Surface Waves and Discontinuities

Abstract

Elsevier, Amsterdam, 1987. 229pp. Price $48.00.The most significant contribution to the recorded ground displacement during an earthquake comes from surface waves propagating along the ground. A consistent mathematical theory describing such surface wave propagation, then, is at the heart of seismology. Since the Earth can be approximated by a layered, elastic half-space that is infinitely deep, this problem can be handled as a special waveguide problem. The theory of open waveguides is of exceptional importance to this effort. Thus a book describing waveguide theory applied to geology really is quite useful and quite logical.The Earth is a highly dispersive system. A temporally short impulse in the source region reaches a distant observer as a disturbance of long duration. The dispersion, which affects surface waves, also is a typical waveguide phenomenon. For this reason, it is not surprising that surface waves are suited to treatment using the theory of waveguides. Such a treatment is presented in great mathematical detail in the book under review. Some general remarks are made about waveguides with discontinuities in the Introduction. In Chap. 2, Malischewsky considers boundary conditions and their seismological interpretation, and concludes with differential operators in closed domains and in open domains.The real mathematical ingenuity begins with Chap. 3, "The Undisturbed Waveguide." After demonstrating that waveguides exhibit an eigenspectrum, Malischewsky shows that the formalism of quantum mechanics is appropriate for their description. The list of topics covered is suggestive of a quantum mechanics text: 3.1. The mode conception .... 3.2.2. The eigenspectrum .... 3.2.3. The eigenfunctions .... 3.2.4. Orthogonality and normalization .... and 3.3.6. The radiation conditions.The mathematics does not stop at this point because Malischewsky selects his models by considering the principle "As simple as possible, as complicated as necessary." Thus Chap. 4, "The Disturbed Waveguide," continues the discussion of the interrelations with quantum theory. The surface and body waves, respectively, of the ideal waveguide can be compared with the eigenstates of an atom. Whereas the eigenstates of an atom are characterized by their energy, for surface waves each "state" is characterized by its phase velocity (eigenvalue). The ionization energy at which the continuum of the atom begins is analogous to the shear velocity of the layered half-space that determines the cutoff of a given Love mode. In one respect, the analogy breaks down: The bound states of an atom are a denumerable infinity, whereas the surface wave modes are finite in number at any given period. However, since one most often concentrates on the lowest level modes in any mathematical analysis, this shortcoming of the analogy is not of great importance. Since the other analogies hold completely, it is useful to introduce the bra-ket formalism in order to apply some methods of quantum theory to surface wave calculations. To illustrate its usefulness, Mali...

Propagative and Evanescent Waves Diffracted by Periodic Surfaces: Perturbation Method

“…In this case it is possible to obtain the solution by a perturbation method inspired by quantum physics methods. Our approach is comparable to that of Malischewsky in the field of seismology [5,6].…”

SWRT: A package for semi-analytical solutions of surface wave propagation, including mode conversion, across transversely aligned vertical discontinuities

“…SWRT is available via GitHub (https://github.com/arjundatta23/sw_reftrans.git) Appendix A: Rayleigh wave expressions in the Green's function method 20 The Rayleigh wave stress-displacement vector defined by Gomberg and Masters (1988) is β = {β 1 , β 2 , β 3 , β 4 }, the four components being proportional to u z , u x , σ zz and σ xz respectively: For the type of problem under consideration, with the plane interface being normal to the x−direction, the required components of traction are σ xz and σ xx . The latter can be written in terms of β 1 and β 2 (with the phase factor omitted for convenience):…”

Contribution of leaky modes in the modal analysis of unbounded problems with perfectly matched layers

“…For unbounded problems, the modal basis is yet more complicated because it involves a continuous sum of modes (radiation modes) in addition to a discrete set (trapped modes, if any). 1 This contribution of the continuum is usually quite difficult to calculate and interpret from a physical point of view. Based on steepest descent techniques, for example, it can be approximated by a discrete set of another class of modes, the so-called leaky modes.…”