An axisymmetric Boussinesq wave

Abstract

An axisymmetric gravity wave, for which each of nonlinearity, dispersion and radial spreading is weak but significant, is determined as a similarity solution with slowly varying amplitude $\frac{3}{4}{\cal S}a $ and length scale l, where $a/d \propto (r/d)^{\frac{2}{3}}, l/d \propto (r/d)^{\frac{1}{3}}, r$ is the radius, d is the depth, and [Sscr ] is the family parameter of the solutions. It is shown that the free-surface displacement η(r,t) is either a wave of elevation (η ≥ 0) or a wave of depression (η ≤ 0) and that (|η|/)^½ satisfies a Painlevé equation that is a nonlinear generalization of Airy's equation. Representative numerical solutions and asymptotic approximations for small and large [Sscr ] are presented. It is shown that the similarity solution conserves energy but not mass, in consequence of which (in order to obtain a complete solution to a well-posed initial-value problem) it must either be accompanied by some other component or components or be driven by a source (or sink) in some interior domain in which the implicit restriction r [Gt ] d is violated. A linear model is developed that is valid for r [lsim ] d and compensates for the mass defect of, and matches, the nonlinear similarity solution for |[Sscr ]| [Lt ] 1.