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The Quarterly Journal of Mechanics and Applied Mathematics 1961 14(3):293-318; doi:10.1093/qjmam/14.3.293
© 1961 by Oxford University Press
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RAYLEIGH WAVES IN TRANSVERSELY ISOTROPIC MEDIA

V. T. BUCHWALD {dagger}

( Department of Mathematics, the Manchester College of Science and Technology )

It has been shown by Syngo (1) that the propagation of Rayleigh waves on a free plane surface of a transversely isotropic medium is only possible if the free plane is either normal or parallel to the direction about which there is symmetry of rotation.

In this paper, the displacements due to surface waves radiating from a given harmonic source are found in terms of double Fourier integrals. In order to satisfy the boundary conditions, additional free vibrations of the semi-infinite medium have to be introduced. These are expressed as double Fourier integrals, the integrands of which contain poles and branch points.

Using the isotropic case as an analogy, the branch points of the integrands are neglected and the double integrals are estimated asymptotically, using a method derived from the one given by Lighthill (2) for triple integrals.

With this technique it is shown that in the first of the two cases suggested by Synge the slowness and wave curves are circles, and that the asymptotic forms of the displacements are, consequently, of a simple nature. The resulting equation for the velocity of Rayleigh waves is the same as that obtained by Stoneley (3) and other writers by the method of plane waves.

The second case is fully anisotropic, when the method of plane waves is insufficient. Equations representing the slowness and wave curves are obtained. These curves are plotted from data obtained numerically for a number of materials. Calculations show that in certain circumstances, in this case as well as in the first case, the variation of wave amplitudes with depth from the free surface is harmonic, in addition to being exponentially damped.


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