Steady Radiating Gravity waves: An Exponential Asymptotics Approach

Abstract

The radiation of steady surface gravity waves by a uniform stream

$$U_{0}$$

U 0

over locally confined (width

$$L$$

L

) smooth topography is analyzed based on potential flow theory. The linear solution to this classical problem is readily found by Fourier transforms, and the nonlinear response has been studied extensively by numerical methods. Here, an asymptotic analysis is made for subcritical flow

$$D/\lambda > 1$$

D / λ > 1

in the low-Froude-number (

$$F^{2} \equiv \lambda /L \ll 1$$

F 2

≡ λ / L ≪ 1

) limit, where

$$\lambda = U_{0}^{2} /g$$

λ =

U

0

2

/ g

is the lengthscale of radiating gravity waves and

$$D$$

D

is the uniform water depth. In this regime, the downstream wave amplitude, although formally exponentially small with respect to

$$F$$

F

, is determined by a fully nonlinear mechanism even for small topography amplitude. It is argued that this mechanism controls the wave response for a broad range of flow conditions, in contrast to linear theory which has very limited validity.