Time-harmonic Point Source Research Articles

Dynamic Green’s Functions for an Infinite Acoustic Field with Multiple Spheres Subjected to the Robin Boundary Conditions

A semi-analytical approach is presented to solve three-dimensional dynamic Green’s function for an infinitely extended acoustic field with multiple spheres subjected to the Robin boundary conditions. The multipole expansions of the acoustic field induced by a time-harmonic point source are expanded with spherical wave functions. As an alternative to the complex addition theorem, the multipole expansion is computed in a straightforward way. By taking the finite number of terms, an algebraic system is constructed and is used to obtain Green’s function. This result of one sphere agrees with the available analytical solution. For the case of more than one sphere, the proposed results are verified by the numerical method such as the boundary element method (BEM). It indicates that the present solution is more accurate than that of the BEM and shows a fast convergence. Finally, the parameter study is performed to explore the influences of the exciting frequency of the point source, the surface admittance, the number and the separation of spheres on the dynamic Green’s functions. The proposed results can be applied to solve the acoustic scattering problems and to increase the application of boundary integral equation method in the way of numerical Green’s functions.

Localized waves in elastic plates with perturbed honeycomb arrays of constraints.

In this paper, we study wave propagation in elastic plates incorporating honeycomb arrays of rigid pins. In particular, we demonstrate that topologically non-trivial band-gaps are obtained by perturbing the honeycomb arrays of pins such that the ratio between the lattice spacing and the distance of pins is less than 3; conversely, a larger ratio would lead to the appearance of trivial stop-bands. For this purpose, we investigate band inversion of modes and calculate the valley Chern numbers associated with the dispersion surfaces near the band opening, since the present problem has analogies with the quantum valley Hall effect. In addition, we determine localized eigenmodes in strips, repeating periodically in one direction, that are subdivided into a topological and a trivial section. Finally, the outcomes of the dispersion analysis are corroborated by numerical simulations, where a time-harmonic point source is applied to a plate with finite arrays of rigid pins to create localized waves immune to backscattering.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.

Elastic Surface Waves Induced by Internal Sources

The paper is focused on the surface wave field induced by an internal time-harmonic point source embedded in the elastic half space. By using the superposition principle, we first analyze the disturbances caused by the embedded source in an unbounded half space. The problem is then reformulated in terms of the discrepant stresses on the surface of a homogeneous half space. Our analysis is based on the hyperbolic- elliptic asymptotic model of surface elastic waves neglecting the contribution of the bulk waves. Explicit results for the contribution of surface waves are obtained, including the arising frequency- dependent factor.

Analytical Lie-algebraic solution of a 3D sound propagation problem in the ocean

The problem of sound propagation in a shallow sea with variable bottom slope is considered. The sound pressure field produced by a time-harmonic point source in such inhomogeneous 3D waveguide is expressed in the form of a modal expansion. The expansion coefficients are computed using the adiabatic mode parabolic equation theory. The mode parabolic equations are solved explicitly, and the analytical expressions for the modal coefficients are obtained using a Lie-algebraic technique.

Imaging of Local Rough Surfaces by the Linear Sampling Method with Near-Field Data

Consider the scattering of a time-harmonic point source from an unbounded rough surface. The surface is assumed to be different from a plane surface over a finite interval, which leads to the situation that the model problem can be reduced to an equivalent boundary value problem with compactly supported boundary data. Well-posedness is thus proved by employing the variational method with classical Fredholm theory. By constructing a modified near-field equation, we then propose a novel sampling-type method for reconstructing the shape and location of the local rough surface, yielding a fast imaging algorithm. Numerical experiments are presented to illustrate the effectiveness of the method.

Coupled mode theory of scattering by a cylindrically symmetric seamount

The coupled-mode equations of Shevchenko are extended to three-dimensional irregular acoustic waveguides. These equations are solved for a model of a cylindrically symmetric seamount and a time-harmonic point source in a horizontally stratified ocean. An algebraic formula for the scattered field beyond the seamount is obtained from this solution. The formula is characterized by a set of infinite-dimensional matrices that are independent of the source. A stable numerical procedure is developed to compute finite-dimensional approximations to these matrices for use in truncated versions of the formula.

High frequency sound propagation in a network of interconnecting streets

We propose a new model for the propagation of acoustic energy from a time-harmonic point source through a network of interconnecting streets in the high frequency regime, in which the wavelength is small compared to typical macro-lengthscales such as street widths/lengths and building heights. Our model, which is based on geometrical acoustics (ray theory), represents the acoustic power flow from the source along any pathway through the network as the integral of a power density over the launch angle of a ray emanating from the source, and takes into account the key phenomena involved in the propagation, namely energy loss by wall absorption, energy redistribution at junctions, and, in 3D, energy loss to the atmosphere. The model predicts strongly anisotropic decay away from the source, with the power flow decaying exponentially in the number of junctions from the source, except along the axial directions of the network, where the decay is algebraic.

The radiation power of elastic waves excited in a solid half-space by a subsurface time-harmonic source

The method of Fourier transforms is used to solve the problem of excitation of longitudinal, transverse, and Rayleigh surface waves by a time-harmonic point source placed in a homogeneous isotropic, perfectly elastic half-space and acting along the normal surface. Expressions for the time-average radiation powers of the aforementioned waves are obtained by the method of radiation reaction without using any approximations. The distribution of radiation power over different types of waves depending on their velocities and the source’s depth is investigated in detail.

Analysis of sponge zones for computational fluid mechanics

The use of sponge regions, or sponge zones, which add the forcing term − σ( q − q ref) to the right-hand-side of the governing equations in computational fluid mechanics as an ad hoc boundary treatment is widespread. They are used to absorb and minimize reflections from computational boundaries and as forcing sponges to introduce prescribed disturbances into a calculation. A less common usage is as a means of extending a calculation from a smaller domain into a larger one, such as in computing the far-field sound generated in a localized region. By analogy to the penalty method of finite elements, the method is placed on a solid foundation, complete with estimates of convergence. The analysis generalizes the work of Israeli and Orszag [M. Israeli, S.A. Orszag, Approximation of radiation boundary conditions, J. Comp. Phys. 41 (1981) 115–135] and confirms their findings when applied as a special case to one-dimensional wave propagation in an absorbing sponge. It is found that the rate of convergence of the actual solution to the target solution, with an appropriate norm, is inversely proportional to the sponge strength. A detailed analysis for acoustic wave propagation in one-dimension verifies the convergence rate given by the general theory. The exponential point-wise convergence derived by Israeli and Orszag in the high-frequency limit is recovered and found to hold over all frequencies. A weakly nonlinear analysis of the method when applied to Burgers’ equation shows similar convergence properties. Three numerical examples are given to confirm the analysis: the acoustic extension of a two-dimensional time-harmonic point source, the acoustic extension of a three-dimensional initial-value problem of a sound pulse, and the introduction of unstable eigenmodes from linear stability theory into a two-dimensional shear layer.

A note on point source diffraction by a wedge

The object of this paper is to give new expressions for the wave field produced when a time harmonic point source is diffracted by a wedge with Dirichlet or Neumann boundary conditions on its faces. The representation of the total field is expressed in terms of quadratures of elementary functions, rather than Bessel functions, which is usual in the literature. An analogous expression is given for the three‐dimensional free‐space Green′s function.

Acoustic Radiation Properties of an Elastic Plate with an Infinite Baffle and Cavity

A cavity (0 < x < a, 0 < y < b, −d < z < 0) set in an otherwise rigid plane surface (z = 0) is covered by a rectangular elastic plate (of dimensions a × b) that is simply supported at its edges. The back face of the cavity is fixed and its side walls are taken to be acoustically soft. Two related problems are analysed. First, the system is considered to be irradiated by a plane time harmonic wave of radian frequency ω and incoming direction −n, and an estimate is sought for the transverse velocity of the plate and for the induced interior pressure. Mean square values are averaged over spatial variables and over all incidence directions n, in order to represent the case of a random external field. The second problem is that of a sound field induced by a time harmonic point source within the cavity and an estimate is given for the mean square velocity of the plate and for the radiated sound power, averaged over a frequency band and with respect to all source positions.

The Numerical Solution of the Three Dimensional Inverse Obstacle Scattering Problem for Point Source Generated Wave Fields

This work is concerned with the inverse problem to retrieve the shape of a three dimensional obstacle from near-field measurements of the scattering of time-harmonic point source waves. A numerical method for solving the inverse obstacle scattering problem is presented, which is achieved by solving an ill-posed linear system and a well-posed minimization problem independently. Such a separate numerical treatment for the ill-posedness and nonlinearity of the inverse problem makes the numerical implementation of the proposed method very easy and fast since there involves only the solution of a small scale minimization problem with one unknown function in the nonlinear optimization step for reconstructing the shape of the obstacle. Another attractive feature of the scheme is that it needs only the knowledge of the near-field measurements of the scattered fields due to point source incident fields at a finite number of incidence and observation points distributed over a limited aperture. Some numerical examples based on synthetic near-field data are given, showing that this method produces reasonable reconstructions for low wavenumbers even when the incidence and observation aperture is as small as the possible ~ /4 solid angle.

Elastic waves in inhomogeneously oriented anisotropic materials

Ray theory is developed for elastic waves propagating in inhomogeneously oriented anisotropic solids. These are materials of uniform density with moduli which are uniform up to a rotation of the underlying crystalline axes about a common direction, the degree of rotation varying smoothly with position. The ordinary differential equations governing the evolution of a ray have a simple form, and involve the angle of deviation between the slowness and wave velocity directions. The general theory is demonstrated for the case of SH waves in a transversely isotropic medium. The equations required for the description of SH Gaussian beams are derived, including the transport equation and the wavefront curvature equation. The theory is combined with an equivalent complex-source representation to generate an approximation to a time harmonic point source.

The fundamental singularity in a shallow ocean with an elastic seabed*

In this paper, the problem of constructing the acoustic pressure field excited by a time-harmonic point source in a shallow ocean over an elastic seabed is considered. This is a transmission boundary value problem for a system of prtial differential equations. Two different expansions for the propagating solutions are derived. The first expansion is obtained by using Hankel transformation. We symbolically compute the Hankel transform of the propagating solution and then perform inversion using Mittag-Leffler decomposition and other spectral analysis techniques. The second expansion is normal mode expansion. We find an appropriate inner product in a suitable function space, which permits us to compute the generalized Fourier coefficients. Numerical computations are presented to verify the orthogonality of the set of eigenfunctions with rewpect to this inner product, and to verify the equivalence of the two expansions.

Excitation of 3D-disturbances in wall-bounded shear flow with adverse pressure gradient

The excitation of 3D disturbances by a time-harmonic Dirac point source at the wall in a decclerated and a flat-plate boundary layer is compared. It is demonstrated that the linearized parallelflow theory including the receptivity problem can capture the main features of unstable 3D wavetrains found in previous experiments. Depending on the exciting frequency, three different types of wavetrains are identified.

Rossby Waves

An asymptotic solution of the linear shallow water equations for small Rossby number is constructed to describe Ross by waves. It leads to a dispersion or eiconal equation for the phase of the waves and a transport equation for their amplitude. It is shown how these equations can be solved by means of rays for both planetary and topographic Rossby waves. The method is illustrated by constructing the wave field produced by a time harmonic point source in fluid of uniform depth. This solution is a Green's function for the equations.

A note on using the fast field program

The fast field program is a numerically efficient algorithm for computation of the sound pressure due to a time harmonic point source above a general boundary in a layered medium. In this method, the inverse Hankel transform for obtaining the pressure is approximated by two Fourier integrals. One of the Fourier integrals was treated as the incoming wave term and neglected in the computation of sound pressure. Actually, no one has theoretically proven that one of the Fourier integrals can be neglected. In this paper, it is shown that the neglected integral is necessary for the computation of low‐frequency sound pressure when a pressure‐release boundary is involved.

Frequency and angle spreading for scattering from a rough ocean bottom

An analysis has been performed of the sound scattered from a Gaussian random rough ocean bottom when radiated from a time-harmonic point source and received by an omnidirectional receiver positioned at the same deep depth. Both source and receiver are moving, at low Mach number, in a plane parallel to the ocean’s surface, with arbitrary independent velocities. The problem is idealized and analyzed as a linear system with stochastically varying components, viz., the propagation distances because of the reflections from the random rough bottom. An expression is developed for the power spectral density of the received signal as a function of the source amplitude function. This expression describes the dependence of the frequency shift of the received signal upon the source and receiver speeds and upon the azimuthal angles, grazing angles, and ranges from source to arbitrary scattering facet of the rough bottom to receiver. [Work supported by ONR/AEAS Program.]

Matched-mode processing schemes of a moving point source

Matched-mode processing schemes are presented to estimate the frequency, velocity, depth, and range information of a time-harmonic point source moving uniformly in stratified oceanic waveguides. Three linear modal filtering approaches (frequency spectral decomposition by a single receiver, spatial spectral decomposition by a horizontal receiving array, and eigenfunction decomposition by a vertical receiving array) are first discussed to provide an overview for possible approaches to localize a moving source. An iterative nonlinear extrapolation algorithm involving only the fast Fourier transform is then employed to reduce the observation time by about an order less than that required by the linear frequency method. Physical insights of the Doppler effect in these inversion procedures are discussed in detail. Based on these insights, practical criteria for choosing processing parameters are established so that they can be applicable to all range-independent waveguides. Numerical results calculated by these methods of both shallow water and deep ocean examples are presented.

Numerical implementation of an adaptive fast-field program for sound propagation in layered media using the chirp z transform

Using the chirp z transform, a numerically efficient adaptive integration algorithm for fast-field programs has been developed. The fast-field program (FFP) is a numerically efficient algorithm for computation of the sound pressure due to a time harmonic point source above a general boundary in a layered medium. The new algorithm allows the user to specify the precise location of the range sample points, or “detectors,” independent of the horizontal wave-number sampling grid. This feature makes the algorithm particularly attractive for use in applications where sampling on a predetermined spatial is required. An example of such an application is the computation of the acoustic frequency response as a function of range. In this case the complex sound pressure must be computed at many frequencies while maintaining a constant range sampling interval. In addition, the algorithm can be used to recompute the sound pressure field with different range resolutions without recomputing the horizontal wave-number spectrum samples. It is also shown that the algorithm can be used to adaptively increase the number of integration points required to evaluate the Sommerfeld integral that results from the FFP type of formulation.