Ring waves on the surface of shear flows: a linear and nonlinear theory

Abstract

A theory is presented which describes the propagation of a ring wave on the surface of a flow which moves with some prescribed velocity profile. The problem is formulated in suitable far-field variables (which give the concentric KdV equation for a stationary flow), but allowance is made for the fact that the wavefront is no longer circular. The leading order of this small-amplitude long-wave theory reduces to a generalized Burns condition which is used to determine the shape of the wavefront. This condition is written as \[ (h^2+h^{\prime 2}\int^1_2dz/[F(z, \theta)]^2=1, \] where F(z, θ) = -1 + {U(z) − c} (h cos θ − h′ sin θ), U(z) is the velocity profile, c is a parameter and the local characteristic coordinate for the wave is ξ = rh(θ) − t. (The Burns condition is interpreted in terms of the finite part of the integral in order to allow the possibility of a critical layer where F(zc, θ) = 0, 0 < zc < 1.) The wavefront is represented by r = constant /h(θ). A model boundary-layer profile, which gives rise to a critical-layer solution, is chosen for U(z). The role of this critical-layer solution, and the general question of upstream propagation, is then examined by constructing a wavefront which is continuous from the downstream to the upstream side. Solutions are presented which demonstrate that a critical layer never appears and so upstream propagation is necessary. These solutions (for various values of surface speed and boundary-layer thickness) are one branch of what we might term the singular solution of the differential equation for h(θ). The other branch corresponds to a solution which has a critical layer for all θ, which would seem to be unphysical since this solution is not an outward propagating ring wave.At the next order we obtain the equation which describes the dominant contribution to the surface wave, in this approximation. The equation is a new form of Korteweg–de Vries equation; the novel feature is the dependence on the polar angle, θ. This equation is not analysed in any detail here, but the connection with plane waves over a shear flow, and with concentric waves in the absence of shear, is made.