Kelvin’s Ship-Wave Pattern

超、凢脫俗 2022-07-13 01:27 35阅读 0赞

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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 #

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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][V] g denoting the acceleration due to gravity, the wave height is approximately given by

where

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

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

These coalesce when

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:

Here [k][]=|k| k=|k|, and V V is the ship velocity (so that [V][V 1]=|V| V=|V|).

The disturbance [z][y][(][Link 1]ρ[,][Link 1]ϕ[)][Link 1] 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)][2.4_v].) Then with the definitions ([36.12.12][]), and the real functions

the disturbance is

See Figure [36.13.1][].

[![See accompanying text][]][See accompanying text 1] Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation ( [36.13.8][]), as a function of [x][y]=ρ[cos][]ϕ x=ρ⁢cos⁡ϕ, [y][]=ρ[sin][]ϕ y=ρ⁢sin⁡ϕ. [![Magnify][]][See accompanying text 1]

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

[© 2010–2016 NIST][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][]

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[V]: http://dlmf.nist.gov/36.13#p1
[36.12.1]: http://dlmf.nist.gov/36.12#E1
[y]: http://dlmf.nist.gov/36.1#p2.t1.r4
[u]: http://dlmf.nist.gov/36.13#E7
[k]: http://dlmf.nist.gov/36.1#p2.t1.r2
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[36.13.8]: http://dlmf.nist.gov/36.13#E8
[cos]: http://dlmf.nist.gov/4.14#E2
[sin]: http://dlmf.nist.gov/4.14#E1
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[1891]: http://dlmf.nist.gov/bib/L#bib1241
[1905]: http://dlmf.nist.gov/bib/L#bib1242
[1960]: http://dlmf.nist.gov/bib/U#bib2283
[1994]: http://dlmf.nist.gov/bib/U#bib2287
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