On the theory of the Kelvin ship-wave source: the near-field convergent expansion of an integral

The velocity potential of the Kelvin ship-wave source is fundamental in the mathematical theory of the wave resistance of ships, but is difficult to evaluate numerically. We shall be concerned with the integral term F(x, ρ, ∝) = ∫∞ -∞ exp {— 1/2ρ cosh (2u — i∝)} cos (x cosh u)du in the source potential, where x and ρ are positive and —1/2π ≼ ∝ ≼ 1/2π, which is difficult to evaluate when x and ρ are small. It will be shown here that F(x, ρ, ∝) = 1/2ƒ(x, ρ, ∝ ) + 1/2ƒ(x, ρ, ─∝) + 1/2ƒ(─x , ρ, ∝) + ½ƒ(─x, ρ, ─∝), where ƒ(x, ρ, ∝) = P0 (x, ρ e-i∝) Σgm (x, ρ ei∝) cm (x, ρ e-i∝) + P1 (x, ρ e-i∝) Σgm (x, ρ ei∝) bm (x, ρ e-i∝) + Σgm (x, ρ ei∝) am (x, ρ e-i∝) In this expression each of the functions gm(x, ρ ei∝), am (x, ρ e-i∝), bm (x, ρ e-i∝), cm(x, ρ e-i∝), satisfies a simple three-term recurrence relation and tends rapidly to 0 for small x and ρ when m → ∞, and the functions P 0 (x, ρ e-i∝) and P1(x, ρ ei∝) are simply related to the parabolic cylinder functions Dv(ζ) respectively, where ζ = — ix(2ρ)-1/2e1/2i∝.