Weak and short waves in inhomogeneous isotropic elastic materials. I.

Abstract

Wave propagation in an elastic material is investigated theoretically. The material is assumed to be isotropic and inhomogeneous and to have a nonlinear stress–strain relation, and the prestrain is assumed to be dynamic and inhomogeneous. Geometrical acoustics is applied, and the wave is assumed to be weak and short and to have an arbitrarily curved wave front. The amplitude is less than the wavelength, and the wavelength is less than the distances over which such wave quantities as prestrain and stress are subject to considerable change. By equating first‐order terms, a principal longitudinal and two principal transverse waves are shown to exist and their speeds of propagation are evaluated. Likewise, by equating second‐order terms, amplitude equations which govern the growth and decay of amplitude are derived. The variation of amplitude depends upon the curvature of the wave front, the velocity gradient and prestrain and material inhomogeneities, the nonlinearity of the stress–strain relation, and the waveform. The results obtained in simple cases are identical with those derived using the method of singular surface.