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P. Borejko, F. Ziegler:
"Recent Advances in the Research on Acoustic and Elastic Waves in a Wedge-Shaped Layer over a Penetrable Bottom";
Talk: International Symposium on Mechanical Waves in Solids, Zhejiang University, Hangzhou, China (invited); 2006-05-15 - 2006-05-18.

Recent Advances in the Research on Acoustic and Elastic Waves in a Wedge-Shaped Layer over a Penetrable Bottom
Piotr Borejko1 and Franz Ziegler1*
1 Department of Civil Engineering, Vienna University of Technology, Wiedner Hauptstr. 8/E206/3, Vienna, A-1040, Austria
Abstract
A wedge-shaped waveguide formed by two inclined planes that intersect along the line of apex is a simple model for analyzing acoustic waves in shallow water with a sloping bottom and seismic waves in a dipping structure. In general, the wave field in such a waveguide consists of two components: the image component, including the direct wave from the source and a finite number of waves arising from the interactions of the former with the wedge boundaries, and the diffracted component due to scattering at the apex. The methods and results of analyses of the wave field in a wedge-shaped layer overlying a penetrable substratum (penetrable wedge) are thus important in underwater acoustics and in seismology, especially in earthquake engineering.
A solution for steady-state waves governed by the Helmholtz equation in a wedge with perfectly reflecting (impenetrable) boundaries (perfect wedge) can be derived by the technique of separation of variables, and the harmonic wave field is deduced by the expansion in eigenfunctions (normal modes), the formalism known as the method of normal modes. This method was thus applied to analyze acoustic waves in a perfect liquid wedge [1] and elastic SH waves in a perfect solid wedge [2]. A perfect wedge overlying a rigid substratum is the simplest model of a dipping structure or shallow water with a sloping bottom, but it is inadequate in that it does not allow for penetration of the substratum, characteristic for a real bedrock or a real ocean floor.
No exact solution exists for acoustic or elastic waves in a penetrable wedge. The method of normal modes and many other analytical methods such as the method of images, the Kontorovich-Lebedev transform, the method of self-similar solutions and some others, cannot be extended to treat waves in a penetrable wedge because "penetrable" boundary conditions preclude the separability of the Helmholtz equation. The wave field in a penetrable wedge was thus analyzed by applying a variety of approximate and/or numerical methods.
The only solution for the wave field in a penetrable wedge, which is exact other than the omission of diffraction from the apex, is that obtained by modifying the method of generalized ray for parallel layers [3]. The essence of the modified method is that the diffracted component (if present at all) is entirely neglected, the image component is analyzed into a complete set of waves, each element of this set is identified as a wave propagating along its specific stationary time path (the first being the wave from the source directly to the receiver, the next two being those reflected once off the either boundary of the wedge, etc.), and the solution for each wave is in the form of a definite integral (a ray integral) taken over a fixed interval in an appropriate local wave slowness.
Applications of the method of generalized ray to non-parallel layers began with the derivation of a solution for the transient elastic SH-wave field in two-layered (a sloping layer over a half-space) [4-6] and three-layered (two sloping layers over a half-space) [7] models. The ray integrals for various primary SH-waves (those with a few reflections or transmissions, or both) were then evaluated to asses the effect of the receiver location (up-dip or down-dip relative to the hypocenter) on the surface response in a two-layered model with a line source in the half-space and in a three-layered model with a line source in the surface layer. Setting up the ray integrals for a complete set of SH-waves, one obtains a complete SH-solution (a sum of all of the ray integrals) which can be applied to study waves of Love type in a sloping layer and to investigate the effect of the sloping interface on the amplification of surface response and the predominant frequencies.
The generalized-ray formalism was applied to derive a solution for the transient elastic P-SV-wave field in a two-layered model with a line source in the surface sloping layer [8]. The ray integrals for various groups of waves (each group consisting of waves with an equal number of reflections) where then evaluated to ascertain the effect of the receiver location on the surface response and to find out that (despite the increasing number of reflections) each group contributed significantly to the response. A complete P-SV-solution can be applied to predict the existence of a phase of Rayleigh type in the surface response curve which becomes dominant at large epicentral ranges.
A ray-integral solution was derived for the transient elastic P-S-wave field from a point source in a sloping layer overlying a half-space [9]. In contrast to the former two two-dimensional models (in which the receiver was restricted to the plane perpendicular to the line of apex), this model is three-dimensional in that the receiver may lie cross-slope as well as down-dip or up-dip of a point source, and an analytical simulation of the seismic field in a dipping structure (based on a complete P-S-solution) is adequate in that it accounts for the typical P and S phases of the field in a real ground, for the three-dimensional propagation effects in a sloping layer, and for seismic penetration of an underlying bedrock.
The method of generalized ray was also applied to derive a complete solution for the acoustic field from a point source in a liquid wedge with a rigid, liquid, or elastic substratum [10, 11]. A sloping layer of fluid overlying a rigid substratum is the simplest model of a shallow-water wedge, and that overlying a penetrable substratum is a more realistic model of a sand-bottom (the case of a liquid substratum) or a rock-bottom (the case of an elastic substratum) shallow-water wedge. The solution was then applied to compute the propagation paths and the pressure response curves, the former illustrate the three-dimensional propagation effects (such as horizontal refraction and backscattering) in a liquid wedge and the latter exhibit ground and water wave phases for the cases of fast-speed liquid and elastic substrata.
References

[1] Buckingham M. J. (1989) Theory of acoustic radiation in corners with homogeneous and mixed perfectly reflecting boundaries. J. Acoust. Soc. Am. 86: 2273-2291.
[2] Hudson J. A. (1963) SH waves in a wedge-shaped medium. Geophys. J. R. astr. Soc. 7: 517-546.
[3] Pao Y.-H. and Gajewski R. R. (1977) The generalized ray-theory and transient responses of layered elastic solids. In Physical Acoustics 13: 184-265, ed. by Mason W. P. and Thurston R. N., New York: Academic Press.
[4] Pao Y.-H. and Ziegler F. (1982) Transient SH-waves in a wedge-shaped layer. Geophys. J. R. astr. Soc. 71: 57-77.
[5] Ziegler F. and Pao Y.-H. (1984) Transient elastic waves in a wedge-shaped layer. Acta Mechanica 52: 133-163.
[6] Ziegler F., Pao Y.-H., and Wang Y.-S. (1985) Transient SH-waves in dipping layers: the buried line-source problem. J. Geophys. 57: 23-32.
[7] Ziegler F., Pao Y.-H., and Wang Y.-S. (1985) Generalized ray-integral representations of transient SH-waves in a multiply layered half-space with dipping structure. Acta Mechanica 56: 1-15.
[8] Borejko P. and Ziegler F. (1991) Seismic waves in layered soil: the generalized ray theory, In Structural Dynamics, Recent Advances, ed. by Schuëller G. I., Berlin: Springer-Verlag, pp. 52-90.
[9] Borejko P. and Ziegler F. (2002) Pulsed asymmetric point force loading of a layered half-space. In Acoustic Interactions with Submerged Elastic Structures, Part IV, ed. by Guran A. et al., New Jersey: World Scientific, pp. 307-388.
[10] Pao Y.-H., Ziegler F., and Wang Y.-S. (1989) Acoustic waves generated by a point source in a sloping fluid layer. J. Acoust. Soc. Am. 85: 1414-1426.
[11] Borejko P., Chen C.-F., and Pao Y.-H. (2001) Application of the method of generalized rays to acoustic waves in a liquid wedge over elastic bottom. J. Comput. Acoust. 9: 41-68.