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Scattering of plane elastic waves at rough surfaces. II

  • Iya Abubakar (a1)

This is a continuation of (1) on the two-dimensional problem of the diffraction of elastic waves by irregular surfaces. The effect of an irregular surface with an isolated irregularity like a trough or ditch on incident plane harmonic P- and SV-waves is discussed. The maximum depth of the ditch is assumed small compared to the wavelength of the incident wave.

It is found that, when either a P- or an SV-wave is incident on such a boundary, besides the specularly reflected P- and SV-waves whose amplitudes are independent of the curvature of the surface there exist scattered waves travelling in various directions. In particular the diffracted zone contains the following second wave-types whose amplitudes are proportional to the depth of the ditch: (i) direct reflected P- and SV-waves, which at large distances appear to diverge from the point of intersection of the axis of symmetry of the ditch and the horizontal plane asymptotic to the boundary if the ratio of the wavelength of the incident wave to the half-width of the ditch is large. If the ratio is small these waves are reflected in the specular directions, (ii) A ‘secondary S-wave’ which finishes as P having travelled most of the way as an SV-wave. Its energy is confined to the neighbourhood of the free surface, (iii) A secondary P-wave which travels along the surface and finally emerges into the medium as an SV-wave at the critical angle for the medium, (iv) Rayleigh waves whose particle motion is in elliptic orbits.

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(1)Abubakar, I.Proc. Cambridge Philos. Soc. 58 (1962), 136.
(2)Jeffreys, H. and Jeffreys, B. S.Methods of mathematical physics, 3rd ed. (Cambridge, 1956).
(3)Lapwood, E. R.Philos. Trans. Boy. Soc. London. Ser. A, 242 (1949), 63.
(4)Van Der Pol, B. and Bremmer, H.Phil. Mag. 24 (1937), 141; 825.
(5)Sommerfeld, A.Ann. Physik, 28 (1909), 665.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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