Nonlinear propagation of surface acoustic waves

Abstract

Some problems relating to the nonlinear propagation of surface acoustic waves on an isotropic homogeneous medium are studied in this thesis using a perturbation approach. The propagation of finite amplitude surface acoustic waves is governed by a system of quasilinear hyperbolic partial differential equations with nonlinear boundary conditions at the free surface. Nonlinearity gives rise to the generation of harmonics and combination frequencies. Since the linearised problem is nondispersive, the phase matching condition is satisfied by all harmonics and also by all combination frequencies generated by interaction between co-directional waves. The coefficient matrices of the inhomogeneous algebraic equations, which arise from the inhomogeneous boundary conditions at each stage of the perturbation expansion, become singular whenever the phase matching condition is satisfied. A solution to these algebraic equations exists only if certain solvability conditions are satisfied. But a direct perturbation method fails to satisfy these solvability conditions.

A singular perturbation method which overcomes the difficulty mentioned above is developed in this thesis. This method is an adaptation of Lighthill’s method of strained coordinates wherein the dependent as well as independent variables are expanded in power series in a perturbation parameter. In the method presented in this thesis, the coefficient functions arising in the expansions of the independent variables are so chosen as to satisfy the solvability conditions. The method also yields solutions that are uniformly valid.

Using this method, three steady state problems (problems which admit periodic progressive wave solutions) and a monochromatic signalling problem are studied. The perturbation expansions are carried out up to the second order in the perturbation parameter. The steady state problems considered are the propagation of surface acoustic waves of a single frequency, and interaction between surface acoustic waves propagating collinearly in the same direction and in opposite directions. Such steady state solutions do exist in the case of propagation of finite amplitude surface acoustic waves since the two-dimensional character of wave motion prevents the formation of shocks provided that the perturbation parameter is sufficiently small [J. S. Kelley and M. H. McMullan, J. Acoust. Soc. Amer., 49, 229–333 (1971)].

In the steady state, every surface acoustic wave is accompanied by harmonics of all orders and also by a zero frequency wave (i.e., a non-propagating permanent deformation). Only the second harmonic and zero frequency terms are manifest in the second order solution. The second harmonic wave has two displacement components, longitudinal and transverse, while the zero frequency term has a component only in the transverse direction. The amplitude ratio of the two components of the second harmonic and their decay profiles are different from those for a Rayleigh wave. The amplitude of both components of the second harmonic and the magnitude of the zero frequency term are proportional to the fundamental frequency and also to the square of the amplitude of the fundamental wave. The constants of proportionality, which are functions of the second and third order elastic constants of the material, are the nonlinearity parameters that characterise the nonlinear propagation of single frequency surface acoustic waves in the material.

Nonlinear interaction between two surface waves of frequencies ?? and ?? produces surface waves of frequencies ?? + ?? and |?? ? ??|. The amplitudes of the sum and difference frequency waves are proportional to the product of the amplitudes of the interacting waves. The amplitude ratio and decay profiles of these waves are functions of the frequency ratio ??/??. The part of the solution corresponding to sum and difference frequency waves, and the part corresponding to the second harmonic and zero frequency waves are interconnected, and one cannot be determined independently of the other. When the interacting waves are travelling in the same direction, the interaction modifies the decay profiles of the second harmonic and zero frequency waves also. When the interacting waves are travelling in opposite directions, the interaction also generates zero, one, or two bulk waves depending on the frequency ratio. The bulk waves generated are pure shear modes with frequency ?? + ??. The propagation vectors of the bulk waves are directed into the solid at different angles to the free surface. The propagation vectors of both bulk waves approach a direction normal to the free surface as ??/?? approaches unity.

For the monochromatic signalling problem, the need to satisfy additional boundary conditions necessitates a further modification of the method of solution. In this case, the amplitude and phase factors of the second harmonic surface wave exhibit an oscillatory variation along the direction of propagation. The rate and nature of this variation are determined solely by the material constants of the medium and not by the amplitude of the fundamental wave. The solution also contains a bulk wave of constant amplitude but varying phase.