Nonlinear surface waves in isotropic elastic substrates with thin films

Abstract

Surface waves in isotropic elastic solids are nondispersive. Considered here is the effect of dispersion, due to a thin solid film on the surface of an isotropic substrate, on nonlinear propagation. The analysis is performed with the theoretical model developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569 (1992)] augmented to include dispersion. The dispersion relation is obtained from linear theory but is otherwise exact. Approximations leading to a relation of the Korteweg–de Vries or Benjamin–Ono type, sometimes proposed for the study of dispersive nonlinear surface waves, are not employed. Both loading and stiffening films are considered. Loading films decrease the surface wave speed. Stiffening films increase the surface wave speed until it equals the shear wave speed in the substrate, where cutoff of the surface wave mode occurs. Computations were performed with elastic moduli for a fused quartz substrate, which exhibits negative nonlinearity in the sense that rarefaction rather than compression shocks form in the horizontal particle velocity waveform. Gold (loading) and alumina (stiffening) films are considered. Weak, moderate, and strong dispersion (corresponding to increasing film thickness) relative to nonlinearity are examined. Stationary wave solutions are obtained for the alumina film. [Work supported by ONR.]