On the Theory of the Kelvin Ship-Wave Source: Asymptotic Expansion of an Integral

The Kelvin ship-wave source is important in the mathematical theory of the wave resistance of ships but its velocity potential is difficult to evaluate numerically. In particular, the integral term F(x, ρ, α) = int^∞_-∞ exp{-1/2ρ \cosh (2u-iα)} \cos(x \cosh u) du in the source potential is difficult to evaluate when x and ρ are positive and small, and when -1/2π <=slant α <=slant 1/2π. In this work we are concerned with the asymptotic expansion of this integral when x^2/4ρ is large while x and ρ are not large, for which case the asymptotic expansion F(x, ρ, α) ~ -π I_0(1/2ρ) Y_0(x)-2π sum^∞_1 I_m(1/2ρ) Y_2m(x) \cos mα in terms of Bessel functions was proposed by Bessho (1964). This expansion has recently been shown to have great computational advantages but has never been proved. (The standard asymptotic theory of integrals, based on Watson's lemma, is not applicable, and the expansion is not of standard form.) In this paper it is shown that the expansion is valid except near α = ±1/2π where an additional term is needed.