Interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations. Forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area, especially in elasticity and acoustics. The theory has indeed been developed primarily in a quantum context; it is increasingly becoming clear, however, that important applications lie in the field of mechanical vibrations and acoustics. We provide an overview over basic concepts in this emerging field of wave chaos. This ranges from ray approximations of the Green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics—both experimentally and from a theoretical/numerical point of view.

https://iopscience.iop.org/article/10.1088/1751-8113/40/50/R01/pdf

Wave chaos in acoustics and elasticity

A robust and efficient technique for predicting the far-field scattering behavior for an arbitrarily-shaped defect in a generally anisotropic medium is presented that can be implemented in a commercial FE package. The spatial size of the modeling domain around the defect is as small as possible to minimize computational expense and a minimum number of models are executed. The method is based on an integral representation of a wave field in a homogeneous anisotropic medium. A plane incident mode is excited by applying suitable forces at nodes on a surface that encloses the scatterer. The scattered wave field is measured at monitoring nodes on a concentric surface and then decomposed into far-field scattering amplitudes of different modes in different directions. Example results for 2D and 3D bulk wave scattering in isotropic material and guided wave scattering are presented. Modeling accuracy is examined in various ways, including a comparison with the analytical solutions and calculation of the energy balance.

A generalized approach for efficient finite element modeling of elastodynamic scattering in two and three dimensions.

An asymptotic Green’s function in homogeneous anisotropic viscoelastic media is derived. The Green’s function in viscoelastic media is formally similar to that in elastic media, but its computation is more involved. The stationary slowness vector is, in general, complex valued and inhomogeneous. Its computation involves finding two independent real-valued unit vectors which specify the directions of its real and imaginary parts and can be done either by iterations or by solving a system of coupled polynomial equations. When the stationary slowness direction is found, all quantities standing in the Green’s function such as the slowness vector, polarization vector, phase and energy velocities and principal curvatures of the slowness surface can readily be calculated. The formulae for the exact and asymptotic Green’s functions are numerically checked against closed-form solutions for isotropic and simple anisotropic, elastic and viscoelastic models. The calculations confirm that the formulae and developed numerical codes are correct. The computation of the P-wave Green’s function in two realistic materials with a rather strong anisotropy and absorption indicates that the asymptotic Green’s function is accurate at distances greater than several wavelengths from the source. The error in the modulus reaches at most 4% at distances greater than 15 wavelengths from the source.

https://www.jstor.org/stable/pdfplus/20209338.pdf

Asymptotic Green's function in homogeneous anisotropic viscoelastic media

Direct and inverse elastic scattering from anisotropic media

A horizontally multilayered Green elastic transversely isotropic half-space is considered as the domain of the boundary value problem involved in this paper, such that the axes of material symmetry of different layers are parallel to the axis of material symmetry of the lowest half-space, which is depthwise. The domain is assumed to be affected by an arbitrary time-harmonic forced vibration due to a rigid rectangular surface foundation. With the use of a potential function method and the Hankel integral transforms, the displacements and stresses Green's functions are determined in each layer. The unknown functions due to integrations in each layer are transformed to the unknown functions of the surface layer with the use of the concept of propagator matrix and the continuity conditions. The mixed boundary conditions at the surface of the whole domain are numerically satisfied with the assumption of piecewise constant distribution of tractions in the contact area. It is numerically shown that the surface displacement and stress boundary conditions are satisfied very well. The vertical and horizontal impedance functions of the rectangular foundation are determined, which may be used as lumped parameters in time-harmonic soil-structure interaction with transversely isotropic horizontally layered domain as the soil. It is shown that the impedance functions determined in this paper coincide with the same functions for the simpler case of isotropic homogeneous half-space as degenerations of this study.

Impedance Functions for Surface Rigid Rectangular Foundations on Transversely Isotropic Multilayer Half-Spaces

A new method for simulating the propagation of pulses radiated by a circular normal ultrasonic transducer which is directly coupled to a homogeneous and isotropic elastic half-space is proposed Both nonuniform and uniform high-frequency asymptotics inside geometrical regions as well as boundary layers (penumbra, an axial region, and a vicinity of the critical rays) have been used to describe the transient field by means of harmonic synthesis The nonuniform asymptotic formulas involving elementary or well-known special functions elucidate the physics of the problem and give explicit dependence of the radiated waves upon the model parameters The formulas are applicable in the radiating near field which is the near-field with the evanescent wave zone excluded The code based on the uniform asymptotics has been tested in all regions against an exact numerical solution It is orders of magnitude faster, but in many realistic cases the accuracy does not suffer The limits of applicability of the model have been established

The high-frequency asymptotic description of pulses radiated by a circular normal transducer into an elastic half-space

The high-frequency asymptotic description of head waves as well as the field inside boundary layers surrounding the critical rays are obtained for two cases: (a) a point source, and (b) a circular transducer, both acting normally on an isotropic and homogeneous elastic half-space. The edge head waves underneath a circular transducer are described by the asymptotics of a higher order compared to those of direct compressional, edge compressional, and shear waves, but are still discernible in the radiating near zone and thus might be useful in nondestructive evaluation of industrial materials. The asymptotic formulas produced involve in geometrical zones elementary functions and inside boundary layers well-known special functions. Therefore, they allow us to elucidate the physics of the problem and can be used in writing computer codes which simulate the radiating near field of a circular transducer orders of magnitude faster than full numerical schemes. The formulas have been tested against exact integral solutions evaluated numerically.

High-frequency asymptotic description of head waves and boundary layers surrounding the critical rays in an elastic half-space

The dynamic time–harmonic Green's tensor for a transversely isotropic homogeneous linearly elastic solid is studied. First, the representation of the Green's tensor as an integral over the unit sphere is obtained. It consists of three parts: quasi–longitudinal (P), shear–horizontal (SH) and quasi–shear (SV). Then, an exact analytical solution for the SH part in terms of elementary functions is derived. The complete far–field asymptotic approximation of P and SV parts is obtained next, using the uniform stationary phase method. For the P wave it involves the leading term of the ray series since there is only one arrival of this wave. The wave surface for the SV wave contains conical points and cuspidal edges. The asymptotic description applicable near these singular directions is derived involving the Airy and Bessel functions. The directions close to the points of tangential contact of the SH and SV sheets of the wave surface are also treated. At the end of the paper numerical results in both frequency and time domain are presented. They show that the agreement between the outputs of the asymptotic and direct numerical codes is very good throughout all regions but the former can be orders of magnitude faster.

Far–field asymptotics of the Green's tensor for a transversely isotropic solid

The problem of propagation of pulses in the radiating near zone of a large circular normal transducer directly coupled to a homogeneous and isotropic elastic half-space is re-visited. It is shown that for certain observation angles the impulse response approach is computationally inefficient. A new method based on the so-called wavefront expansions of the impulse response is developed instead. The expansions are obtained by the analytical harmonic synthesis of the high-frequency asymptotics of the transducer field. Unlike these asymptotics the wavefront expansions are expressed in terms of elementary functions only. The direct P, edge P and S waves as well as the transition regions (penumbra and axial region) are described. The uniform asymptotic expansions applicable throughout the radiating near zone are derived as well. The code based on the time convolution of the pressure input function with the wavefront expansions is compared to a direct numerical code. It is thousands of times faster but practical...

On the radiation of ultrasound into an isotropic elastic half-space via wavefront expansions of the impulse response

A new fast method for simulating the propagation of pulses radiated by a rectangular normal ultrasonic transducer which is directly coupled to an isotropic and homogeneous elastic half-space is proposed. First, the so-called two-tier approach introduced in Fradkin et al. [“The radiating near-field asymptotics of a time-harmonic circular normal ultrasonic transducer in an elastic half-space,” J. Acoust. Soc. Am. 104, 1178–1187 (1998)] and the uniform stationary phase method are used to obtain both nonuniform and uniform high-frequency asymptotics of the time-harmonic field. Then, the transient field is described by means of harmonic synthesis. The nonuniform asymptotics elucidate the physics and all the asymptotics give explicit dependence of the radiated waves on model parameters. The formulas are applicable in the radiating near field that is the near field with the evanescent wave zone excluded. The asymptotics involve in geometrical regions elementary and inside boundary layers, well-known special func...

Dmitri Gridin

Papers

The high-frequency asymptotic description of pulses radiated by a circular normal transducer into an elastic half-space

High-frequency asymptotic description of head waves and boundary layers surrounding the critical rays in an elastic half-space

Far–field asymptotics of the Green's tensor for a transversely isotropic solid

On the radiation of ultrasound into an isotropic elastic half-space via wavefront expansions of the impulse response

A fast method for simulating the propagation of pulses radiated by a rectangular normal transducer into an elastic half-space