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Exact solution for the displacement components in P₁P₂P₁, (S₁S₂S₁)_SV, (S₁S2S₁)_SH head waves due to an impulsive double couple source

DEPARTMENT OF GEOPHYSICS AND GEOPHYSICAL ENGINEERING SAINT LOUIS UNIVERSITY, ST. LOUIS, MISSOURI

Abstract

Exact integral expressions for the displacement components as a function of azimuth, in P₁P₂P₁, (S₁S₂S₁)_SV, and (S₁S₂S₁)_SH head waves due to an impulsive double-couple source of arbitrary orientation in two semi-infinite media separated by a plane horizontal boundary, and the effect of free surface on it, have been obtained. The time variation of the source is assumed to be represented by the Heaviside unit function. The response due to an arbitrary source excitation may, however, be obtained from this solution by superposition. Necessary formulae for the operational interpretation of integral equations occurring in the problem are given.

Illustrative examples showing the waveform in P₁P₂P₁, (S₁S₂S₁)_SV, and (S₁S₂S₁)_SH head-wave displacement components for a particular fault and a suitably chosen earth model are given. Plots of the projection of particle trajectory for P waves in u_r-u_z plane, and for S waves in u-u_r and u-u_z planes are given for different azimuths; where u_r, u, u_z are the components of displacement in a cylindrical coordinate system (r, , z).