## Abstract

One considers an elastic halfspace with depth (x_{1}) dependent density Q and Lamé moduli (λ,m̈). Impulsive stresses τli = δ(x_{2}, x_{3})δ(t) for x_{1} = 0 are applied with displacement responses u_{i} = g_{i}(t, x_{2}, x_{3}) at x_{1} = 0 (i = 1, 2, 3). Let v_{i}(t, x_{1}) = ∫∫u_{i}dx_{2}dx_{3} (i = 1, 2 is enough) and set w(t, x_{1}) = ∫∫x_{2}u_{1}dx_{2}dx_{3}. One obtains a system of 3 differential equations for v_{1}, v_{2}, and w to which the spectral techniques of inverse scattering theory are applied as in [25]. The inverse problems for the uncoupled v_{i} can then be solved to produce 2 functions A_{1} and A_{2} involving (Q, λ, μ) as functions of “bound” variables y_{1} and y_{2} containing (Q, λ, μ), between which a relation then is determined. Analysis of the coupled equation for w then leads to a Fredholm integral equation whose solution provides an additional relation between (Q, λ, μ) from which (Q, λ, μ) can be determined as functions of x_{1}. The integral equation is reduced to a Volterra type equation by results and techniques of transmutation and then solved by a modification of standard techniques. A number of features and results of independent mathematical interest arise from the transmutation theory.

Original language | English (US) |
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Pages (from-to) | 33-73 |

Number of pages | 41 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - 1982 |