 What is Acoustic Cavitation? 39 5.2 Types of Cavitation 40 5.3 Cavitation Thresholds 40 5.4 Cavitation Summary 42 6. Experiments 44 6.1 Apparatus 44 6.2 Beam Characterization 53 6.3 The Relative Importance of Streaming and Radiation Pressure . . . 56 6.4 Acoustic Forces Due to Reflection 62 6.5 Linearity of Torque vs. Power Input 66 6.6 Optimizing Position 68 6.7 Optimum Disk Thickness 74 6.8 Scaling of Rotation Rate with Disk Diameter 79 6.9 Rotation Rate versus Acoustic Power Input 90 7. Discussion and Conclusions 94 7.1 Radiation Pressure Causes the Disk to Rotate 94 7.2 Optimization Experiments 96 7.3 Summary 102 7.4 What Remains to be Done? 103 iv BIBLIOGRAPHY 1 \u00b0 5 APPENDIX I.-Lagrangian and Eulerian Coordinate Systems 107 APPENDIX II-Exact Calculations of Eulerian Radiation Pressure 110 APPENDIX III.-List of Symbols 118 v L I S T O F T A B L E S 1. Exact Dimensions of Disks of Varying Thickness 75 2. Dimensions of Disks of Various Diameters 81 3. Optimum Position for Disks of Different Diameters 84 4. Transducer Input Voltages for Standard Torque 88 vi L I S T O F F I G U R E S 1. The Rotating Disk Intensity Meter 2 2. Some Disks Used in These Experiments 3 3. Sound Beam Laterally Confined by a Rigid Wall 12 4. The Physical Meaning of Rayleigh Pressure 14 5. The Physical Meaning of Langevin Pressure 15 6. A Perfectly Absorbing Target 17 7. Acoustic Forces on a Reflecting Disk 19 8. Acoustic Forces on a Sound Absorbing Disk 20 9. Reflection and Refraction of an Acoustic Beam at a Liquid-Solid Interface . . 21 10. How the Angle of Incidence is related to the Impact Parameter, d . . . . . . 23 11. Acoustic Streaming Between Two Parallel Walls 31 12. Velocity Profiles for Acoustic Streaming Between Two Parallel Walls 33 13. Fluid Flow Patterns Near a Rotating Disk 37 14. The Tank 45 15. Power Delivered to a 500 Load vs. Amplifier Output Voltage 47 16. The Electronics Used to Generate and Monitor the Acoustic Beam 48 17. The 3-Axis Positioner 48 18. A 25 mm Diameter Disk Mounted in the Apparatus 50 19. The Beam Absorber 51 vii 20. The Tank and Assembled Apparatus 52 21. The Speed Measuring System 52 22. Starch Plate Photography System 53 23. Beam Profile Determined from Starch Plate Measurements 55 24. Some Exposed Starch Plates 55 25. Arresting Streaming with an Acoustically Transparent Membrane 57 26. Saran Wrap Screen in Place 57 27. The Screen Tilted Near a Disk 58 28. The Cylindrical Saran Wrap Screen 58 29. RPM vs. Power for PVC Disk, with and without Cylindrical Screen 59 30. Streaming along the Sides of a Disk . 60 31. RPM vs. Power: Thick Nylon Disk, with and without Screen 61 32. Notched Disk in an Acoustic Beam 62 33. Disk Thickness and Diameter 63 34. The Notched Disk 64 35. Copper Disks Mounted in Tandem 65 36. Setup for the Third Experiment 67 37. Power Needed to Maintain 202 RPM vs. Number of Small Disks on the Shaft . 67 38. RPM vs. Disk Height 69 39. RPM vs. Lateral Position in Sound Beam 70 40. RPM vs. Distance to Transducer Face 71 41. Optimum Position for a 2.5 cm Diameter Nylon Disk 72 42. Optimum Height vs. Distance to Transducer Face 72 43. RPM vs. Lateral Position at Various Distances From the Transducer 73 44. Some Disks of Various Thicknesses 74 45. RPM vs. Disk Thickness at Various Power Levels 76 viii 46. RPM vs. Power for Various Disk Thicknesses 77 47. RPM vs. Distance to the Transducer for a Thick Disk 78 48. Some Disks of Various Diameters 79 49. Disks Mounted With Their Bottoms at the Same Point in the Sound Beam . . 80 50. RPM vs. Disk Diameter at various Power Levels 82 51. RPM vs. Disk Diameter at 2W and 5W 83 52. Optimum Axle Height vs. Disk Diameter 85 53. RPM vs. Disk Diameter with Disks in Optimum Position 86 54. Two Disks Mounted in Tandem 87 55. RPM vs. Disk Diameter at Constant Torque 89 56. RPM vs. Power for Disks D6 and D9 91 57. RPM vs. Power for Disk D6 92 58. RPM vs. Power for Disk D9 93 59. Sound Capture by a Disk 96 60. Angles of Incidence for a 2.5 cm Diameter Disk in Optimum Position 97 61. Disk in a Non-Uniform Acoustic Beam 98 62. 6\/R vs. Power for Disks D6 and D9 102 63. The Relationship Between Eulerian and Lagrangian Coordinates 108 ix A C K N O W L E D G E M E N T I owe thanks to many people who helped to make this research pos-sible. It is my special pleasure to thank: My supervisor, Dr. Frank Curzon, for his support and encourage-ment over the past two years. John Koblanski for devising the meter that this research is based on, and for his help. Al Cheuck for his technical assistance. And Jack Bosma for maintaining the best student machine shop I have seen anywhere, and for giving me many useful ideas. Last, but not least, a special thank-you to Catherine for making re-turning to school especially worthwhile. x C H A P T E R 1-Introduction 1.1 What This Thesis is About, and Why it Matters In this thesis, we investigate the properties of a novel device for measuring the intensity of an ultrasonic beam with the aim of clarifying its principle of operation, and studying the scaling laws which will be important in its optimization. Ultrasonic measurements are important in many industrial processes where a com-mon method of measuring the density of a slurry is to measure the attenuation of an ultrasonic beam passing through it . An example of a process requiring such measure-ments is the making of paper where it is crucial to maintain the proper concentration of pulp fibers in the slurry. Another area where it is important to measure ultrasonic beam intensities is in the calibration of medical ultrasound equipment. Several types of ultrasonic intensity meters are currently available. These can be divided into three main groups: [Zieniuk and Chivers 1976] 1. ) Transducers such as calorimeters where the energy of the beam is measured directly. 2. ) Transducers which measure acoustic pressure, velocity, or displacement such as piezoelectric, magnetostrictive, or capacitive transducers. 1 3.) Transducers which are based on nonlinear effects in the ultrasonic field such as streaming, which is the steady flow of a fluid caused by an ultrasonic field, or radiation pressure, which is the force exerted by an ultrasonic field on an object or interface in a fluid. The commonest example of this type of transducer is the radiation pressure balance commonly used to calibrate medical equipment. L E N S D I S K FIGURE 1-The Rotating Disk Intensity Meter The subject of this thesis is a meter devised by John Koblanski of Ocean Ecology Ltd., Vancouver B.C. This meter falls into the third category above, and consists of a disk mounted on a shaft which is free to rotate on fine bearings, and a lens to focus the ultrasonic beam on the edge of the disk [Koblanski 1983]. The arrangement is shown in Figure 1. When an ultrasonic beam is focussed through the lens, the disk experiences a torque and begins to rotate. There are a great many factors important to the performance of this device: these include the viscosity, speed of sound, and coefficient of sound absorption for 2 THE QUALITY OF THIS MICROFICHE IS HEAVILY DEPENDENT UPON THE QUALITY OF THE THESIS SUBMITTED FOR MICROFILMING. UNFORTUNATELY THE COLOURED ILLUSTRATIONS OF THIS THESIS CAN ONLY YIELD DIFFERENT TONES OF GREY. LA QUALITE DE CETTE MICROFICHE DEPEND GRANDEMENT DE LA QUALITE DE LA THESES SOUMISE AU MICROFILMAGE. MALHEUREUSEMENT, LES DIFFERENTES ILLUSTRATIONS EN COULEURS DE CETTE THESES NE PEUVENT DONNER QUE DES TEINTES DE GRIS. ' 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 FIGURE 2-Some Disks Used in These Experiments the fluid in which it operates; the materials that the disk and lens are made of, the quality of the bearing surfaces, and several geometric factors such as the shape and focal length of the lens, the diameter and thickness of the disk, the surface condition of the disk, and the positions of the lens and disk relative to each other as well as to the ultrasonic beam. It is the purpose of the research reported in this thesis, first to determine what causes the torque on the disk, whether it is streaming of the fluid, or radiation pressure, and secondly to investigate the effect of varying some of the geometric factors with a view to the optimization of this meter. The effect of varying the disk dimensions and changing the relative position of disk and lens is reported here. Some of the disks used are shown in Figure 2 . 3 1.2 Thesis Organization In order to understand this meter, we need to know a little bit about acoustics. In particular we need to understand acoustic radiation pressure, by which an ultrasonic beam directly causes a force on a solid object or interface in the fluid, and acoustic streaming where an ultrasonic beam causes flow in the fluid through which it propagates. The flowing fluid can then impinge on a solid object with a resulting force. These topics compose Chapters 2 and 3 of this thesis. The torque on our disk arising from its interaction with the ultrasonic field is bal-anced by viscous drag on the disk. Chapter 4 is therefore devoted to a brief study of the viscous drag on a rotating disk. Another topic that deserves mention is acoustic cavitation, the creation of cavities in a liquid by an intense ultrasonic field. These cavities interfere with the propagation of sound waves. They can also collapse with enough vigour to damage equipment or grow into bubbles that stick to surfaces and add drag to moving objects. Acoustic cavitation is the subject of Chapter 5. In Chapter 6 we report on the design of our apparatus, some experiments done to characterize the ultrasonic field, experiments done to clarify which mechanism is respon-sible for the rotation of our disk, and experiments done to find some scaling laws for this meter. Chapter 7 discusses these experimental results and summarizes the conclusions reached in the course of research for this thesis. In particular, the original contributions made by the author are given in section 7.3. 4 C H A P T E R 2-Radiation Pressure 2.1 What is Radiation Pressure? The propagation of acoustic waves in a fluid is intrinsically a nonlinear phenomenon. The equations of fluid motion include nonlinear convective terms, and the equation of state for the fluid is also, in general, nonlinear. Although these equations governing the motion of fluids can be linearized, and the resulting 'small amplitude equations' can be used to solve many practical problems where the finite amplitude of the sound waves can be ignored, much interesting physics is missed by this approach. Among the effects that arise because of the nonlinearities present in sound propagation are the formation of shock waves [Beyer 1974], the nonlinear interaction of waves [Ingard and Pridmore-Brown 1956], and two of the topics discussed here, acoustic radiation pressure, and acoustic streaming. To begin then, we need to take the equations governing fluid flow. Given a fluid through which a harmonic plane acoustic wave is propagating, we can then calculate a value for the time averaged pressure (P) in the beam. This pressure is different from the equilibrium hydrostatic pressure Pn and it depends on the boundary conditions we impose on the fluid. The different values of (P) we obtain for different boundary conditions cor-respond to pressures that could be measured in a fluid in different physical circumstances. 5 2.2 Equations of Fluid Motion In Eulerian coordinates')\", there are three equations that describe all of the allowed motions of a fluid including the propagation of acoustic waves [Temkin 1981]. These are the equation of continuity: ^ + v . P u = o (1) The force equation: and the equation of state: P = P(\/> , S) (3) where p is the fluid density, U is the fluid velocity, F represents all of the forces acting on elements of the fluid including body forces such as gravity and internal viscous forces, P is fluid pressure, and S the entropy of the fluid. D\/Dt is the total or material derivative which measures the changes occurring in a fluid element as it moves with the fluid flow %. For investigating the phenomenon of radiation pressure we shall consider soundwaves propagating adiabatically through an ideal nonviscous gas. In this case, the equation of state is given by: f See Appendix I for a description of Eulerian and Lagrangian coordinates D t % The total derivative of a quantity Q is given by: ^ = dQ\/dt + U VQ 6 where 7 is the ratio of specific heats Cp\/Cv. Assuming (4) for our equation of state is not nearly as restrictive as it might at first seem. Firstly, many real gases behave much like ideal gases over a large range of conditions. Secondly, Equation (4) can be used as an equation of state for almost any isentropic liquid with very little modification [Beyer 1974, p.98-99]. In an isentropic liquid, we know that the pressure P must be some function of the density p. This equation of state can be written as a Taylor series expanded about the point p \u2014 po as follows: where A , B , C , . . . are constants to be determined for each particular gas. The adiabatic ideal gas equation of state (4) which we would like to use can be expanded by the binomial expansion to yield: P = P o + A \/P ~ Po\\ B (p- p0\\ V po J 2! V Po J (5) P = P o (6) If we compare (5) and (6) term by term,we obtain the relationships: A <-> 7 P 0 (7) and: B <-\u2022 7(7 - 1 ) P 0 (8) hence we can make the direct correspondence: B \/ A ^ (7 - 1) (9) 7 As long as the third and higher terms of (5) are very small in comparison to the first two terms, we can use (4) to describe, liquids as well as gases if we replace 7 by the experi-mentally determined parameter ((B\/A) + l ) . For almost all real liquids at any attainable intensity, the maximum condensation ((p \u2014 po) \/ Po) is less than about IO - 4 [Coppens et.al. 1965] so that even with C\/A given approximately by:[Hagelberg, Holton,Kao 1967] the cubic and higher order terms can be ignored. For water at 20 C and atmospheric pressure, B\/A has been measured to be 5.0 [Beyer I960]. 2.3 Calculating Radiation Pressure in One Dimension In one dimension, the equations of motion (l) and (2) become: and au av BP Each of these equations has one nonlinear term. In combination with the nonlinear equa-tion of state (4) , these equations form a system of three differential equations which can be solved by the method of successive approximations. In this method, p,U, and P are each expressed as power series in the small parameter e and then substituted into (4) , (11) , and (12). It is convenient to pick: t = U m a i \/ c 0 (13) 8 where U m a j ; is the maximum fluid velocity, and c0 is the speed of sound. This choice for e has the desired property of going to zero for infinitesimal waves. As epsilon is an arbitrary parameter, we must require that the coefficient of each power of e is identically zero so that the solution is independent of the way we pick e. In this way we obtain an infinite hierarchy of systems of ordinary differential equations.The first set is: \" \u00bb ^ r + ^7=\u00b0 <14> Dt \\p0 PO J 1 1 These first equations can be recognized as the linearized equations of classical acoustics. The second set of equations provides the first nonlinear correction to (14), (15), and (16). These equations are: au2 ap2 dv1 av1 p\u00b0-bT + = ~P1^T ~ PoVl~dT {17) dp2 d(p0V2) = d ( P i U i ) , g ] dt dx dx { J d [ P 2 !P2)^d(1PlPl l{l + l)pi2} ^ _ l l \u00a3 l _ ^ l \\ (19) d t \\ p 0 po j d t \\ P 0 p o 2 po2) ^ x X p o Po J 1 ' These second order equations are exactly the same as the corresponding first order equa-tions above except that while (14) , (15) , and (16) are homogeneous, these equations each have a forcing term on the RHS which is completely determined once the first order equations are solved. In fact, all of the higher order sets of equations are of this form with different forcing terms on their RHS. When these equations are solved for an initially 9 harmonic plane wave disturbance, the time-averaged Eulerian pressure can be calculated to be: The detailed calculations can be found in Appendix II. Some authors explain alternative methods for solving (4), (11), and (12) including solution by the method of characteristics [Blackstock 1962], or time averaging the equations of motion [Chu and Apfel 1982]. They obtain the same result for (20). 2.4 Range of Validity for Our Solution There have been several assumptions made in attaining the result (20) which should be pointed out. Firstly, we have assumed that the sound wave propagates adiabatically through a nonviscous fluid which obeys the ideal gas equation of state. For the assumption of adiabatic behaviour to hold, we require that both the thermal conductivity, \/c , of the fluid, and the frequency, UJ, of the sound be reasonably small. In air, deviation from adiabatic behaviour is not observed until frequencies on the order of 108 Hz are attained [Randall 1951]. The assumption that the fluid is nonviscous is equivalent to requiring that the sound not be significantly attenuated in the region of interest. We can therefore only apply this theory in cases where the absorption length for sound is greater than the scale length of our problem. As we saw above, many real gases and liquids approximately satisfy the ideal gas equation of state (4) under conditions typical for sound propagation. As long as we pick a value of 7 appropriate for the fluid in question, this assumption is not too restrictive. (20) 10 We made other approximations in solving the equations to calculate (P). Foremost among these was the assumption that e is small enough that terms containing c3 and higher powers of e could be neglected. With e defined in (13) it is clear that this limits our theory to cases where the intensity of the sound beam is low enough that the maximum fluid particle velocity is much less than the speed of sound in the fluid. Another limit on the size of the region of interest, which depends on the intensity of the beam is that the fluid particle velocity must remain a single valued function of position. As this theory now stands, it predicts that the harmonic content of a beam increases as it propagates, and that the resulting steepening of the wavefront will continue until the wave 'breaks' at a distance, XSF> from its source given by: [Beyer 1974, p. 104] What really happens is that a shock wave forms before the fluid velocity can become double valued, something that this theory does not include. The distance (21) is commonly referred to as the shock-formation distance, XSF - Our theory is only valid for distances less than XSF- At 1.0 MHz , and 1 atmosphere pressure, at an acoustic Mach number U m a i \/ c o = 0.046 x 10~3, the shock formation distance in water is 148 cm [Beyer 1974, p.105]. A final tacit assumption which also limits the range of intensities for which this theory is valid is that the fluid remains continuous. If the sound intensity is too large, the pressure will be extremely low in the regions where the fluid is rarefied. In extreme cases, voids can form. This situation is discussed more fully in Chapter 5 on cavitation. To ensure that the fluid remains homogeneous, the sound intensity must be kept below the cavitation threshold for that fluid. In the case that the sound originates from a moving 11 piston, Blackstock [Blackstock 1962] calculates that the piston velocity must also satisfy: so that the fluid will remain in contact with the piston. 2.5 Different Types of Radiation Pressure k , ^ ^ W A L L \\ F =
-AREA FIGURE 3-Sound Beam Laterally Confined by a Rigid Wall We have calculated an average pressure, but should be sure that we understand what it means physically. (21) is the average pressure that would be recorded by a microscopic hydrophone fixed in space, and so small as not to affect the motion of the fluid. It is also the pressure that would act on a rigid wall laterally confining the sound beam as shown in Figure 3 . (21) is not the pressure that would act on the surface of an object partially blocking the sound beam. To calculate the radiation pressure on a completely absorbing target which does block the sound beam, it is best to use Lagrangian coordinates. By definition, the surface 12 of a completely absorbing target exactly follows the motion of the fluid layer immediately next to it. The absorbing target thus feels the Lagrangian pressure on its surface so the radiation pressure on such a target is given by the time averaged Lagrangian pressure P^). This quantity can be calculated using one of the same methods used for calculating the average Eulerian pressure (P) but beginning with the fluid equations of motion in Lagrangian coordinates [Blackstock 1962]. The result is:
= Po + t^PocW (23) So far, we have considered the one-dimensional case where the sound beam extends infinitely laterally. If the beam is finite, the force on an obstacle in the beam depends on whether fluid is allowed to flow into and out of the region of the beam, or is confined by impermeable walls. When the fluid is confined, the preceeding analysis holds true. It is customary to define the 'Rayleigh pressure', P R a to be: P R a = ( P L ) - P 0 (24) P R a is the net force per unit area acting on a perfectly absorbing target one side of which is exposed to a laterally confined sound beam, and the other to fluid at rest. This situation is shown in Figure 4 . If the sound beam is not laterally confined and fluid is allowed to flow in a direction perpendicular to the wavevector, the force on an absorbing target will be different from (24). This is because, given the chance, a fluid will flow in such a way as to reduce pressure gradients. For the sake of clarity, assume that there is a sharp boundary between the beam and the surrounding fluid although this is not a necessary condition [Beissner 1982]. As the fluid surrounding the beam is undisturbed, it will be at pressure P 0 . We have seen above that the fluid inside the beam is pressing outward as if it were pressurized to the 13 PERFECT ABSORBER \\|-| mill iiilllllllllllllilllill SOUND BEAM COMPLETELY FILLS SPACE BETWEEN WALLS iiiijiiiiiiiiiiiiiiiiimiiiiiii J^^A R E A WALL ^ STILL FLUID, P=fJ FIGURE 4-The Physical Meaning of Rayleigh Pressure Eulerian average pressure ( P ) . Fluid will flow into or out of the beam until equilibrium is established at which point the average Eulerian pressure just inside the beam will equal P o , the pressure just outside the beam. The fluid flow changes the density of fluid in the beam effectively changing the base pressure in the beam from P o to P 0 where P 0 is given by: P 0 = P 0 - fc^ii = Po + ^Pocle2 (450) 117 A P P E N D I X III. List of Symbols Amplitude of sound beam Bulk Viscosity Speed of transverse waves in solid Parameter of nonlinearity Speed of sound in fluid Speed of longitudinal waves in solid Specific heat, constant pressure Specific heat, constant volume Saturation concentration of dissolved gas Concentration of dissolved gas at infinity \"Impact parameter \" (fig. 10) Total derivative 118 Acoustic energy density (sec. 2.5) Force on a fluid element (sec. 2.2) Driving force for streaming (sec. 3.4) Sound Intensity (sec. 3.4) Momentum density in sound beam (sec. 2.6) Momentum density in reflected beam (sec. 2.7) Sound wave number (sec. 2.6) Sound wave vector (sec. 2.6) Scale length (sec. 3.4) Torque (sec. 4.1) Momentum (sec. 2.6) Hydrostatic pressure (sec. 2.2) Average Eulerian pressure (sec. 2.1) Average Lagrangian pressure (sec. 2.5) Langevin radiation pressure (sec. 2.5) Rayleigh radiation pressure (sec. 2.5) Blake cavitation threshold (sec. 5.3) Threshold for rectified diffusion (sec. 5.3) 119 Vapour pressure of liquid Equilibrium hydrostatic pressure Effective hydrostatic pressure, unconfined sound beam First terms in a series expansion of P Disk Radius Reflected intensity Reynolds number Reynolds number for a rotating disk Equilibrium bubble radius Frequency dependent term, viscous fluid eqn. of state Entropy of fluid Transmitted beam intensity , Scale velocity Fluid velocity Maximum fluid velocity in a harmonic wave First terms in a series expansion of U Amplitude reflection coefficient for ultrasound Complex conjugate of V 120 Rate of deformation tensor Shock formation distance Acoustic impedances defined in eqn. (43) Sound absorption coefficient Ratio of specific heats Cp\/Cv Angle of refraction, transverse waves Boundary layer thickness Expansion parameter, method of successive approximations Angle of incidence Critical angle for refraction Angle of refraction, longitudinal waves in solid Thermal conductivity Coefficient of shear viscosity Kinematic viscosity Displacement of fluid particle from equilibrium Displacement amplitude of harmonic wave Fluid density Equilibrium fluid density 121 Po,Pi-,P2 \u2022 \u2022 \u2022 First terms in the series expansion of p (sec. 2.3) cr Surface tension (sec. 5.3) Stress tensor (sec. 3.2) bj Angular frequency (sec. 2.3) ( \u2022 \u2022 \u2022 ) Time average 1 sec. 3.3) 122 ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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