Volume 106, Issue 1, July 1999
 GENERAL LINEAR ACOUSTICS [20]


Highresolution finitevolume methods for acoustic waves in periodic and random media
View Description Hide DescriptionHighresolution numerical methods originally developed for shock capturing in the context of nonlinear conservation laws are found to be very useful for solving acoustics problems in rapidly varying heterogeneous media. These methods are based on solving Riemann problems at the interface between grid cells, which resolve waves into transmitted and reflected components at each interface. The wavepropagation method developed in R. J. LeVeque [J. Comput. Phys. 131, 327–353 (1997)] and implemented in the CLAWPACKsoftware package is tested on several acoustics problems with periodic or random media in one and two space dimensions. A new limiter function is presented for solving problems in a periodic medium where numerical instabilities are observed with standard limiters.

Diffraction of an acoustic plane wave by a rectangular hole in an infinitely large rigid screen
View Description Hide DescriptionThe method of the Kobayashi potential (KP) is applied to evaluate an acoustic field diffracted by a rectangular hole in an infinite rigid screen. The screen thickness is assumed to be negligibly small. The KP method yields an eigenfunction expansion of the present geometry, and the expansion coefficients are obtained from the matrix equation. The intensity pattern of the far diffracted field, the velocity distribution on the hole, and the transmission coefficient are numerically evaluated. The intensity pattern results are compared with those obtained from the Kirchhoff’s diffraction integral. The agreement between them is fairly good. Since no existing method has provided transmission coefficients for a rectangular hole in a rigid screen, these results are compared with the total scattering cross section of the soft disk, which is analogous to a circular hole in a rigid screen. Their frequency characteristics are found to show a similar variation.

Threedimensional transducer voltage in anisotropic materials characterization
View Description Hide DescriptionAir and fluidcoupled ultrasound has been used to characterize experimentally the elastic behavior of composite materials by employing a 3D voltage calculation to model transmission or reflection experiments. With sound propagation along material symmetry directions, integration over the incidentplane angle alone is generally sufficient to model the transducer voltage accurately, where a saddlepoint calculation is used to evaluate the dependence of the diffraction integral on the outofincidentplane angle . In general material directions, however, this integration must be extended to account for dependent variations in the reflection or transmission coefficient that give the scattering a strong, asymmetrical dependence on . This article presents theoretical and experimental results to illustrate this effect and explores the relationship between the 2D and 3D calculations. Experimental results demonstrating viscoelastic property reconstruction in composite plates in both aircoupled and liquidcoupled measurements are presented. It is shown that the influence of the dependent integration on the voltage, embodied in the 3D calculation, is particularly strong when the incident angles are small and the wave paths are large, as typically experienced in aircoupled measurements.

Vortical and acoustical mode coupling inside a twodimensional cavity with transpiring walls
View Description Hide DescriptionIn a long, low aspect ratio, twodimensional cavity, where gaseous motion is permitted along transpiring walls, a timedependent field is established when low amplitude, sinusoidal pressure oscillations with nonzero mean are introduced. An accurate solution is extracted here for the timedependent field by way of small parameter perturbations. Contingent upon small pressurewave amplitudes, Navier–Stokes equations are linearized to the order of the mean flowMach number to furnish interaction equations governing the unsteady field. The latter is decomposed into acoustic and solenoidal fields coupled through Dirichlettype boundary conditions. Solving for the solenoidal field from the momentum equation employs separation of variables and multiple scale expansions based on a careful choice of an inner scale. In fact, the unique inner scale used in the twovariable derivative expansion method is original in the sense that it stems from an unconventional, nonlinear variable transformation. A uniformly valid solution is formulated subsequently for the temporal field. This explicit solution discloses the character of the acoustic boundary layer evolving from damped traveling waves. The rate of decay is found to depend on a viscosity parameter, revealing that deeper penetration of rotational waves is possible at low viscosity. Characterization of the boundary layer region is covered in addition to a standard error analysis. In closing, results are verified through comparisons to accurate numerical predictions.

Lowfrequency shear wave propagation in periodic systems of alternating solid and viscous fluid layers
View Description Hide DescriptionWaves in periodic layered systems at low frequencies can be studied using an asymptotic analysis of the Rytov’s exact dispersion equations. Since the wavelength of the shear wave in the fluid (viscous skin depth) is much smaller than the wavelength of the shear wave in the solid, the presence of viscous fluid layers requires a consideration of higher terms in the asymptotic expansions. For a shear wave with the directions of propagation and of particle motion in the bedding plane, the attenuation (inverse Q) obtained by this procedure is where is frequency, is weighted average density of the solid/fluid system, is the thickness of fluid layers, is fluid volume fraction, i.e., the “porosity,” is solid shear modulus, and and are the density and viscosity of the fluid, respectively. The term proportional to is responsible for the viscous shear relaxation, while the term proportional to accounts for the viscoinertial (poroelastic) attenuation of Biot’s type. This result shows that the characteristic frequencies of viscoelastic poroelastic and scattering attenuation mechanisms obey the relation which explains why the viscoelastic and poroelastic mechanisms are usually treated separately in the context of macroscopic theories that imply The poroelastic mechanism dominates over the viscoelastic one when the frequencyindependent parameter and vice versa.

Lowfrequency slowwave dispersion computations by compoundmatrix propagation
View Description Hide DescriptionSlow PSV modes whose horizontal slowness tends to infinity while the horizontal wave number tends to zero, as the frequency tends to zero, exist in certain laterally homogeneous fluidsolid media. These modes can be characterized by an asymptotic analysis of the dispersion function. Only certain powers of frequency are possible for the asymptotic increase of the horizontal slowness as the frequency tends to zero: −1/3, −1/2, −3/5, and −2/3. In order to investigate the accuracy of the asymptotic predictions, dispersioncurve computations by propagator techniques are attempted for media composed of homogeneous fluid and solid layers. However, numerical precision is lost by cancellation effects for the elements of the solidlayer compoundmatrix propagators that are involved. Guided by the asymptotic growth of these compoundmatrix elements, cancellationfree expressions are derived for applications to the slow modes at very low frequencies. The harmful contributions causing loss of numerical precision are eliminated analytically. To demonstrate the success, a numerical case study is performed leading to conjectures about the nexttoleading terms in lowfrequency asymptotic expansions of the slowmode modal slownesses.

Greenspan acoustic viscometer: Numerical calculations of fields and ductend effects
View Description Hide DescriptionInertial and resistive end corrections for the Greenspan acoustic viscometer were computed using a boundaryintegralequation technique for determination of the acoustic field. Viscouseffects were estimated using a boundarylayer approximation. The results apply to a circular duct coupling two concentric chambers and to ducts terminated by infinite plane baffles. The effects of rounding the sharp edge at the duct end were investigated and found to be described by simple scaling relations.

Wave motion in an isotropic elastic layer generated by a timeharmonic point load of arbitrary direction
View Description Hide DescriptionWave motion in an infinite elastic layer due to the application of a timeharmonic point load of arbitrary direction, applied either internally or on one of the faces of the layer, is expressed as a sum of four expansions in Lambwave modes and horizontally polarized wave modes. The point load is decomposed into components normal and parallel to the plate faces. Each of these cases is decomposed into a symmetric and an antisymmetric loading case, relative to the midplane of the layer. The displacement solutions for the symmetric and antisymmetric cases are expressed as expansions of symmetric and antisymmetric modes, respectively. Appropriate orthogonality relations are derived from reciprocity considerations. Elastodynamic reciprocity is also used in conjunction with dummy wave modes to obtain the coefficients in the wavemode expansion.
