On the Lamb problem: forced vibrations in a homogeneous and isotropic elastic half-space
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O R I G I NA L
B. F. Apostol
On the Lamb problem: forced vibrations in a homogeneous and isotropic elastic half-space
Received: 11 February 2020 / Accepted: 23 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The problem of vibrations generated in a homogeneous and isotropic elastic half-space by spatially concentrated forces, known in Seismology as (part of) the Lamb problem, is formulated here in terms of Helmholtz potentials of the elastic displacement. The method is based on time Fourier transforms, spatial Fourier transforms with respect to the coordinates parallel to the surface (in-plane Fourier transforms) and generalized wave equations, which include the surface values of the functions and their derivatives. This formulation provides a formal general solution to the problem of forced elastic vibrations in the homogeneous and isotropic half-space. Explicit results are given for forces derived from a gradient, localized at an inner point in the half-space, which correspond to a scalar seismic moment of the seismic sources. Similarly, explicit results are given for a surface force perpendicular to the surface and localized at a point on the surface. Both harmonic time dependence and time δ-pulses are considered (where δ stands for the Dirac delta function). It is shown that a δ-like time dependence of the forces generates transient perturbations which are vanishing in time, such that they cannot be viewed properly as vibrations. The particularities of the generation and the propagation of the seismic waves and the effects of the inclusion of the boundary conditions are discussed, as well as the role played by the eigenmodes of the homogeneous and isotropic elastic half-space. Similarly, the distinction is highlighted between the transient regime of wave propagation prior to the establishment of the elastic vibrations and the stationary-wave regime. Keywords Lamb problem · Half-space · Vibrations · Eigenmodes Mathematics Subject Classification
35L05 · 35L67 · 74J05 · 74J15 · 74J70 · 86
1 Introduction The generation and the propagation of the seismic waves is a basic problem in mathematical Seismology. It gives information about the processes occurring in an earthquake focus, about the inner structure of the Earth and the effects the seismic waves have on the Earth’s surface. In addressing this problem, it is convenient to approximate the Earth by a half-space bounded by a plane surface; by another useful simplification, the Earth is viewed as a homogeneous and isotropic elastic solid, though the problem may be more complex, involving, for instance, anisotropic stratified structures, or functionally graded media. Similarly, the seismic source may exhibit the complex structure of a moving dislocation, a crack or even multiple cracks. Usually, “elementary” earthquakes are considered, produced by sources localized beneath the Earth’s surface, such that, for long distances, we may consider point seismic sources, i.e. sources represented by spatial δ-functions, or derivatives of
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