Kelvin’s Ship-Wave Pattern

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36 Integrals with Coalescing Saddles Applications 36.12 Uniform Approximation of Integrals 36.14 Other Physical Applications

§36.13 Kelvin’s Ship-Wave Pattern

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

36.13.1	z(ϕ,ρ)=∫π/2−π/2cos(ρcos(θ+ϕ)cos2θ)dθ, z⁢(ϕ,ρ)=∫-π/2π/2cos⁡(ρ⁢cos⁡(θ+ϕ)cos2⁡θ)⁢dθ,

where

36.13.2	ρ=gr/V2. ρ=g⁢r/V2.

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

36.13.3	f(θ,ϕ)=−cos(θ+ϕ)cos2θ. f⁡(θ,ϕ)=-cos⁡(θ+ϕ)cos2⁡θ.

When  ρ>1 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) θ+⁢(ϕ)	=12(arcsin(3sinϕ)−ϕ), =12⁢(arcsin⁡(3⁢sin⁡ϕ)-ϕ),
θ−(ϕ) θ-⁢(ϕ)	=12(π−ϕ−arcsin(3sinϕ)). =12⁢(π-ϕ-arcsin⁡(3⁢sin⁡ϕ)).

These coalesce when

36.13.5	|ϕ|=ϕc=arcsin(13)=19∘.47122. |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 ωas a function of wavevector  k k:

36.13.6	ω(k)=gk−−√+V⋅k. ω⁡(k)=g⁢k+V⋅k.

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

The disturbance  z(ρ,ϕ) 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) θ±⁢(ϕ) are real for  |ϕ|<ϕc |ϕ|<ϕc and complex for  |ϕ|>ϕc |ϕ|>ϕc. (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions

36.13.7	u(ϕ) u⁡(ϕ)	=Δ1/2(ϕ)2−−−−−−−√⎛⎝⎜1f''+(ϕ)−−−−−√+1−f''−(ϕ)−−−−−−−√⎞⎠⎟, =Δ1/2⁢(ϕ)2⁢(1f+′′⁢(ϕ)+1-f-′′⁢(ϕ)),
v(ϕ) v⁡(ϕ)	=12Δ1/2(ϕ)−−−−−−−−√⎛⎝⎜1f''+(ϕ)−−−−−√−1−f''−(ϕ)−−−−−−−√⎞⎠⎟, =12⁢Δ1/2⁢(ϕ)⁢(1f+′′⁢(ϕ)-1-f-′′⁢(ϕ)),

the disturbance is

36.13.8	z(ρ,ϕ)=2π(ρ−1/3u(ϕ)cos(ρf˜(ϕ))Ai(−ρ2/3Δ(ϕ))(1+O(1/ρ))+ρ−2/3v(ϕ)sin(ρf˜(ϕ))Ai′(−ρ2/3Δ(ϕ))(1+O(1/ρ))), z⁢(ρ,ϕ)=2⁢π⁢(ρ-1/3⁢u⁡(ϕ)⁢cos⁡(ρ⁢f~⁢(ϕ))⁢Ai⁡(-ρ2/3⁢Δ⁢(ϕ))⁢(1+O⁡(1/ρ))+ρ-2/3⁢v⁡(ϕ)⁢sin⁡(ρ⁢f~⁢(ϕ))⁢Ai′⁡(-ρ2/3⁢Δ⁢(ϕ))⁢(1+O⁡(1/ρ))),
ρ→∞ ρ→∞.

See Figure 36.13.1.

Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation ( 36.13.8), as a function of  x=ρcosϕ x=ρ⁢cos⁡ϕ,  y=ρsinϕ y=ρ⁢sin⁡ϕ. 

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

© 2010–2016 NIST /  Privacy Policy /  Disclaimer /  Feedback; Version 1.0.13; Release date 2016-09-16. A  printed companion is available. 36.12 Uniform Approximation of Integrals 36.14 Other Physical Applications