Experimental Study of Sound Waves in Sandy Sediment

Transcription

1 Approved for public release; distribution is unlimited. Experimental Study of Sound Waves in Sandy Sediment by Michael W. Yargus Technical Report AL-UW TR 030 May 003 Applied hysics Laboratory University of Washington 03 NE 40th Street Seattle, Washington Contract N

2 Experimental Study of Sound Waves in Sandy Sediment Michael W. Yargus A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of hilosophy University of Washington 003 rogram Authorized to Offer Degree: Electrical Engineering

3 University of Washington Graduate School This is to certify that I have examined this copy of a doctoral dissertation by Michael W. Yargus and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. Chair of Supervisory Committee: Darrell R. Jackson Reading Committee: Darrell R. Jackson Kevin L. Williams Terry E. Eart Date:

4 In presenting this dissertation in partial fulfillment of the requirements for the Doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of the dissertation is alloable only for scholarly purposes, consistent ith fair use as prescribed in the U.S. Copyright La. Requests for copying or reproduction of this dissertation may be referred to roquest Information and Learning, 300 North eeb Road, Ann Arbor, MI , to hom the author has granted the right to reproduce and sell (a) copies of the manuscript in microform and/or (b) printed copies of the manuscript made from microform. Signature Date

5 University of Washington Abstract Experimental Study of Sound Waves in Sandy Sediment Michael W. Yargus Chair of the Supervisory Committee: rofessor Darrell R. Jackson Electrical Engineering This dissertation describes experiments intended to help understand the physics of sound (compressional aves) propagating through sandy sediments (unconsolidated porous media). The theory (using a lumped parameter model) and measurements (using a reflection ratio technique) includes derivations and measurements of acoustic impedances, effective densities, ave speeds (phase velocities), effective pressures, mode shapes, pressure reflection coefficients, and material moduli. The results sho the acoustic impedance divided by the phase velocity, rendering an effective density, is less than the total density of the sediment (effective density 89% 3% of total). The results also sho the fluid in the sediment oscillates back-and-forth. 0.4 times farther than the sand in the sediment (mode shape) during the passing of a sound ave. These facts suggest the existence of Biot aves (to compressional aves) in atersaturated sand.

6 TABLE OF CONTENTS age List of Figures iv List of Tables vii Chapter : INTRODUCTION Outline of Dissertation Background History Observations of Biot Waves Controversy Controversy arameters and Assumptions Ne Contributions of this Dissertation Chapter : MATHEMATICAL THEORY Differential Equations of Motion and Boundary Conditions Lumped arameter Model Reflection and Transmission ressure Coefficients for a Sound Wave in Water Incident Upon a Sand Interface (Open ore) Transmission and Reflection Coefficients of a Fast Wave from Sand to Water (Open ore) Transmission and Reflection Coefficients of a Slo Wave from Sand to Water (Open ore) Reflection and Transmission Coefficients of a Fast Wave from Sand to Non-orous (Closed ore) Interface Reflection and Transmission Coefficients of a Slo Wave from Sand to Non-orous (Closed ore) Interface Reflection and Transmission Coefficients of an Incident Wave from a Non-orous Material to Sand i

7 .9 Summary of ressure Coefficients Chapter 3: ACOUSTIC MEASUREMENTS Specifications Reflection Equations Frequency Measurements Chapter 4: ANALYSIS Fast and Slo Wave Impedances Wave Speeds and Modal Masses Effective ressures A Second Look at the roperties of the Fast Wave Mode Shapes Tortuosity and Viscosity Moduli Open ore ressure Reflection Coefficients Chapter 5: DISCUSSION Comparisons Simplified Equations of Motion Future Work Conclusions REFERENCES Appendix A: IMEDANCES OF ACRYLIC AND ETHAFOAM A. Impedance of Acrylic A. Impedance of Ethafoam Appendix B: NUMERICAL CHECKS Appendix C: ERROR ANALYSIS Appendix D: SINGLE DEGREE-OF-FREEDOM LUMED ARAMETER MODEL D. Homogeneous (No Boundary Effects) ii

8 D. Boundary Appendix E: DYNAMIC DESIGN ANALYSIS METHOD Appendix F: LANS OF TRAY AND LID Appendix G: ULSE FORMATION Appendix H: SAND SIE iii

9 LIST OF FIGURES age. Mass-Spring-Force Mechanical Model of the Biot Medium Free-Body Diagram of the Boundary Rayleigh Notation of Figure Free-Body Diagram of Fast Wave Mode Shape Location of F Using Orthogonality Modal Displacement (u ) and Average article Displacement (u fast ) for the Fast Wave Mechanical System Equivalent to Fig.. But in Terms of Modal arameters Effective Forces and Model Used to Find Acoustic Impedance of Biot Medium Free Body Diagram of Incident, Reflected, and Transmitted Waves Fast Wave Incident on Sand-Water Interface Free-Body Diagram of Incident and Reflected Forces on Sand-Water Interface Fast Wave Incident on Non-Biot Material Incident Wave from Non-Biot Material Schematic for Sand-Water Reflection (r 3 ), and Acrylic-Sand Reflection (r s ) Sand-Water Interface (r 3 ) Acrylic-Sand Interface (r s ) Schematic for Acrylic-Air Reflection (r air ) Acrylic-Air Interface (r air ) Schematic for Sand-Ethafoam Refection (r 3E ) and Water-Acrylic Reflection (r a ) Sand-Ethafoam Interface (r 3E ) Schematic for Sand-Air Reflection (r 3air ) iv

10 3.8 Sand-Air Interface (r 3air ) Impedances of Fast and Slo Waves Modal Masses (Effective Densities) Fast and Slo Wave Speeds Effective ressures Sediment Displacement Model for Fast Wave Mode Shapes, Fluid Amplitude Divided by Sand Amplitude Intermediate Moduli Mass Factor ressure Reflection Coefficient from Sandy Sediment A. Schematic for Acrylic-Air Transfer Function A. Schematic for Acrylic-Water Transfer Function A.3 Impedance of Acrylic A.4 Schematic for Acrylic-Ethafoam Transfer Function A.5 Impedance of Ethafoam C. ulse Reflection from Water-Air Interface ith Envelope Used for Bias Error D. Mass-Spring Model D. Movements of To Masses (u - and u 0 ) and the Difference of Their Movements E. Cover age of NAVSHIS E. Authorization of NAVSHIS E.3 Author s age of NAVSHIS E.4 Modal Analysis of NAVSHIS F. Assembly Vie of Equipment in Tank F. lan of Tray F.3 lan of Lid F.4 rofile Vie of Foundation v

11 F.5 lan Vie of Foundation G. Desired Received ulse G. Frequency Contents of Received and Excitation Signals G.3 Forcing Function Fed Into Transmitting Hydrophone G.4 Actual Received Signal vi

12 LIST OF TABLES age. ressure Coefficients, Open ore, Sand-Water Interface ressure Coefficients, Closed ore, Sand-NonBiot Interface Williams arameters Comparisons at 50 khz arameter Units B. Comparisons of Calculated ressure Coefficients B. Comparisons of Calculated Mode Shapes H. Sand Size Distribution vii

13 ACKNOWLEDGMENTS I ould like to thank my supervisory committee for their ork. --Darrell R. Jackson, Electrical Engineering, Chairperson --Miqin hang, Materials Science and Engineering, Graduate School Representative --Akira Ishimaru, Electrical Engineering --Yasuo Kuga, Electrical Engineering --Terry E. Eart, Oceanography --Kevin L. Williams, Oceanography I ould like to compliment rofessor Terry Eart and rofessor Eric Thorsos for their insight in the design and fabrication of the acoustic tank and measurement system that as ell suited for this experiment. I ould like to thank the Naval Surface Warfare Center for supporting this Extended Term Training. viii

14 Chapter : INTRODUCTION This dissertation implements theory and procedures designed to measure physical parameters of acoustic aves (also called sound aves, compressional aves, bulk compressional aves, and dilatational aves) in sandy sediments. This experimental study as done to help understand the mechanics of sound propagating through ater-saturated sand and in particular to look for effects only seen in porous media. These effects include different particle oscillation amplitudes of the sand and fluid (pore fluid in the sediment), a sound ave not using the total mass of the sediment, and the existence of to compressional aves ith different speeds. These are characteristics of the to (fast and slo) Biot aves. Sound ave measurements reported in this dissertation are obtained through a reflection ratio technique, a ne technique that uses a reference surface and a pressure release surface to measure coherently averaged pressure reflection coefficients from a surface of interest. In the first of to parts, a signal of interest (reflected from the surface of interest) is divided by a reflected reference signal. The reference signal is a measure the sound ave being transmitted, and by dividing the reference signal into the signal of interest a transfer function is obtained. This has the advantage of setting a nearly perfect time reference so the reflections can be averaged coherently. In addition, the to signals from the transmitter do not have to be identical (the ringing of the transmitting hydrophone as actively canceled and the signal periodically needed to be reshaped ). In the second part, the transfer function from the surface of interest is divided by a transfer function from a pressure release surface. The pressure release surface is set up so that the ratio of the to transfer functions leaves only the pressure reflection coefficient. This technique has the advantage of measuring a pressure reflection coefficient ithout needing to kno the acoustic properties (dissipation, damping, spreading loss, etc.) of the media the sound travels through and ithout needing to calibrate the transducers (receiver and transmitter). With the reflection ratio technique, coherently averaged pressure reflection coefficients are measured ithout calibrating the transducers and ithout use of

15 geometric correction factors. By knoing pressure reflection coefficients, acoustic impedances of the sand can be obtained. The mathematical theory is based on a lumped parameter model, a ne model that visibly shos the physics of a Biot medium and facilitates use of free-body diagrams to find the distribution of pressures and the relative velocities beteen particles and acoustic aves, e.g., sand particle velocity to fast ave effective velocity. By knoing the fast and slo ave effective velocities, rigorous derivations of the fast and slo ave acoustic impedances are obtained for the first time.. OUTLINE OF DISSERTATION This chapter includes: () A background of hy knoledge of the physics of sound aves propagating through sand is of practical importance. () A short history of sound propagating through saturated porous materials. (3) revious observations of Biot aves. (4) A look at to contemporary controversies. One is hether slo aves exist in unconsolidated porous materials, i.e., sediments, and another is hether the claimed compression aves in sediments at 00 m/s are slo Biot aves or artifacts from scattering. (5) A list of key parameters and assumptions that play a role in the dynamic response of sediments. (6) A list of the ne contributions published in this dissertation. The second chapter of this dissertation goes through the mathematical theory and discusses a lumped parameter model. The third chapter discusses the acoustic measurements and the reflection ratio technique. The fourth chapter discusses the analysis and results; the last chapter gives a discussion ith conclusions.. BACKGROUND Understanding the physics of sound propagation into a sandy bottom of a shallo ater environment is driven by a need to improve techniques for finding and classifying objects that are buried. Sonar, traditionally used to locate objects in ater or map the ater-sediment interface, can be used to locate objects such as mines, cables, pipelines and containers buried in sediment. The principle reason acoustic ave propagation and

16 3 scattering have been idely used to study processes and detect objects in the ocean is the limited range of optical and other electromagnetic ave propagation underater. For a ide range of scientific and engineering problems dealing ith the ocean environment, acoustic ave propagation and scattering is the only remote sensing technology available. A buried target can be illuminated ith maximum intensity by a sonar signal normally incident on the sediment interface. Hoever, the time taken to detect buried objects can become prohibitively large if only normal incidence is used. Using shallo incidence grazing angles can speed up detection operations considerably. For sand sediments, the compressional sound speed in the sediment is faster than in the ater and the ave transmitted into the sediment is refracted at a smaller grazing angle than in the ater. At the critical angle, typically at grazing angles of about 0 to 30 degrees, the refracted compressional ave is parallel to the interface and no longer propagates into the sediment. There is significant interest in mechanisms by hich a compressional ave in the ater can be coupled into the sediment at angles belo the critical angle ith energy sufficient enough for finding buried objects [4, 7, 34]. It has been proposed that the slo Biot ave may provide such a mechanism [34]. This dissertation explores normally incident aves in an attempt to understand the nature of sound propagating through sandy sediments. The results of this ork are relevant to the general problem including incidence at angles smaller than the critical angle..3 HISTORY A complete theory for acoustic ave propagation in porous media as developed by Biot [3, 4, 5] and presented in a series of papers. Biot has developed a comprehensive theory for the static and dynamic response of linear, porous materials containing compressible fluid. This theory as derived by adding inertia terms into the equation of consolidation developed in earlier theories. Biot s theory predicts that in the absence of boundaries, three kinds of body aves, to dilatational and one shear, may exist in a fluid saturated porous medium [36].

17 4 Stoll and Kan [38] derived the complex reflection coefficient of plane acoustic aves from a poro-elastic sediment half-space. The boundary condition model is based on the classical ork of Biot that, as previously stated, predicts three different body aves in the sediment. As a consequence, hen plane aves in ater are incident upon a ater-sediment interface, as many as three aves can be generated in the sediment, depending on the angle of incidence. Stoll and Kan used potentials in their derivation of the boundary conditions. Johnson and lona rerote Stoll and Kan s boundary conditions using displacements as it is simpler than (although equivalent to) the potential derivation [0]. Johnson and lona s notation for the boundary conditions produces a symmetric determinant hen solving for the eigenvalues and eigenvectors, hich can be modeled into a physically realizable system. Also the density terms in Johnson and lona s determinant add up to the density of the ater-saturated sand. Neither of these are true for Stoll and Kan s notation and both are important for the lumped parameter model to be developed here..4 OBSERVATIONS OF BIOT WAVES lona observed a second bulk compressional ave that propagated at speeds approximately 5% of the speed of the normal bulk compressional ave in a fluid-saturated porous medium [9]. This observation as made using an ultrasonic immersion technique and a fluid-saturated porous medium consisting of ater and sintered glass spheres. The excitation of the slo ave in the solid as shon to be consistent ith the principles of mode conversion and refraction at plane liquid/solid interfaces. To lona s knoledge, this type of bulk ave had not been previously observed at ultrasonic frequencies. Biot s theory has been used in several applications of acoustic ave propagation in porous media, such as superfluid/superleak systems [8], slo aves and the consolidation transition [9, 0], gels [], bones [5], porous solids [5, 3, 30], marine sediments [7, 35, 4, 43, 45], pressure diffusion through porous media [8], and air-saturated consolidated porous media [6].

18 5 Johnson et al. [] observed slo compression aves in consolidated fluidsaturated porous media, e.g., fused glass beads (called Ridgefield Sandstone even though it as man-made) and ceramic ater filters (QF-0), and calculated for the first time all of the input parameters necessary for a complete description of the acoustic properties of all the modes over the entire frequency spectrum. All observed slo Biot aves have been in consolidated porous media. There have been no uncontested observations of a slo Biot ave in an unconsolidated, e.g., sandy, medium. See Controversy- that follos..5 CONTROVERSY- (Do slo Biot aves exist in sediments?) Even though Biot theory predicts both fast and slo compressional aves, only the fast compressional ave, the ave of the first kind, and the shear ave have been observed in the vast majority of porous systems. Johnson and lona, ho observed slo Biot aves in ater-saturated porous media here the solids are bonded together, have looked for slo Biot aves in unconsolidated ater saturated glass beads here they conclude both experimentally and theoretically there is only one compressional ave [0]. Simpson et al., [34] did an experiment shoing the ave fields in sand ith smooth and rough interfaces at all angles of incidence. They concluded, This data does not support the Biot medium model predictions of a second sloer compressional ave in the unconsolidated ater saturated porous medium. Seifert et al., [33] did an experiment ith sand saturated ith fluids of different viscosities (ater, to different silicon oils, and castor oil) to measure attenuation. They stated, Biot s theory shos that acoustic aves create relative motion beteen the fluid and the solid matrix due to inertial effects, resulting in viscous dissipation of acoustic energy. But they concluded The attenuation shos no correlation ith the viscosity of the different pore fluids and thus theories that depend upon fluid flo cannot explain these data. They conclude Biot theory does not explain attenuations measured in fluid filled sand.

19 6 On the pro side, Stoll et al., [38] say Biot aves exists but, Waves of the second kind are difficult to observe in sediments here ater is the saturant because of their high attenuation and the ay in hich energy is partitioned at the interface..6 CONTROVERSY- (Compressional aves at 00 m/s are from slo Biot or scattering?) Recent experimental results (Boyle and Chotiros 99 [6], Chotiros 995 [9]) reveal acoustic penetration from ater into sandy sediments at grazing angles belo the compressional critical angle in relation to the mean surface. These authors interpret their results to indicate the excitation of a Biot slo ave at 00 m/s in the sediment. But numerous authors disagree. Analytical derivations sho that roughness of the ater-sediment interface causes propagation of acoustical energy from ater into the sediment at grazing angles belo the compressional critical grazing angle, and simulations indicate that the experimental results can be explained in terms of diffraction of an ordinary longitudinal ave. By modeling sand as a fluid, and including a small amount of roughness, the simulation results match the acoustical penetration experimental results by Boyle and Chotiros (99), both in magnitude and in arrival time [8, 40]. Mellema [7] demonstrated that significant energy is scattered across a rough fluid-solid interface at shallo grazing angles and that the received intensity can be accurately modeled using first order perturbation theory. For the roughened case, it is clear that the later time arrival is best described by an assumed ray path in the ater to a point above the buried hydrophone, scattered from the roughened interface, and then propagation nearly vertically to the buried hydrophone at the speed of the ordinary compressional ave (680 m/s). This can have the appearance of a ave traveling at 00 m/s [34]. Recent field measurements [7, 6, 34] have strongly supported the conclusion that subcritical penetration in sands is due to scattering, not Biot slo aves. To match the 00 m/s slo compression ave using Biot theory, Chotiros [9] used structural and material moduli of sand different than previous authors. Stoll [39] and other authors used moduli that predict a slo ave having a speed of roughly 400

20 7 m/s. Stoll argues Chotiros... is not justified in his claim that a Biot ave of the second kind, ith a ave speed of 00 m/s, had been detected in a near-bottom sand deposit. This opinion is based on the fact that the author (Chotiros) has used an unacceptable value for the bulk modulus of the skeletal frame and a questionable value for the bulk modulus of the individual grains. Chotiros [] replied that the approach advocated by Stoll to calculate bulk moduli orks ell for a porous materials ith cemented frames, such as fused glass beads and sandstone, but it does not ork for sand. Chotiros also contended that the measured reflection coefficient for normal incidence on a ater-sand interface does not match elementary, Rayleigh, viscoelastic reflection theory but his parameters using Biot theory do produce a match ith the measured coefficient..7 ARAMETERS AND ASSUMTIONS A survey of the literature suggests that there are a number of parameters that play a principal role in controlling the dynamic response of saturated sediments. Of these, Stoll suggests the folloing may be important (not necessarily in the order listed) [37]: a. dynamic strain amplitude, b. porosity, c. static intergranular stress, d. gradation of sand size and grain shape, e. material properties of individual grains, f. degree and kind of lithification (bonding of particles), g. structure as determined by the mode of deposition. In his classic development, Biot produced constitutive relationships for fluidsaturated granular media and folloed these ith an analysis of elastic ave propagation in such media by means of a Lagrangian formulation. The assumptions underlying this derivation are that []: a. The medium is isotropic, quasi-homogeneous and that the porosity is uniform throughout;

21 8 b. The pore size is very much less than the avelengths of interest; c. Scattering, in the sense of diffraction around individual grains or particles, can be ignored; d. ore alls are impervious and the pore size is concentrated around an average value; e. Fluid is compressible; f. Fluid may flo relative to the solid..8 NEW CONTRIBUTIONS OF THIS DISSERTATION The contributions of this dissertation are both theoretical and experimental. On the theoretical side the primary contributions include: () Making a lumped parameter model from the continuous parameter Biot theory, fig... () Deriving the fast and slo acoustic impedances, eqs. (-35) and (-37). (3) Deriving pressure reflection and transmission coefficients from impedances and effective pressures, tables. and.. (4) Deriving mode shapes from effective pressures and effective densities, eqs. (4-7) and (4-8). On the experimental side the primary contributions include: (5) A reflection ratio technique in hich reflection amplitudes are divided leaving only information (pressure reflection coefficient) about the surface, e.g., eq. (4-), ithout calibrating the transducers or using geometric corrections. (6) Measuring the fast and slo ave acoustic impedances, fig. 4.. (7) Measuring the fast and slo effective densities, fig. 4.. (8) Bounding the slo acoustic ave speed in an unconsolidated medium, fig (9) Measuring the fast and slo ave effective pressures from an open pore boundary condition, fig (0) Measuring the fast and slo ave mode shapes, fig () And last, measuring the generalized elastic coefficients (intermediate moduli), fig. 4.7.

22 9 Chapter : MATHEMATICAL THEORY This chapter revies the continuous parameter equations of motion and boundary conditions, and then develops a lumped parameter model from hich the fast and slo impedances, moment arms, effective densities, and effective pressures are derived. Then pressure coefficients are derived from impedances and effective pressures. Only normally incident pressure aves, aves propagating perpendicular to the sediment boundary, are considered in the folloing continuous parameter revie and lumped parameter development. This consideration simplifies the equations by removing shear aves (and coupling beteen shear and pressure aves), and is sufficient for analysis of the experimental data discussed in the next chapter.. DIFFERENTIAL EQUATIONS OF MOTION AND BOUNDARY CONDITIONS From Johnson and lona [0], eq. (a) and (b), the homogeneous differential equations of motion ith aves propagating in the z-direction ith no shear displacements are: us z u f Q z - r u& - r u&& Fu& - Fu& 0 (-) s f f s u f R z us Q z - r u& - r u&& - Fu& Fu& 0. (-) s f f s Eq. (-) is the equation of force on sand particles per unit area (area of sand plus area of fluid in sediment) and eq. (-) is the force equation on fluid particles per unit area. The unknons, u s and u f, are displacements (in the z-direction) of the sand and fluid particles., Q, and R are the generalized elastic coefficients (intermediate moduli). ( - b ) Ê K Á - b - Ë K K - b - K b s b s ˆ K K s b K Ks b K s K b 4 3 N (-3)

23 0 Ê Kb ˆ Á - b - K K b s Q Ë Kb K - b - b K K b K s R Kb K - b - b K K s s s s s (-4) (-5) Where b is the porosity (volume of fluid per unit volume of sediment), K b is the bulk modulus of the sand s skeletal frame, K s is the bulk modulus of the sand, K is the bulk modulus of the ater, and N is the shear modulus of the skeletal frame. The density terms are: ( - b ) rs ( a ) br f r (-6) - r abr f (-7) ( a ) br f r -. (-8) - Where r s is the density of the sand grains, r f is density of ater, and a is a virtual mass constant, or tortuosity, expressing the apparent increase in the mass of the fluid. As a fluid particle oscillates in the z-direction it must translate slightly in the x- and y- directions to go around sand particles causing it to appear more massive. The viscous term is: 3 Ê Ê p ˆ ˆ - i J Ák expái3 hb k Ë Ë 4 F (-9) 4k Ê Ê p ˆ ˆ Ê Ê p ˆ ˆ kj 0 Ák expái3 ij Ák expái3 Ë Ë 4 Ë Ë 4 r f k a. (-0) h The viscous term, F, is equivalent to bf, eq. (4.3) in [4], here b is defined by eq. (6.8) in [3]. In eqs. (-9) and (-0), h is the viscosity of ater, k is the permeability of the medium, J and J 0 are Bessel functions, and a is the average radius of the pore holes.

24 Solving eqs. (-) and (-) for plane aves, assume the solid displacements and fluid displacements are: ( i( t kz) ) u s Aexp - (-) ( i( t kz) ) u f B exp -. (-) Here, k is ave number, t is time, A is the sand displacement amplitude and B is the fluid displacement amplitude. ut eqs. (-) and (-) into (-) and (-) and express the results in matrix form: Èk Í ÍÎ k - Q - r r if - if k k Q - R - r r - i F ÏA Ï0 Ì Ì, (-3) i F ÓB Ó0 k È Í ÎQ Q ÏA Ì R ÓB È Í r Í Í r Î F - i F i r r F i ÏA Ì. (-4) F - i ÓB The square matrix on the right hand side of eq. (-4) is the density matrix and ill be referred to later. Solve for the eigenvalues (ave numbers squared) and eigenvectors (mode shapes). This ill give displacements of the sand and fluid in terms of the ave numbers and the mode shapes. ( i( t - k z) ) A exp( i( t k )) u s A exp - z (-5) ( i( t - k z) ) B exp( i( t k )) u f B exp - z (-6) Where the ave fast and slo numbers are: ith a R - Q b - r k c - b - 4 b a - 4ac, b a - 4ac, ( - Rr Qr ) if ( R Q) 3 ( r r - r ) i F ( r - r - r ). k - b (-7) The mode shapes are (ater particle displacement divided by sand particle displacement):

25 F i R k F i Q k F i Q k F i k A B r r r r (-8) F i R k F i Q k F i Q k F i k A B r r r r (-9) The boundary conditions for the ater-sediment (open pore) interface are taken from Johnson et al. [] and specialized to normal incidence for hich there is no excitation of shear aves. Combine equations (.7), (.8), (.9), (.0), (.), and (.5) from ref. [] and get: z u K z u K z u R z u Q z u Q z u r i f s f s. (-0) Eq. (-0) says the force in the sand per unit area of interface (note: not the pressure in the sand because area is of sand and pore ater) plus the force in the sediment fluid per unit area of interface is equal to the pressure in the ater above the sediment. Johnson s W is the same as this dissertation s u i u r. The variables u i and u r are the incident and reflected displacements, assumed to be of the form: ( ) ) ( exp z k t i A u i i - (-) ( ) ) ( exp z k t i A u r r. (-) Combine equations (.8), (.0), and (.3) from ref. [] and get: ˆ Á Ë Ê z u K z u K z u R z u Q r i f s b. (-3) Eq. (-3) says the force in the sediment fluid at the interface is equal to the force in the ater above the sediment fluid per unit area of interface. From equations (.) and (.5) from ref. []: ( ) r i f s u u u u - b b. (-4) Eq. (-4) says the displacement of sand times the area of sand per unit area of interface plus the displacement of sediment fluid times the area of fluid per unit area of interface is equal to the displacements of the incident plus reflected aves in the ater per unit area. This is the so-called open pore boundary condition. Eq. (-4) can be reritten:

26 3 here u u u u, (-5) fast slo ( A ( - ) B b ) exp( i( t k )) z i r u fast b - (-6) ( A ( - ) B b ) exp( i( t k )) u slo b -. (-7) z Eq. (-6) is the average of the sand and fluid particle displacements during the passing of a fast ave. Both previous equations ill be used later to derive participation factors. The closed pore boundary condition for displacements ill be used later and is: here at z 0: u u i u r, (-7a) ( A A ) i t) ( B B ) exp( i ) u us u f exp( t (-7b) This section (equations of motion and boundary conditions) has given the ave numbers, mode shapes, and the boundary conditions for a continuous, degrees-offreedom, Biot medium having a planar interface and normal incidence. The next section develops the lumped parameter model.. LUMED ARAMETER MODEL The lumped parameter method developed here as strongly influenced by the Dynamic Design Analysis Method (DDAM) introduced by Belsheim and O Hara []. DDAM is used to analyze non-contact, bomb shock on Naval ships. See Appendix E. DDAM requires a physical system to be engineered into a lumped mass-spring model. Then the equations of motion are ritten in matrix form from the mass-spring model. The natural frequencies and mode shapes are calculated from the determinant of the eigenvalue problem, and the determinant is alays symmetrical. Likeise, even though Biot s equations of motion are for continuous parameters, the determinant for the ave numbers is symmetrical using Johnson and lona s notation. The inspiration for the lumped parameter model as the question Can Biot s continuous parameter, symmetric determinate be reverse-engineered (through trial-and-error) into a lumped mass-spring

27 4 model? A lumped parameter model has the advantage of alloing easy visualization of movements and draing free body diagrams of the forces. The continuous medium homogeneous differential equations of motion and boundary conditions eqs. (-3), (-0), (-3), and (-4) can be described by a lumped mass, linear spring, concentrated force, viscous damp-pot, rigid-bar mechanical system, see Fig.. and Appendix D. r u s A b - Q _ i 4Q _ R - Q B ater sediment z 0 u f r - r 4 r r - r - Q F 4Q R - Q r - r 4r r - r - Q F 4Q R - Q Figure. Mass-Spring-Force Mechanical Model of the Biot Medium In going from the continuous model to the lumped model, stiffness (N/m) and modulus (a), mass (kg) and density (kg/m 3 ), force (N) and pressure (a), viscous damping (Ns/m) and body viscous damping (Ns/m 4 ), difference in displacement of to points (m) and strain (m/m), acoustic impedance ( a s/m ) and mechanical impedance

28 5 (Ns/m) are used interchangeably. Hopefully this does not cause too much confusion. oint A in Fig.. is the location of the sand (displacement of point A equals the sand particle displacement) and point B is the location of the fluid in the sediment (also the displacement of point B equals the sediment fluid particle displacement). i and r are the incident and reflected forces. To sho that the boundary conditions of the mechanical system of Fig.. are equivalent to Johnson et al. [] boundary conditions dra a free body diagram at z 0, see Fig... In Fig.. the positive direction is don. Eq. (-0) is the sum of forces in the vertical direction. Eq. (-3) is the sum of moments about point A. The incident and reflected displacements, u i and u r, are related to the solid and fluid displacements, u s and u f, by eq. (-4). Fig..3 is equivalent to Fig.. but in Rayleigh notation, i.e., e it suppressed (ref. [3], section 70, eq. ()). r K ur z A b i K ui z B z 0 - us / ( - Q) z Ê u u ˆ Á s f u f - Q - ( R - Q) Ë z z z Figure. Free-Body Diagram of the Boundary

29 6 ik k A r A b -ik k A i i( Q)(k A k A ) / iq(k A k A k B k B ) B z 0 i(r - Q)(k B k B ) Figure.3 Rayleigh Notation of Figure. F A d B A B u s u f / / i( Q)k A iqk (A B ) i(r Q)k B Figure.4 Free-Body Diagram of Fast Wave Mode Shape

30 7 A force can be positioned that ill only excite the fast ave. As ill be seen, this is a consequence of mode orthogonality. See Fig..4. From Fig..4, sum forces in the vertical direction: ( k A Qk A Qk B Rk ) F. (-8) i B The sum of the moments around point A in Fig..4 is: ( Qk A Rk ) d. (-9) F i B From equations (-8) and (-9) the moment arm for a force to excite only mode is: QA RB d. (-30) A QA QB RB Another ay to find d is by knoing that a force applied at d, the moment arm for mode (slo ave), ill cause no ork at d, i.e., the translation at d in the z- direction is zero. See Fig..5. A d (-3) A - B d F -A A d F B B Figure.5 Location of F Using Orthogonality These to relationships for d, equations (-30) and (-3), are equivalent to the orthogonality relationship:

31 8 0 Ó Ì Ï Í Î È Ó Ì Ï B A R Q Q B A T. (-3) From Fig..4 the velocity at point d is: ( ) [ ] ) ( B A i t u i t u i v f s d d d d - -. (-33) The impedance of first mode (fast ave) is: v F (-34) ( ) [ ]. ) ( ææ RB QA B A B Q R A Q k (-35) Similarly: B A A RB QB QA A RB QA - d, (-36) ( ) [ ] ææ ) ( RB B QA A B Q R A Q k. (-37) Because: k K (-38) K k ˆ Á Ë Ê r, (-39) the modal stiffnesses (moduli) and masses (densities) of the first and second modes are: ( ) [ ] ) ( RB B QA A B Q R A Q K (-40) ( ) [ ] ) ( RB B QA A B Q R A Q K (-4) ( ) [ ] ) ( RB B QA A B Q R A Q k ˆ Á Ë Ê r (-4)

32 9 [( Q) A ( R Q) B ] Ê k ˆ r Á. (-43) Ë A QA B RB The folloing to equations are taken from the matrix equation of (-4): ÈÊ F ˆ F ˆ ( A QB) c ÍÁ r - i A Á r i B ÎË Ë ÈÊ F ˆ ( QA RB) c ÍÁ r i A Á r - i B ÎË Ë Ê Ê F ˆ (-44). (-45) Add equations (-44) and (-45) and put in the numerators of (-4) and (-43). Also multiply equation (-44) by A and (-45) by B, add, and put in the denominators of (- 4) and (-43). The results are the modal masses or the effective masses in terms of density (and viscous) parameters: [( r r ) A ( r r ) B ] r (-46) F ra r A B rb - i ( A - B ) [( r r ) A ( r r ) B ] r. (-47) F ra r A B rb - i ( A - B ) These modal masses, eqs. (-46) and (-47), agree ith the Dynamic Design Analysis Method (DDAM), see Appendix E. Also, Clough and enzien [4] p. 559, sho that the sum of the modal masses is equal to the sum of all the terms in the mass matrix, eq. (-4), i.e., r r r r r (-b)r s br f r. Where r is the total density of the ater filled sand. With Johnson and lona s form of the Biot equations, eqs. (-) and (-), the modal masses add up to the total mass, i.e., r r r. The terms that link particle displacements to effective (or modal) displacements are called participation factors in this dissertation. For a single degree-of-freedom system, of course, particle displacements and effective displacements are equal. Transforming average particle displacement of the sand and fluid during the passing of a fast ave, eq. (-6), into a lumped parameter model, Fig..6, shos the average particle

33 0 displacement is located at a distance b from point A. In Fig..6 points and are the locations of the first and second modes and points A and B are the locations of the sand and fluid particles. The symbols u and u are the effective or modal displacements. The fast ave participation factor is: ~ u u d d fast - b. - d And likeise the slo ave participation factor is: ~ u u slo b - d d - d. (-48) (-49) b d d A B u u fast Figure.6 Modal Displacement (u ) and Average article Displacement (u fast ) for the Fast Wave

34 Fig..7 is a mechanical system equivalent to Fig.. but in terms of modal parameters. The acoustic impedances for Biot aves are fully described by the impedances and the impedance moment arms, hich are modal parameters independent of the loading of the mechanical system, i.e., they are homogeneous parameters (in a differential equation sense). The terms that link loading (boundary conditions) and modal parameters are the modal forces or effective forces. In the open pore boundary mentioned earlier, the ater above the sediment and the fluid in the sediment are in direct contact. This is the most commonly used boundary condition for sediments. The terms that link external forces to forces on the first and second mode are called effective forces. The incident and reflected forces act at a distance b from point A for an open pore interface, see Fig..7. Let the incident and red A b i r K d B K r r K K r r Figure.7 Mechanical System Equivalent to Fig.. But In Terms of Modal arameters

35 flected forces equal one then from Fig..8, is the effective force on mode and is the effective force on mode. d - b (-50) d - d b - d. (-5) d - d b A B d d Figure.8 Effective Forces and Note that, numerically, open pore effective forces are equal to participation factors even though their meanings are different. For a closed pore boundary the sediment is in contact ith a non-porous material, like acrylic. In a closed pore boundary the sand and fluid particles and the fast and slo modal displacements move ith the same amplitude at the interface, and the modal points and move equally, see eq. (-7b). The forces needed to move these points equally are the closed pore effective forces. ( c ) (-5) ( c ) (-53) The next section calculates the pressure reflection and transmission coefficients using impedances and effective forces.

36 3.3 REFLECTION AND TRANSMISSION RESSURE COEFFICIENTS FOR A SOUND WAVE IN WATER INCIDENT UON A SAND INTERFACE (OEN ORE) This section finds the reflection and transmission coefficients for various boundary conditions of a medium that supports Biot aves. If the distance from the first to second modes, points and, are normalized to one, the normalized distances from an applied unit force (the force is applied at a position to make the forces on and equal to and ) to the locations of the modes are numerically equal to the normalized effective forces, and, see Fig..9. These normalized forces and normalized distances ill be assigned the same symbols, and. The impedance of the Biot medium is found by applying a force of at a distance from the left in Fig..9. The springs in the figure are labeled as impedances, (units of Ns/m in lumped system). Technically, the spring constants should be ω (N/m), but the frequency terms are dropped. Velocities of the first and second modes are: v v. (-54), The velocity at the point of unit force is: v v v. (-55) The impedance seen by an incident ave from ater to a Biot medium is: v. (-56) The pressure reflection coefficient of a sound ave off a Biot medium (ater to sand) is (here is the impedance of ater): R R. (-57)

37 4 unit force ater sediment unit distance v v The pressure transmission coefficient for the fast and slo aves combined is: T Figure.9 Model Used to Find Acoustic Impedance of Biot Medium s T s. (-58) From the free-body diagram of Fig..0 the fast ave transmission coefficient is: R (Reflected) (Incident) ater sediment T s (Transmitted Fast) T s (Transmitted Slo) Figure.0 Free Body Diagram of Incident, Reflected, and Transmitted Waves

38 5 T s T T. (-59) And the slo ave transmission coefficient is: T s T T. (-60).4 TRANSMISSION AND REFLECTION COEFFICIENTS OF A FAST WAVE FROM SAND TO WATER (OEN ORE) The impedance diagram for a fast ave in sand incident on a ater interface is shon in Fig... The ater particle velocity at a point from the left from unit force at the left is: v. (-6) The slo ave particle velocity is: v. (-6) Therefore the fast ave particle velocity from a unit force at the fast ave is: v v v. (-63) The impedance of a fast ave incident on a sand-ater interface is: v. (-64) The transmitted pressure coefficient from a fast ave into the ater and into the slo ave is: & T

39 6 & T. (-65) From Fig.. the transmitted ave into the ater from an incident fast ave is: & T T T. (-66) The reflected pressure coefficient (reflected fast ave from an incident fast ave, open pore, ater interface) is: () R () R. (-67) (fast) ater sand Figure. Fast Wave Incident on Sand-Water Interface

40 7 T T & (Transmitted) (Incident, Fast) () R (Reflected Fast) ater sediment T & (Reflected Slo) Figure. Free-Body Diagram of Incident and Reflected Forces on Sand-Water Interface The reflected slo ave pressure coefficient is: R ( ) R ( ) T &. (-68).5 TRANSMISSION AND REFLECTION COEFFICIENTS OF A SLOW WAVE FROM SAND TO WATER (OEN ORE) The impedance of a slo ave incident on a sand-ater interface is:. (-69) The transmitted pressure coefficient from a slo ave into the ater and into the fast ave is:

41 8 & T & T. (-70) The transmitted ave into the ater from an incident slo ave is: & T T T. (-7) The reflected pressure coefficient is (reflected slo ave from an incident slo ave, open pore, ater interface): () R () R. (-7) The reflected fast ave pressure coefficient: & ) ( T R ) ( R. (-73).6 REFLECTION AND TRANSMISSION COEFFICIENTS OF A FAST WAVE FROM SAND TO NON-OROUS (CLOSED ORE) INTERFACE Figure.3 is the impedance diagram for a fast ave incident on a non-porous interface that has a closed pore surface (e.g., sand on acrylic, sand on Ethafoam, etc.). The sand particles, ater particles, first mode, and second mode all have the same displacement on a closed pore surface; therefore the fast and slo ave particle velocities, v and v, are the same at the interface. Impedance seen by fast ave ( c is impedance of a closed pore, non-porous medium):

42 9. (-74) The fast, pressure reflection coefficient from a fast incident ave off a closed pore surface is: R R (c) (c) c c. (-75) The transmitted coefficient from a fast incident ave on a closed pore surface is: T c T c c c c. (-76) c (fast) sand c closed pore material Figure.3 Fast Wave Incident on Non-Biot Material The slo, pressure reflection coefficient from a fast incident ave off a closed pore surface is: ( ) R c

43 30 R ( c) (-77) c.7 REFLECTION AND TRANSMISSION COEFFICIENTS OF A SLOW WAVE FROM SAND TO NON-OROUS (CLOSED ORE) INTERFACE The impedance seen by slo ave is:. (-78) The slo, pressure reflection coefficient from a slo incident ave off a closed pore surface is: R R (c) (c) c c. (-79) The transmitted coefficient from a slo incident ave on a closed pore surface is: T T c c c c c. (-80) The fast, pressure reflection coefficient from a slo incident ave off a closed pore surface is: R c R ( c ) ( c). (-8) c

44 3.8 REFLECTION AND TRANSMISSION COEFFICIENTS OF AN INCIDENT WAVE FROM A NON-OROUS MATERIAL TO SAND Figure.4 is the impedance diagram for a fast ave incident on sand from a closed pore surface. The impedance seen by the incident ave is: The transmission of the fast ave is: T c The transmission of the slo ave is: T c. (-8) Reflection back into the closed pore material: R cc c c c c. (-83). (-84). (-85) sand closed pore material Figure.4 Incident Wave from Non-Biot Material

45 3.9 SUMMARY OF RESSURE COEFFICIENTS The Tables. and. summarize the pressure coefficients. The energy balances for these coefficients, i.e., pressure times velocity of the incident ave equals pressure ( ) ( ) ( R ) ( R ) times velocity of the response aves, e.g., T. Also the pres- sures on each side of the interface are equal, i.e., one plus reflected equals transmitted, ( ) ( ) R e.g., R T (given ). Appendix B numerically checks parts of Table.. The closed pore table (Table.) can also be calculated from the open pore table by using the effective pressures for closed pore, eqs. (-5) and (-53). ressure coefficients are used in the next chapters here different combinations of acoustic measurements are taken to extract impedances, effective densities, ave speeds, effective pressures, mode shapes, and intermediate moduli. Only four of the 8 pressure coefficients (R cc, T c, T c, and R ) derived are used in this experiment. The other 4 coefficients are included only for completeness.

46 33 Table. ressure Coefficients, Open ore, Sand-Water Interface Response Wave Fast Slo Water Incident Wave Fast ) ( R ) ( R T Slo ) ( R () R T Water T T R 33

47 34 Table. ressure Coefficients, Closed ore, Sand-NonBiot Interface Response Wave Fast Slo Closed ore Incident Wave Fast (c) R c c ) ( R c c T c c c Slo ) ( R c c (c) R c c T c c c Closed ore c c T c c T c c cc R 34

48 35 Chapter 3: ACOUSTIC MEASUREMENTS The Applied hysics Laboratory has an acoustic ater tank system suitable for measuring pressure amplitudes of aves transmitted and reflected through various media. In this chapter, specifications are revieed, reflection equations are derived, and frequency measurements are discussed. 3. SECIFICATIONS The ater tank is.6 m deep,.3 m diameter, and holds 5000 liters of ater. The tank has a lid that can be sealed and a vacuum pump can be used to de-gas the contents in the tank. The tank has a piping and filter system that can recirculate and purify the ater. A resistivity probe monitors purity and indicates hen the filters need to be changed. With the de-gassing and purifying systems the ater is pure and the sand is bubble free. The tank has a positioning system for the transmitter and receiver ith an accuracy of about 5 µm and a computer based measuring system including a data acquisition system and aveform generator. The data acquisition employs a bit analog-to-digital converter and samples at 0 MHz. An overhead electric crane handles all heavy items going in and coming out of the tank. The pulses ere shaped so that the residual ringing of the transducer as actively canceled (see Appendix G). Return pulses ere indoed ith a Gaussian indo 30 µs ide. Measurements of each surface ere taken at 00 different locations ith 00 samples at each location. The ra data shon in Figs. 3., 3.4, 3.6, and 3.8 are the coherent time averages of 00 samples at 00 locations (0,000 total samples). The sand used as Ottaa sand, ith an average size of 79 µm and a standard deviation of about 00 µm. Appendix H has a more detailed analysis of sand size and composition.