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36 Integrals with Coalescing SaddlesApplications

§36.13 Kelvin’s Ship-Wave Pattern

A ship moving with constant speed V on deep water generates a surface gravity wave. In a reference frame where the ship is at rest we use polar coordinates r and ϕ with ϕ=0 in the direction of the velocity of the water relative to the ship. Then with g denoting the acceleration due to gravity, the wave height is approximately given by

36.13.1 z(ϕ,ρ)=π/2π/2cos(ρcos(θ+ϕ)cos2θ)dθ,


36.13.2 ρ=gr/V2.

The integral is of the form of the real part of (36.12.1) with y=ϕ, u=θ, g=1, k=ρ, and

36.13.3 f(θ,ϕ)=cos(θ+ϕ)cos2θ.

When ρ>1, that is, everywhere except close to the ship, the integrand oscillates rapidly. There are two stationary points, given by

36.13.4 θ+(ϕ) =12(arcsin(3sinϕ)ϕ),
θ(ϕ) =12(πϕarcsin(3sinϕ)).

These coalesce when

36.13.5 |ϕ|=ϕc=arcsin(13)=19.47122.

This is the angle of the familiar V-shaped wake. The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency ω as a function of wavevector 𝐤:

36.13.6 ω(𝐤)=gk+𝐕𝐤.

Here k=|𝐤|, and 𝐕 is the ship velocity (so that V=|𝐕|).

The disturbance z(ρ,ϕ) can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that θ±(ϕ) are real for |ϕ|<ϕc and complex for |ϕ|>ϕc. (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions

36.13.7 u(ϕ) =Δ1/2(ϕ)2(1f+′′(ϕ)+1f′′(ϕ)),
v(ϕ) =12Δ1/2(ϕ)(1f+′′(ϕ)1f′′(ϕ)),

the disturbance is

36.13.8 z(ρ,ϕ)=2π(ρ1/3u(ϕ)cos(ρf~(ϕ))Ai(ρ2/3Δ(ϕ))(1+O(1/ρ))+ρ2/3v(ϕ)sin(ρf~(ϕ))Ai(ρ2/3Δ(ϕ))(1+O(1/ρ))),

See Figure 36.13.1.

See accompanying text
Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of x=ρcosϕ, y=ρsinϕ. Magnify

For further information see Lord Kelvin (1891, 1905) and Ursell (1960, 1994).