§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 $\varphi$ with $\varphi =0$ 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		$$z(\varphi ,\rho )={\int }_{-\pi /2}^{\pi /2}\cos \left(\rho \frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }\right)d\theta ,$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $(a,b)$: open interval and $z$: real parameter 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		$$\rho =gr/V^2.$$
ⓘ 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 $y=\varphi$, $u=\theta$, $g=1$, $k=\rho$, and

36.13.3		$$f(\theta ,\varphi )=-\frac{\cos \left(\theta +\varphi \right)}{{\cos }^2\theta }.$$
ⓘ Symbols: $\cos z$: 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		${\theta }_{+}(\varphi )$	$=\frac{1}{2}(\arcsin \left(3\sin \varphi \right)-\varphi ),$
		${\theta }_{-}(\varphi )$	$=\frac{1}{2}(\pi -\varphi -\arcsin \left(3\sin \varphi \right)).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function and $\sin z$: 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		$$|\varphi |={\varphi }_c=\arcsin \left(\frac{1}{3}\right)={19}^{\circ }.47122.$$
ⓘ Symbols: $\arcsin z$: 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 $z(\rho ,\varphi )$ 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 ${\theta }_{\pm }(\varphi )$ are real for and complex for $|\varphi |>{\varphi }_c$. (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions

36.13.7		$u(\varphi )$	$=\sqrt{{\displaystyle \frac{{\mathrm{\Delta }}^{1/2}(\varphi )}{2}}}\left({\displaystyle \frac{1}{\sqrt{f_{+}^{\prime \prime }(\varphi )}}}+{\displaystyle \frac{1}{\sqrt{-f_{-}^{\prime \prime }(\varphi )}}}\right),$
		$v(\varphi )$	$=\sqrt{{\displaystyle \frac{1}{2{\mathrm{\Delta }}^{1/2}(\varphi )}}}\left({\displaystyle \frac{1}{\sqrt{f_{+}^{\prime \prime }(\varphi )}}}-{\displaystyle \frac{1}{\sqrt{-f_{-}^{\prime \prime }(\varphi )}}}\right),$
ⓘ Defines: $u(\varphi )$: function (locally) and $v(\varphi )$: 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		$$z(\rho ,\varphi )=2\pi (\begin{array}{l}{\rho }^{-1/3}u(\varphi )\cos \left(\rho \tilde{f}(\varphi )\right)\mathrm{Ai}\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))\\ +{\rho }^{-2/3}v(\varphi )\sin \left(\rho \tilde{f}(\varphi )\right){\mathrm{Ai}}^{\prime }\left(-{\rho }^{2/3}\mathrm{\Delta }(\varphi )\right)(1+O\left(1/\rho \right))),\end{array}$$
$\rho \to \mathrm{\infty }$.
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $(a,b)$: open interval, $\sin z$: sine function, $z$: real parameter, $u(\varphi )$: function and $v(\varphi )$: function Referenced by: Figure 36.13.1, Figure 36.13.1 Permalink: http://dlmf.nist.gov/36.13.E8 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

See Figure 36.13.1.

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