Radiation pattern of Rayleigh waves from a fault of arbitrary dip and direction of motion in a homogeneous medium

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Abstract
Expressions for the displacements in the body waves radiated in an unbounded, homogeneous elastic medium by dipolar point sources of arbitrary orientation may be readily derived in Cartesian coordinates from formulae given by Love. The free-surface boundary conditions are, however, most conveniently expressed in terms of Sezawa's cylindrical wave functions. The necessary transformation between the two representations is provided by the Sommerfeld integral and others that may be derived from it by differentiations with respect to the radial and axial (vertical) coordinates. By this means the total radiation field (direct plus surface reflected) is expressed in terms of integrals of cylindrical wave functions. The Rayleigh wave component may then be separated out by calculating the residue at the Rayleigh pole of the integrand. The azimuthal dependence of the Rayleigh wave displacements appears as the sum of three terms, one independent of the azimuth angle, φ, another depending upon sin φ and cos φ, and a third depending upon sin 2φ and cos 2φ. The coefficients of these terms are functions of the direction cosines of the normal to the fault plane and the direction of the relative displacement vector in the fault plane. Equations are presented for sources of both single and double couple types. The effect of fault propagation with finite velocity over a finite distance may be included by multiplying these expressions by the finite source factor previously derived by Ben-Menahem. Polar plots of the amplitude and initial phase are presented for single and double-couple representations of a number of different types of faults. It is noted that for one certain orientation a shallow double-couple source generates no Rayleigh waves.

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