# §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 $\phi$ with $\phi=0$ 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 $z(\phi,\rho)=\int_{-\pi/2}^{\pi/2}\cos\left(\rho\frac{\cos\left(\theta+\phi% \right)}{{\cos}^{2}\theta}\right)\mathrm{d}\theta,$ ⓘ Defines: $z(\NVar{\phi},\NVar{\rho})$: wave height approximant (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/36.13.E1 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

where

 36.13.2 $\rho=\ifrac{gr}{V^{2}}.$ ⓘ Symbols: $V$: speed and $g$: gravity Permalink: http://dlmf.nist.gov/36.13.E2 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

The integral is of the form of the real part of (36.12.1) with $y=\phi$, $u=\theta$, $g=1$, $k=\rho$, and

 36.13.3 $f(\theta,\phi)=-\frac{\cos\left(\theta+\phi\right)}{{\cos}^{2}\theta}.$ ⓘ Symbols: $\cos\NVar{z}$: cosine function Permalink: http://dlmf.nist.gov/36.13.E3 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

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

 36.13.4 $\displaystyle\theta_{+}(\phi)$ $\displaystyle=\tfrac{1}{2}(\operatorname{arcsin}\left(3\sin\phi\right)-\phi),$ $\displaystyle\theta_{-}(\phi)$ $\displaystyle=\tfrac{1}{2}(\pi-\phi-\operatorname{arcsin}\left(3\sin\phi\right% )).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{arcsin}\NVar{z}$: arcsine function and $\sin\NVar{z}$: sine function Permalink: http://dlmf.nist.gov/36.13.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.13 and Ch.36

These coalesce when

 36.13.5 $|\phi|=\phi_{c}=\operatorname{arcsin}\left(\tfrac{1}{3}\right)=19^{\circ}.47122.$ ⓘ Symbols: $\operatorname{arcsin}\NVar{z}$: arcsine function Permalink: http://dlmf.nist.gov/36.13.E5 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

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 $\omega$ as a function of wavevector $\mathbf{k}$:

 36.13.6 $\omega(\mathbf{k})=\sqrt{gk}+\mathbf{V}\cdot\mathbf{k}.$ ⓘ Symbols: $k$: variable and $g$: gravity Permalink: http://dlmf.nist.gov/36.13.E6 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

Here $k=|\mathbf{k}|$, and $\mathbf{V}$ is the ship velocity (so that $\mathrm{V}=|\mathbf{V}|$).

The disturbance $z(\rho,\phi)$ 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 $\theta_{\pm}(\phi)$ are real for $|\phi|<\phi_{c}$ and complex for $|\phi|>\phi_{c}$. (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions

 36.13.7 $\displaystyle u(\phi)$ $\displaystyle=\sqrt{\dfrac{\Delta^{1/2}(\phi)}{2}}\left(\dfrac{1}{\sqrt{f_{+}^% {\prime\prime}(\phi)}}+\dfrac{1}{\sqrt{-f_{-}^{\prime\prime}(\phi)}}\right),$ $\displaystyle v(\phi)$ $\displaystyle=\sqrt{\dfrac{1}{2\Delta^{1/2}(\phi)}}\left(\dfrac{1}{\sqrt{f_{+}% ^{\prime\prime}(\phi)}}-\dfrac{1}{\sqrt{-f_{-}^{\prime\prime}(\phi)}}\right),$ ⓘ Defines: $u(\phi)$: function (locally) and $v(\phi)$: function (locally) Permalink: http://dlmf.nist.gov/36.13.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.13 and Ch.36

the disturbance is

 36.13.8 $z(\rho,\phi)=2\pi\left(\rho^{-1/3}u(\phi)\cos\left(\rho\widetilde{f}(\phi)% \right)\mathrm{Ai}\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho\right% ))+\rho^{-2/3}v(\phi)\sin\left(\rho\widetilde{f}(\phi)\right)\mathrm{Ai}'\left% (-\rho^{2/3}\Delta(\phi)\right)\*(1+O\left(1/\rho\right))\right),$ $\rho\to\infty$.

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=ρ⁢cos⁡ϕ, y=ρ⁢sin⁡ϕ. Magnify

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