Journal of Theoretical and Computational Acoustics

After the parabolic equation method was initially applied to scalar wave propagation problems in ocean acoustics and seismology, it took more than a decade before there was any substantial progress in extending this approach to problems involving solid layers. Some of the key steps in the development of the elastic parabolic equation are discussed here. The first breakthrough came in 1985 with the discovery that changing to an unconventional set of dependent variables makes it possible to factor the operator in the elastic wave equation into a product of outgoing and incoming operators. This innovation, which included an approach for handling fluid-solid interfaces, was utilized in the first successful implementations of the elastic parabolic equation less than five years later. A series of papers during that period addressed the issues of accuracy and stability, which require special attention relative to the scalar case. During the 1990s, the self-starter made it possible to handle all types of waves, rotated rational approximations of the operator square root made it possible to handle relatively thin solid layers, and there was some progress in the accurate treatment of sloping interfaces. During the next decade, an improved formulation and approach for handling interfaces facilitated the treatment of piecewise continuous depth dependence and sloping interfaces. During the last 10 years, the accuracy of the elastic parabolic equation was improved and tested for problems involving sloping interfaces and boundaries, and this approach was applied to Arctic acoustics and other problems involving thin layers. After decades of development, the elastic parabolic equation has become a useful tool for a wide range of problems in seismology, seismo-acoustics, and Arctic acoustics, but possible directions for further work are discussed.

A possible medium is considered with a large number of diverse but possibly similar relaxation processes. The equations for single spatially located relaxation processes of general type are developed and extended to a large spatially extended system. Each process is characterized by a relaxation time and a relaxation strength and is excited by the local pressure in an incident acoustic wave. A constant frequency model is initially assumed with complex amplitudes associated with acoustic pressure and relaxation responses. A smearing process results in the collective relaxation responses being regarded as a continuous function of relaxation times. An expression for entropy perturbation is developed in terms of the relaxation responses. The equations of fluid mechanics then lead to an expression for the pressure perturbation in terms of the dilatation and the relaxation processes. The extended result yields a time-domain wave equation for propagation through an inhomogeneous medium with distributed relaxation processes. Expressions for frequency-dependent attenuation and phase velocity are derived in which relaxation is characterized by a single function giving strength as a continuous function of relaxation time. Possible choices for this function are discussed, and it is shown that some choices lead to an attenuation varying nearly as a power of the frequency over any fixed and possibly large range of frequencies. A model set of parameters leads to good agreement with attenuation and phase velocity measurements for a suspension of human red blood cells in saline solution, as communicated to the authors by Treeby and analogous to those reported by Treeby, Zhang, Thomas, and Cox.

The numerical solution of the elastodynamic problem with kinematic boundary conditions is considered using mixed finite elements for the space discretization and a staggered leap-frog scheme for the discretization in time. The stability of the numerical scheme is shown under the usual CFL condition. Using the general form of Robin-type boundary conditions some cases of debonding and the resulting acoustic emission are studied. The methodology presented finds applications to geophysics such as in seismic waves simulation with dynamic rupture and energy release. In this paper, we focus on problems of fracture occurring at the interface of composite materials. Our goal is to study both the mechanism of crack initiation and propagation, as well as the acoustic emission of the released structure-borne energy which may be used in structural health monitoring and prognosis applications.

Surface waves have been extensively studied in earthquake seismology. Surface waves are trapped near an infinitely large surface. The displacements decay exponentially with depth. These waves are also named Rayleigh and Love waves. Surface waves are also used for nondestructive testing of surface defects. Similar waves exist in finite width three-dimensional plates. In this case, displacements are no longer constant in the direction perpendicular to the wave propagation plane. Wave energy could still be trapped near the edge of the three-dimensional plate, and hence the term edge waves. These waves are thus different to the two-dimensional Rayleigh and Love waves. This paper presents a numerical model to study dispersion properties of edge waves in plates. A two-dimensional semi-analytical finite element method is developed, and the problem is closed by a perfectly matched layer adjacent to the edge. The numerical model is validated by comparing with available analytical and numerical solutions in the literature. On this basis, higher order edge waves and mode shapes are presented for a three-dimensional plate. The characteristics of the presented edge wave modes could be used in nondestructive testing applications.

The application of a boundary element technique in combination with a contour integral approach to the numerical analysis of acoustic resonances in exterior configurations is investigated in this paper. Similar to the boundary element analysis of exterior acoustic radiation or scattering problems, spurious eigenfrequencies also turn up in the boundary element solution to exterior acoustic resonance problems. To filter out the spurious eigenfrequencies, the Burton–Miller-type combined formulation is employed to shift them from the real axis to the complex domain. The shifting effect brought by the combined formulation with different types of coupling parameters is investigated. Unlike in acoustic radiation and scattering analyses for which $-\mathrm{\text{i}}/k$ is suggested as the coupling parameter, it will be shown that the coupling parameter specified as $\beta \cdot \mathrm{\text{i}}/k$ with $\beta >0$ (the time-dependent term herein is ${\mathrm{\text{e}}}^{-\mathrm{\text{i}}\omega t}$) is more desirable in distinguishing the spurious eigenfrequencies in the boundary element analysis of exterior acoustic resonances.

Due to the dispersion of normal modes in shallow water, there exist regular striations on the interference spectrogram in space-frequency domain. Based on the range-frequency domain striations, waveguide invariant can be estimated with the prior knowledge of source range. Utilizing striations along the array, Rouseff and Zurk [J. Acoust. Soc. Amer. 130 (2011) EL76–EL81] proposed the striation-based beamforming (SBF), which shows a theoretical possibility to estimate waveguide invariant that is independent of source range. To make this beamforming technique more practical, its application requirements and performance limitation should be defined. These two issues are researched with the phase and amplitude relationships deduced by the frequency-shift compensation theory. It shows the source whose spectrum varies slowly and received signals that contain the complete information of transmission time are keys for applying SBF to real work. The azimuth limitation can lead to the degradation of array gain and distortion of interference structure. Matching parameters of waveguide invariant and source range, respectively, two modified striation-based beamformers are developed to eliminate the azimuth limitation. Simulation and experimental results are presented.