# §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 $z(\phi,\rho)=\int_{-\pi/2}^{\pi/2}\mathop{\cos\/}\nolimits\!\left(\rho\frac{% \mathop{\cos\/}\nolimits\!\left(\theta+\phi\right)}{{\mathop{\cos\/}\nolimits^% {2}}\theta}\right)d\theta,$

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

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 $f(\theta,\phi)=-\frac{\mathop{\cos\/}\nolimits\!\left(\theta+\phi\right)}{{% \mathop{\cos\/}\nolimits^{2}}\theta}.$ Symbols: $\mathop{\cos\/}\nolimits z$: cosine function Permalink: http://dlmf.nist.gov/36.13.E3 Encodings: TeX, pMML, png

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 $\displaystyle\theta_{+}(\phi)$ $\displaystyle=\tfrac{1}{2}(\mathop{\mathrm{arcsin}\/}\nolimits\!\left(3\mathop% {\sin\/}\nolimits\phi\right)-\phi),$ $\displaystyle\theta_{-}(\phi)$ $\displaystyle=\tfrac{1}{2}(\pi-\phi-\mathop{\mathrm{arcsin}\/}\nolimits\!\left% (3\mathop{\sin\/}\nolimits\phi\right)).$

These coalesce when

 36.13.5 $|\phi|=\phi_{c}=\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\tfrac{1}{3}\right)% =19^{\circ}.47122.$ Symbols: $\mathop{\mathrm{arcsin}\/}\nolimits z$: arcsine function Permalink: http://dlmf.nist.gov/36.13.E5 Encodings: TeX, pMML, png

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

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

the disturbance is

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

See Figure 36.13.1.

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