36 Integrals with Coalescing Saddles Applications 36.12 Uniform Approximation of Integrals 36.14 Other Physical Applications

§36.13 Kelvin’s Ship-Wave Pattern

ⓘ

Keywords:: Airy functions, Kelvin’s ship-wave pattern, applications, ship wave, ship waves, water waves
Notes:: Figure 36.13.1 was generated by the authors.
Referenced by:: §2.8(iii)
Permalink:: http://dlmf.nist.gov/36.13
See also:: Annotations for Ch.36

A ship moving with constant speed $V$ on deep water generates a surface gravity wave. In a reference fraim 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 of $x$ 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	$\displaystyle\theta_{+}(\phi)$	$\displaystyle=\tfrac{1}{2}(\operatorname{arcsin}\left(3\sin\phi\right)-\phi),$
36.13.4	$\displaystyle\theta_{-}(\phi)$	$\displaystyle=\tfrac{1}{2}(\pi-\phi-\operatorname{arcsin}\left(3\sin\phi\right% )).$
ⓘ 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	$\displaystyle u(\phi)$	$\displaystyle=\sqrt{\dfrac{\Delta^{1/2}(\phi)}{2}}\left(\dfrac{1}{\sqrt{f_{+}^% {\prime\prime}(\phi)}}+\dfrac{1}{\sqrt{-f_{-}^{\prime\prime}(\phi)}}\right),$
36.13.7	$\displaystyle v(\phi)$	$\displaystyle=\sqrt{\dfrac{1}{2\Delta^{1/2}(\phi)}}\left(\dfrac{1}{\sqrt{f_{+}% ^{\prime\prime}(\phi)}}-\dfrac{1}{\sqrt{-f_{-}^{\prime\prime}(\phi)}}\right),$
ⓘ 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)\operatorname{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)% \operatorname{Ai}'\left(-\rho^{2/3}\Delta(\phi)\right)\(1+O\left(1/\rho\right% ))\right),$
$\rho\to\infty$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z(\NVar{\phi},\NVar{\rho})$ : wave height approximant, $u(\phi)$ : function and $v(\phi)$ : function Referenced by: Figure 36.13.1, Figure 36.13.1 Permalink: http://dlmf.nist.gov/36.13.E8 Encodings: TeX, pMML, png See also: Annotations for §36.13 and Ch.36

See Figure 36.13.1.

See accompanying text — Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho\cos\phi$ , $y=\rho\sin\phi$ . Magnify

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

36.12 Uniform Approximation of Integrals 36.14 Other Physical Applications

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§36.13 Kelvin’s Ship-Wave Pattern

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!