Self-Focussing of Coupled Guided Acoustic Waves at a Nonlinear Resonance

Abstract

Guided acoustic waves, especially surface acoustic waves and waves in elastic plates, have attracted considerable interest for a number of years, mainly because of their applicability in signal processing devices¹. Due to the concentration of energy at the waveguide, nonlinear effects are readily observable¹. This has stimulated a large number of theoretical investigations on various effects of nonlinearity on the propagation of guided acoustic waves. In certain geometries with material properties homogeneous along an axis vertical to the direction of propagation, the shear-horizontal component of the displacement field decouples from the components of sagittal polarization. In the presence of nonlinearity, acoustic waves of purely sagittal polarization can still exist. However, shear-horizontal acoustic waves can generate sagittal components of the displacement field via the second-order nonlinearity even in isotropic elastic media^{2, 3}. In derivations of evolution equations for slow variations of the envelope of shear-horizontal guided acoustic waves, these sagittal components may usually be eliminated adiabatically giving rise to an effective third-order nonlinearity in the evolution equation of the nonlinear Schrödinger type³. In situations where the second harmonic of a sinusoidal shear-horizontal wave with wavenumber q has frequency equal or close to a sagittal guided mode with wavenumber 2q, an adiabatic elimination is no longer possible. This nonlinear resonance of shear-horizontal and sagittal waves has been discussed in more detail in Ref. 4 for the case of plate modes. The purpose of this contribution is to incorporate dispersion and diffraction into the evolution equations for the envelopes of the two nonlinearly coupled carrier waves and to discuss, on the basis of these equations, the possibility of the formation of coupled solitary channels in this system.