A note on an integral associated with the Kelvin ship-wave pattern

Abstract

The velocity potential in the Kelvin ship-wave source can be partly expressed in terms of space derivatives of the single integral \[F(x,\rho,\alpha)=\int_{-\infty}^\infty \exp\,[-\frac{1}{2}\rho \cosh (2u-i\alpha)] \cos (x\cosh u)\,du,\] where $(x, \rho, \alpha)$ are cylindrical polar coordinates with origin based at the source and $-\pi/2\leq\alpha\leq\pi/2$. An asymptotic expansion of $F(x,\rho,\alpha)$ when $x$ and $\rho$ are small, but such that $M\equiv x^2/(4\rho)$ is large, was given using a non-rigorous approach by Bessho in 1964 as a sum involving products of Bessel functions. This expansion, together with an additional integral term, was subsequently proved by Ursell in 1988. Our aim here is to present an alternative asymptotic procedure for the case of large $M$. The resulting expansion consists of three distinct parts: a convergent sum involving the Struve functions, an asymptotic series and an exponentially small saddle-point contribution. Numerical computations are carried out to verify the accuracy of our expansion.