Finite-amplitude acoustic pulses in a waveguide layer with longitudinal inhomogeneity

This paper considers the propagation of a weakly nonlinear acoustic pulse in a slightly curved waveguide layer which is strongly inhomogeneous in the transverse direction and weakly inhomogeneous in the longitudinal direction. The basic system of hydrodynamic equations reduces to a nonlinear wave equation, whose coefficients are determined using the equations of state of the medium. It is established that as the adiabatic exponent passes through the value $\gamma$ = 3/2, the nature of the pulse propagation changes: for large values of $\gamma$, the medium is focusing, and for smaller values, it is defocusing. It is shown that the pulse propagation process is characterized by three scales: the high-frequency filling is modulated by the envelope, whose evolution, in turn, is determined by the moderate-rate evolution of the envelope phase and slow amplitude variation. A generalized nonlinear Schrödinger equation with the coefficients dependent on the longitudinal coordinate is derived for the pulse envelope. An explicit soliton solution of this equation is constructed for some types of longitudinal inhomogeneity.