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

TL;DR

This paper develops an alternative asymptotic expansion for a specific integral related to the Kelvin ship-wave pattern, involving Struve functions, an asymptotic series, and saddle-point contributions, verified through numerical analysis.

Contribution

It introduces a new asymptotic method for large parameter M, providing a detailed expansion involving Struve functions and saddle-point analysis, improving upon previous approaches.

Findings

Derived a convergent sum with Struve functions

Established an asymptotic series for the integral

Validated the expansion with numerical computations

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.