SUMMARY
The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.
For the scalar problem of the diffraction of a plane wave by a prolate spheroid the exact solution is known, and by expanding this in ascending powers of ka, where k is the wave number and 2a is the interfocal distance, the Rayleigh series for both the "soft" and "hard" bodies are obtained up to and including terms in (ka)6. The corresponding results for an oblate spheroid can be deduced by a trivial change of parameters. Some particular cases are examined.
Consideration is given to the scattering of a plane wave by N cylinders equispaced in a row. The problems associated with scatterers, both "soft" and "hard" in the acoustical sense, are treated. An application of Green's theorem together with the appropriate boundary condition on the cylinders leads to a set of simultaneous integral equations in the unknown function on the cylinders.Solutions in the form of series in powers of a small parameter δ (essentially the ratio of cylinder dimension to wavelength) are assumed. In the case of elliptic cylinders, the integral equations are reduced to sets of linear algebraic equations. Only for the first term in the solution for "soft" cylinders is it necessary to solve N simultaneous equations in N unknowns; all other equations involve essentially only one unknown. Far-fields and scattering cross sections are calculated. The case of two "soft" cylinders is given particular attention.Conditions for justification of the neglect of higher-order terms are discussed. It is found that all terms but the first (in either problem) may be neglected if [Formula: see text] and (N–1)/(ka) is sufficiently small. (Here a is the spacing between centers of adjacent cylinders, and k is the wave number.) For this reason these solutions are most useful when the number of cylinders is small.
SummaryThe problem of scalar Dirichlet diffraction of a plane wave by an elliptic disc is discussed. A scheme is given whereby the low frequency expansion of the scattered field may be readily obtained. Series expansions are obtained for the far-field amplitude up to and including the second order in the wave number. The first two terms of the scattering cross-section are also derived.
Abstract
It is shown that the plane-wave picture of a leaky mode proposed by Burg, Ewing, Press and Stulkin (1951) yields the accepted period equation for leaky modes in a water layer a half-space. The resultant mode is formed by an inhomogeneous wave with real frequency and complex wave number and phase velocity. Another form of mode considered is that formed by a homogeneous wave in the guide with real phase velocity and complex frequency and wave number. The phase-velocity dispersion curve for this case is appropriate for determining shear-wave coupling to PL waves. The procedures of the article could be readily extended to the more complicated case of a solid layer over a half-space. It is also demonstrated that the derivative of the real part of angular frequency with respect to the real part of the wave number is a good approximation to the group velocity for leaky modes with low losses.
Analysis of the interaction of a cylindrical wave impinging on a cylindrical cavity is presented. It is assumed that a line source is located an arbitrary distance from the cavity and that its strength varies harmonically in time. The resulting dynamic stress concentration factors at the cavity wall are determined by considering the wave-diffraction effects. Numerical results indicate that the dynamic stress concentration factors around the cavity are dependent upon (a) distance from the source to the cavity, (b) wave number, and (c) the Poisson ratio of the medium. At high wave number (high frequency), the response to an incident cylindrical wave becomes almost identical with the response to an incident plane wave. At low wave number, however, the response departs drastically from all previous investigations where the incident wave was assumed to be a plane wave. Stress concentration factors substantially higher than those determined in earlier studies were noted in the present analysis.
Partition of unity method for Helmholtz equation: q-convergence for plane-wave and wave-band local bases