A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media

We introduce a high-order discontinuous Galerkin (dG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in coupled elastic–acoustic media. A velocity–strain formulation is used, which allows for the solution of the acoustic and elastic wave equations within the same unified framework. Careful attention is directed at the derivation of a numerical flux that preserves high-order accuracy in the presence of material discontinuities, including elastic–acoustic interfaces. Explicit expressions for the 3D upwind numerical flux, derived as an exact solution for the relevant Riemann problem, are provided. The method supports h-non-conforming meshes, which are particularly effective at allowing local adaptation of the mesh size to resolve strong contrasts in the local wavelength, as well as dynamic adaptivity to track solution features. The use of high-order elements controls numerical dispersion, enabling propagation over many wave periods. We prove consistency and stability of the proposed dG scheme. To study the numerical accuracy and convergence of the proposed method, we compare against analytical solutions for wave propagation problems with interfaces, including Rayleigh, Lamb, Scholte, and Stoneley waves as well as plane waves impinging on an elastic–acoustic interface. Spectral rates of convergence are demonstrated for these problems, which include a non-conforming mesh case. Finally, we present scalability results for a parallel implementation of the proposed high-order dG scheme for large-scale seismic wave propagation in a simplified earth model, demonstrating high parallel efficiency for strong scaling to the full size of the Jaguar Cray XT5 supercomputer.     

Using streamlines to visualize acoustic energy flow across boundaries

For spherical waves that radiate from a point source in a homogeneous fluid and propagate across a plane boundary into a dissimilar homogeneous fluid, the acoustic field may differ significantly from the geometric acoustic approximation if either the source or receiver is near the interface (in acoustic wavelengths) or if the stationary phase path is near the critical angle. In such cases, the entire acoustic field must be considered, including inhomogeneous waves associated with diffraction (i.e., those components that vanish with increasing frequency). The energy flow from a continuous-wave monopole point source across the boundary is visualized by tracing acoustic streamlines: those curves whose tangent at every point is parallel to the local acoustic intensity vector, averaged over a wave cycle. It is seen that the acoustic energy flow is not always in line with the "Snell's law" or stationary phase path. Also, plots of acoustic energy streamlines do not display unusual behavior in the vicinity of the critical angle. Finally, it is shown that there exists a law of refraction of acoustic energy streamlines at boundaries with density discontinuities analogous to Snell's law of refraction of ray paths across sound speed discontinuities. Examples include water-to-seabed transmission and water-to-air transmission.     

Evidence of Doppler-shifted Bragg scattering in the vertical plane by ocean surface waves

A set of narrowband tones (280, 370, 535, and 695 Hz) were transmitted by an acoustic source mounted on the ocean floor in 10 m deep water and received by a 64-element hydrophone line array lying on the ocean bottom 1.25 km away. Beamformer output in the vertical plane for the received acoustic tones shows evidence of Doppler-shifted Bragg scattering of the transmitted acoustic signals by the ocean surface waves. The received, scattered signals show dependence on the ocean surface wave frequencies and wavenumber vectors, as well as on acoustic frequencies and acoustic mode wavenumbers. Sidebands in the beamformer output are offset in frequency by amounts corresponding to ocean surface wave frequencies. Deviations in vertical arrival angle from specular reflection agree with those predicted by the Bragg condition through first-order perturbation theory using measured directional surface wave spectra and acoustic modes measured by the horizontal hydrophone array.     

Phase velocities of plane waves in a pipe with a flow whose velocity varies along the radius

The phase velocities of plane waves in a pipe filled with a moving acoustic medium are studied for different laws of flow velocity variation along the pipe radius. The wave equation is solved by the discretization method, which breaks the entire pipe volume into individual cylinders under the assumption that, within each of the cylinders, the flow velocity of the medium is constant. This approach makes it possible to reduce the solution to the wave problem to solving Helmholtz equations for individual cylinders. Based on boundary conditions satisfied at the boundaries between neighboring cylinders, a homogeneous system of linear algebraic equations is obtained. From this system, with the use of the scattering matrices, a simple dispersion equation is derived for determining the phase velocities of plane waves. The stability of the numerical solution to the dispersion equation with respect to the number of cylinders is investigated. The phase velocities of quasi-homogeneous and inhomogeneous waves in a pipe are numerically calculated and analyzed for different velocities of a moving medium and different laws of flow velocity variation along the radius. It is shown that the variation that occurs in the phase velocity of a homogeneous plane wave in a pipe due to the motion of the medium is identical to the mean flow velocity for different laws of flow velocity variation along the radius. For inhomogeneous plane waves, the phase velocity increment exceeds the mean flow velocity several times and depends on both the law of wave amplitude distribution along the radius and the law of the flow velocity variation along the radius.     

Reflection of sound at area expansions in a flow duct

An analytical model for scattering at area discontinuities and sharp edges in flow ducts and pipes is presented. The application we have in mind is large industrial duct systems, where sound attenuation by reactive and absorptive baffle silencers is of great importance. Such devices commonly have a rectangular cross-section, so the model is chosen as two-dimensional. Earlier solutions to this problem are reviewed in the paper. The modelling of the flow conditions downstream of the area expansion, with and without extended edges, and its implications for the resulting acoustic modes are discussed. Here, the scattering problem is solved with the Wiener-Hopf technique, and a Kutta condition is applied at the edge. The solution of the wave equation downstream of the expansion includes hydrodynamic waves, of which one is a growing wave. Theoretical results are compared with experimental data for the reflection coefficient for the plane wave, at frequencies below the cut-on for higher order modes. Influence of the interaction between the sound field and the flow field is discussed. A region where the reflection coefficient is strongly Strouhal number dependent is found.     

An Investigation of the Effects of Internal Waves on Sound Propagation in a Stratified Medium with a Sloping Bed

Internal waves usually cause temporal and spatial changes of density and consequently affect the acoustic wave propagation in the ocean. The purpose of this study is a laboratory investigation of the effects of internal waves generated by oscillation of a cylinder in a large stratified glass tank with a sloping bed on the sound waves propagation. Results showed that sound waves are affected by internal waves that depend on the slope angle to the direction of internal wave propagation angle ratio. When the ratio is subcritical or supercritical, the acoustic signal is much reduced as compared to the case with no sloped bottom. This can be explained in terms of the internal waves energy reaching the sloped bed and their reflections.     

Finite difference computation of acoustic scattering by small surface inhomogeneities and discontinuities

The use of finite difference schemes to compute the scattering of acoustic waves by surfaces made up of different materials with sharp surface discontinuities at the joints would, invariably, result in the generations of spurious reflected waves of numerical origin. Spurious scattered waves are produced even if a high-order scheme capable of resolving and supporting the propagation of the incident wave is used. This problem is of practical importance in jet engine duct acoustic computation. In this work, the basic reason for the generation of spurious numerical waves is first examined. It is known that when the governing partial differential equations of acoustics are discretized, one should only use the long waves of the computational scheme to represent or simulate the physical waves. The short waves of the computational scheme have entirely different propagation characteristics. They are the spurious numerical waves. A method by which high wave number components (short waves) in the wave scattering process is intentionally removed so as to minimize the scattering of spurious numerical waves is proposed. This method is implemented in several examples from computational aeroacoustics to illustrate its effectiveness, accuracy and efficiency. This method is also employed to compute the scattering of acoustic waves by scatterers, such as rigid wall acoustic liner splices, with width smaller than the computational mesh size. Good results are obtained when comparing with computed results using much smaller mesh size. The method is further extended for applications to computations of acoustic wave reflection and scattering by very small surface inhomogeneities with simple geometries.     

Spectral and modal formulations for the Doppler-shifted field scattered by an object moving in a stratified medium

Spectral and normal mode formulations for the three-dimensional field scattered by an object moving in a stratified medium are derived using full-field wave theory. The derivations are based on Green's theorem for the time-domain scalar wave equation and account for Doppler effects induced by target motion as well as source and receiver motion. The formulations are valid when multiple scattering between the object and waveguide boundaries can be neglected, and the scattered field can be expressed as a linear function of the object's plane wave scattering function. The advantage of the spectral formulation is that it incorporates the entire wave number spectrum, including evanescent waves, and therefore can potentially be used at much closer ranges to the target than the modal formulation. The normal mode formulation is more computationally efficient but is limited to longer ranges. For a monochromatic source that excites N incident modes in the waveguide, there will be roughly N2 distinct harmonic components in the scattered field. The Doppler shifts in the scattered field are highly dependent upon the waveguide environment, target shape, and measurement geometry. The Doppler effects are illustrated through a number of canonical examples.     

Theoretical and numerical calculations for the time-averaged acoustic force and torque acting on a rigid cylinder of arbitrary size in a low viscosity fluid

In this paper, theoretical calculations as well as numerical simulations are performed for the time-averaged acoustic force and torque on a rigid cylinder of arbitrary size in a fluid with low viscosity, i.e., the acoustic boundary layer is thin compared to the cylinder radius. An exact analytical solution and its approximation are proposed in the form of an infinite series including Bessel functions. These solutions can be evaluated easily by a mathematical software package such as mathematica and matlab. Three types of incident waves, plane traveling wave, plane standing wave, and dual orthogonal standing waves, are investigated in detail. It is found that for a small particle, the viscous effects for an incident standing wave may be neglected but those for an incident traveling wave are notable. A nonzero viscous torque is experienced by the rigid cylinder when subjected to dual orthogonal standing waves with a phase shift even when the cylinder is located at equilibrium positions without imposed acoustic forces. Furthermore, numerical simulations are carried out based on the FVM algorithm to verify the proposed theoretical formulas. The theoretical results and the numerical ones agree with each other very well in all the cases considered.     

Acoustic wave scattering from a composed shell reinforced by a single rib: characteristics of peripheral waves

The steady-state axisymmetrical problem of a plane acoustic wave scattering from a composed shell is considered. The shell has a cylindrical part and two hemispherical endcaps. The rib is a ring of rectangular cross-section that divides the shell into two equal parts. The motion of the shell and the rib is described by the equations of elasticity theory, and the liquid is described by the Helmholtz equation. The solution is obtained numerically by a coupled finite element/boundary element model. Two peripheral waves are generated in the shell: the membrane S0 wave and the bending type water-borne A wave. The form function, acoustic spectrogram and dispersion curves of the phase velocities are presented, and the effect of the rib on the peripheral waves is discussed.     

Simulation of the vector wave field of a low-frequency sound source in the ocean: Some important features

An approach to the simulation of low frequency vector wave fields in stratified media (mainly in the ocean) is considered. The approach is characterized by an improved stability with respect to dividing the medium into many layers of arbitrary thickness. The model for the sound field of a point source is based on an integral representation of two-dimensional, cylindrically symmetric vector wave fields in inhomogeneous media, so that the contributions of all types of waves are included automatically. The model medium is subdivided into N horizontally homogeneous layers for which 4(N?1) equations are formulated to satisfy the boundary conditions between adjacent layers. The method of the generalized Schmidt matrix is used to obtain the coefficients of the equations; these coefficients are substituted into the expressions (of the Fourier-Bessel integral type) for the local parameters of the field. The latter are calculated according to the numerical procedure, and the results are used to model the distributions of the acoustic pressure and the horizontal and vertical components of the particle velocity in liquid and elastic media. The instability of the calculation procedure may result in a disagreement between the model and the exact solution. However, the disagreement is shown to occur mainly in models containing excessively thick layers. A way for improving the stability of the numerical model is suggested. The simulation results are compared with the exact analytical solution for the simplest example and with the results obtained according to the commonly used generalized matrix procedure (the benchmark problem). The examples of the practical application of the model for investigating more complex seismoacoustic wave fields in the ocean are presented.     

Underwater acoustic scattering from a radially layered cylindrical obstacle in a 3D ocean waveguide

A solution based on coupled mode expansions is presented for the 3D problem of acoustic scattering from a radially layered penetrable cylindrical obstacle in a shallow-water plane-horizontal waveguide. Each cylindrical ring is characterized by a general, vertical sound speed and density profile (ssdp), the ocean environment around the obstacle can be also considered horizontally stratified with a depth-arbitrary ssdp, and the bottom is assumed to be rigid. The total acoustic field generated by an harmonic point source is represented as a normal-mode series expansion. The expansion coefficients are calculated exploiting the matching conditions at the cylindrical interfaces, which results in an infinite linear system. The system is appropriately truncated and numerically solved by using a recursive relation, which involves the unknown coefficients of two successive rings. Results concerning the transmission loss outside and inside obstacles consisting of three cylindrical rings are given for a typical depth-dependent ocean sound-speed profile. The presented solution can serve as a benchmark solution to the general problem of 3D acoustic scattering from axisymmetric inhomogeneities in ocean waveguides at low frequencies.     

Acoustic propagation from a spiral wave front source in an ocean environment

A spiral wave front source generates a pressure field that has a phase that depends linearly on the azimuthal angle at which it is measured. This differs from a point source that has a phase that is constant with direction. The spiral wave front source has been developed for use in navigation; however, very little work has been done to model this source in an ocean environment. To this end, the spiral wave front analogue of the acoustic point source is developed and is shown to be related to the point source through a simple transformation. This makes it possible to transform the point source solution in a particular ocean environment into the solution for a spiral source in the same environment. Applications of this transformation are presented for a spiral source near the ocean surface and seafloor as well as for the more general case of propagation in a horizontally stratified waveguide.     

Elastohydrodynamic problems in quasicrystal elasticity theory and wave propagation

Elastohydrodynamic problems of decagonal quasicrystals are analysed where the phonon field obeys wave equation and the phason field obeys diffusive wave equation. Basic equations are solved in the quasiperiodic plane and periodic plane, respectively. Final governing equations of dynamic behaviours of decagonal quasicrystals are obtained. A general solution is derived in terms of introduced three auxiliary functions, where two individually satisfy a fourth-order partial differential equation and one satisfies a second-order hyperbolic diffusion equation. Using the derived governing equations, elastic waves propagating in the quasiperiodic plane and a plane containing the period axis are analysed. Secular equations are obtained. It is found that differing from conventional crystals, at least four branches of elastic waves exist when the phonon–phason coupling is present. Moreover, acoustic waves have attenuation during wave propagation. Phason fluctuations exhibit exponential decaying behaviour due to kinematic viscosity and damping. The phase speeds are isotropic in the quasiperiodic plane and anisotropic in a plane with the periodic axis. The section of the slowness surfaces is plotted.     

On coupled fields in warm stratified plasmas

This paper deals with small-amplitude coupled electromagnetic and electron acoustic waves, as described by Maxwell's equation and single-fluid hydrodynamics, in a horizontally stratified, continuously varying, warm electron plasma. First-order coupled wave equations are used to investigate the fields in a coupling region, and the results are compared with those obtained from second-order coupled wave equations. All field components are found to be finite in the coupling region.Nedlands, Western Australia, 6009.     

The far field asymptotics in the problem of diffraction of an acoustic plane wave by an impedance cone

The work deals with the far field asymptotics of the classical solution for the problem of diffraction by an impedance cone. The incident acoustic plane wave completely illuminates the semi-infinite conical surface. The scattered field contains different components in the asymptotics, namely, the spherical wave from the vertex of the cone, the reflected waves, and, under some conditions, also the surface waves of Rayleigh type. We give integral representations for the scattering diagram of the spherical wave. The uniform (with respect to the observation direction) asymptotic expression for the wave field is also addressed and described by the parabolic cylinder ansatz. Dedicated to the memory of Vladimir Borovikov     

Axial time-averaged acoustic radiation force on a cylinder in a nonviscous fluid revisited

Objective

The present research examines the acoustic radiation force of axisymmetric waves incident upon a cylinder of circular surface immersed in a nonviscous fluid. The attempt here is to unify the various treatments of radiation force on a cylinder with arbitrary radius and provide a formulation suitable for any axisymmetric incident wave.

Method and results

Analytical equations are derived for the acoustic scattering field and the axial acoustic radiation force. A general formulation for the radiation force function, which is the radiation force per unit energy density per unit cross-sectional surface, is derived. Specialized forms of the radiation force function are provided for several types of incident waves including plane progressive, plane standing, plane quasi-standing, cylindrical progressive diverging, cylindrical progressive converging and cylindrical standing and quasi-standing diverging waves (with an extension to the case of spherical standing and quasi-standing diverging waves incident upon a sphere).

Significance and some potential applications

This study may be helpful essentially due to its inherent value as a canonical problem in physical acoustics. Potential applications include particle manipulation of cylindrical shaped structures in biomedicine, micro-gravity environments, fluid dynamics properties of cylindrical capillary bridges, and the micro-fabrication of new cylindrical crystals to better control light beams.     

Exact solution of three-dimensional acoustic field in a wedge with perfectly reflecting boundaries

An exact approach is presented to compute the three-dimensional(3D) acoustic field in a homogeneous wedge-shaped ocean with perfectly reflecting boundaries. This approach applies the Fourier synthesis technique, which reduces a 3D point-source ideal wedge problem into a sequence of two-dimensional(2D) line-source ideal wedge problems, whose analytical solution is well established. A comparison of numerical efficiency is provided between this solution and the solution proposed by Buckingham,which is obtained by a sequence of integral transforms. The details of numerical implementation of these two solutions are also given. To validate the present approach and at the same time compare numerical efficiency between this approach and Buckingham's analytical solution, two numerical examples are considered. One is the Acoustical Society of America(ASA) benchmark wedge problem and the other is a wide-angle wedge problem. Numerical results indicate that the present approach is efficient and capable of providing accurate 3D acoustic field results for arbitrary receiver locations, and hence can serve as a benchmark model for sound propagation in a homogeneous wedge-shaped ocean.     

Physical optics theory of resistive surfaces

The physical optics current is obtained from the exact solution of the scattering problem of plane waves by a resistive surface. The edge point method is used for the determination of the physical optics surface current. The derived physical optics integral by considering the new surface current enables one to evaluate the scattering problems by various resistive surfaces with edge discontinuities. The method is applied to the diffraction problem of plane waves by a concave cylindrical reflector. The scattered fields are examined numerically.     

A vertical eigenfunction expansion for the propagation of sound in a downward-refracting atmosphere over a complex impedance plane

The propagation of sound in a stratified downward-refracting atmosphere over a complex impedance plane is studied. The problem is solved by separating the wave equation into vertical and horizontal parts. The vertical part has non-self-adjoint boundary conditions, so that the well-known expansion in orthonormal eigenfunctions cannot be used. Instead, a less widely known eigenfunction expansion for non-self-adjoint ordinary differential operators is employed. As in the self-adjoint case, this expansion separates the acoustic field into a ducted part, expressed as a sum over modes which decrease exponentially with height, and an upwardly propagating part, expressed as an integral over modes which are asymptotically (with height) plane waves. The eigenvalues associated with the modes in this eigenfunction expansion are, in general, complex valued. A technique is introduced which expresses the non-self-adjoint problem as a perturbation of a self-adjoint one, allowing one to efficiently find the complex eigenvalues without having to resort to searches in the complex plane. Finally, an application is made to a model for the nighttime boundary layer.