High-frequency nonlinear acoustic beams and wave packets

Gaussian beams and wave packets are formally equivalent to high-frequency solutions of the linear wave equation with complex-valued phase functions. The theory of weakly nonlinear high-frequency waves is extended in this paper to allow for complex phase solutions. The procedure is similar to nonlinear geometrical optics and the nonlinearity causes the phase to vary, leading to the development of weak shocks. However, unlike standard geometrical optics that describes the evolution of wave fronts, complex phase solutions correspond to Gaussian decay away from a central ray, and the associated curvatures are complex valued. Geometrical singularities due to caustics and loci do not occur, and the only singularities in the theory are purely nonlinear. The theory is developed specifically for nonlinear acoustic waves in a homogeneous inviscid fluid. The time taken for nonlinear singularities to develop is compared for a real spherical wave front and a Gaussian beam with Gaussian radius equal to the real radius of curvature of the wave front. The time to blowup for the Gaussian beam is intermediate between the shortest possible, which is for the converging wave front, and the longest possible, for the diverging front.