§36.13 Kelvin’s Ship-Wave Pattern

Notes:: Figure 36.13.1 was generated by the authors.
Keywords:: Airy functions, Kelvin’s ship-wave pattern, ship wave, water waves
Referenced by:: §2.8(iii)
Permalink:: http://dlmf.nist.gov/36.13

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

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $(a,b)$ : open interval and $z$ : real parameter
Permalink:: http://dlmf.nist.gov/36.13.E1
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where

36.13.2 $\rho=\ifrac{gr}{V^{2}}.$

Symbols:: $V$ : speed and $g$ : gravity
Permalink:: http://dlmf.nist.gov/36.13.E2
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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
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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

$\theta _{{+}}(\phi)=\tfrac{1}{2}(\mathop{\mathrm{arcsin}\/}\nolimits\!\left(3\mathop{\sin\/}\nolimits\phi\right)-\phi),$

$\theta _{{-}}(\phi)=\tfrac{1}{2}(\pi-\phi-\mathop{\mathrm{arcsin}\/}\nolimits\!\left(3\mathop{\sin\/}\nolimits\phi\right)).$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and $\mathop{\sin\/}\nolimits z$ : sine function
Permalink:: http://dlmf.nist.gov/36.13.E4
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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
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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

$u(\phi)=\sqrt{\dfrac{\Delta^{{1/2}}(\phi)}{2}}\left(\dfrac{1}{\sqrt{f_{{+}}^{{\prime\prime}}(\phi)}}+\dfrac{1}{\sqrt{-f_{{-}}^{{\prime\prime}}(\phi)}}\right),$

$v(\phi)=\sqrt{\dfrac{1}{2\Delta^{{1/2}}(\phi)}}\left(\dfrac{1}{\sqrt{f_{{+}}^{{\prime\prime}}(\phi)}}-\dfrac{1}{\sqrt{-f_{{-}}^{{\prime\prime}}(\phi)}}\right),$

Symbols:: $u(\phi)$ : function and $v(\phi)$ : function
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the disturbance is

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, $(a,b)$ : open interval, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : real parameter, $u(\phi)$ : function and $v(\phi)$ : function
Referenced by:: Figure 36.13.1, Figure 36.13.1
Permalink:: http://dlmf.nist.gov/36.13.E8
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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 $x=\rho\mathop{\cos\/}\nolimits\phi$ , $y=\rho\mathop{\sin\/}\nolimits\phi$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $y$ : real parameter and $x$ : real parameter
Referenced by:: §36.13, §36.13
Permalink:: http://dlmf.nist.gov/36.13.F1
Encodings:: pdf, png

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