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Water Waves and the Korteweg–de Vries Equation

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

A  wave is usually defined as the propagation of a disturbance in a medium.

The simplest example is the exponential or sinusoidal wave which has the form

$$ \begin{aligned}[b] u(x,t) &= a \operatorname{Re} \exp\left[i(kx-\omega t)\right] \\ &=a\cos(kx-\omega t)\:, \end{aligned} $$
(1)

where a is called the amplitude, Re stands for the real part, \( \smash{ k \: (= \frac{2\pi}{\lambda}) } \) is called the wavenumber and λ is called the wavelength of the wave, and ω is called the frequency and it is a definite function of the wavenumber k and hence, \( { \omega=\omega(k) } \) is determined by the particular equation of the problem. The quantity \( { \theta=kx-\omega t } \) is called the phase of the wave so that a wave of a constant phase propagate with \( { kx-\omega t=\text{constant} } \).

The mathematical relation

$$ \begin{aligned}[b] \omega &= \omega(k)\quad\text{or} \\ D(\omega,k)&=0\:, \end{aligned} $$
(2)

is called the dispersion relation...

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Abbreviations

Axisymmetric (concentric) KdV equation:

This is a partial differential equation for the free surface \( { \eta(R,t) } \) in the form \( \smash{ (2\eta_{R}+\frac{1}{R}\,\eta+3\eta\eta_{\xi})+\frac{1}{3}\,\eta_{\xi\xi\xi}=0 } \).

Benjamin–Feir instability of water waves:

This describes the instability of nonlinear water waves.

Bernoulli's equation:

This is a partial differential equation which determines the pressure in terms of the velocity potential in the form \( \phi_{t} + \smash{\frac{1}{2}(\nabla\phi)^{2} + \frac{P}{\rho}} + gz = 0 \), where ϕ is the velocity potential, P is the pressure, ρ is the density and g is the acceleration due to gravity.

Boussinesq equation:

This is a nonlinear partial differential equation in shallow water of depth h given by

$$ u_{tt}-c^{2}u_{xx}+\frac{1}{2}\big(u^{2}\big)_{xx}=\frac{1}{3}\, h^{2}u_{xxtt}\:, \\ \text{where } \; c^{2}=\sqrt{gh}\:.$$
Cnoidal waves:

Waves are represented by the Jacobian elliptic function \( { cn(z,m) } \).

Continuity equation:

This is an equation describing the conservation of mass of a fluid. More precisely, this equation for an incompressible fluid is \( { \mathit{div} \: \mathbf{u}=\frac{\partial u}{\partial x}} \allowbreak {+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0 } \), where \( { \mathbf{u}=(u,v,w) } \) is the velocity field, and \( { \mathbf{x}=(x,y,z) } \).

Continuum hypothesis:

It requires that the velocity \( { \mathbf{u}=} \allowbreak {(u,v,w) } \), pressure p and density ρ are continuous functions of position \( { \mathbf{x}=(x,y,z) } \) and time t.

Crapper's nonlinear capillary waves:

Pure progressive capillary waves of arbitrary amplitude.

Dispersion relation:

A mathematical relation between the wavenumber, frequency and/or the amplitude of a wave.

Euler equations:

This is a nonlinear partial differential equation for an inviscid incompressible fluid flow governed by the velocity field \( { \mathbf{u}=(u,v,w) } \) and pressure \( { P(\mathbf{x},t) } \) under the external force \( { \mathbf{F} } \). More precisely, \( { \frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\frac{1}{\rho}\,\nabla P+\mathbf{F} } \), where ρ is the constant density of the fluid.

Group velocity:

The velocity defined by the derivative of the frequency with respect to the wavenumber \( { (c_{g}=\mskip2mu\mathrm{d}\omega/\text{d} k) } \).

Johnson's equation:

This is a nearly concentric KdV equation for \( { \eta(R,\xi,\theta) } \) in cylindrical polar coordinates in the form

$$ \left(2\eta_{R}+\frac{1}{R}\,\eta+3\eta\eta_{\xi}+\frac{1}{3}\,\eta_{\xi\xi\xi}+\frac{1}{R^{2}}\,\eta_{\theta\theta}\right)=0\:.$$
Kadomtsev–Petviashvili (KP) equation:

This is a two‐dimensional KdV equation in the form

$$ \left(2\eta_{t}+3\eta\eta_{\xi}+\frac{1}{3}\,\eta_{\xi\xi\xi}\right)_{\xi}+\eta_{yy}=0\:.$$
KdV–Burgers equation:

This is a nonlinear partial differential equation in the form

$$ \eta_{t}+c_{0}\,\eta_{x}+d\,\eta\,\eta_{x}+\mu\,\eta_{xxx}-\nu\,\eta_{xx}=0\:,\quad\\ \text{where }\mu=\frac{1}{6}\, c_{0}h_{0}^{2}\:. $$
Korteweg–de Vries (KdV) equation:

This is a nonlinear partial differential equation for a solitary wave (or soliton). This equation in a shallow water of depth h is governed by the free surface elevation \( { \eta(x,t) } \) in the form \( { \frac{\partial\eta}{\partial t}+c(1+\frac{3}{2h}\,\eta)\eta_{x}+\left(\frac{ch^{2}}{6}\right)\eta_{xxx}=0 } \), where \( \smash{ c} \allowbreak \smash{=\sqrt{gh} } \) is the shallow water speed.

Laplace equation:

This is partial differential equation of the form \( { \nabla^{2}\phi=\phi_{xx}+\phi_{yy}+\phi_{zz} } \) where the \( { \phi=} \allowbreak {\phi(x,y,z)} \) is the potential.

Linear dispersion relation:

A mathematical relation between the wavenumber k and the frequency ω of waves \( { (\omega=\omega(k)) } \).

Linear dispersive waves:

Waves with the given dispersion relation between the wavenumber and the frequency.

Linear Schrödinger equation:

This can be written in the form \( { i\, a_{t}+\frac{1}{2}\,\omega^{\prime\prime}(k)\frac{\partial^{2}a}{\partial x^{2}}=0 } \), where \( { a=a(x,t) } \) is the amplitude and \( { \omega=\omega(k) } \).

Linear wave equation in Cartesian coordinates:

In three dimensions this can be written in the form \( { u_{tt}=c^{2}\,\nabla^{2}u } \), where c is a constant and \( {\nabla^{2}=} \allowbreak {\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial}{\partial z^{2}} } \) is the three‐dimensional Laplacian.

Linear wave equation in cylindrical polar coordinates:

This can be written in the form \( { u_{tt}=u_{rr}+\frac{1}{r}\, u_{r}} \allowbreak {+\frac{1}{r^{2}}\, u_{\theta\theta}} \).

Navier–Stokes equations:

This is a nonlinear partial differential equation for an incompressible and viscous fluid flow governed by the velocity field \( { \mathbf{u}=(u,v,w) } \) and pressure \( { P(\mathbf{x},t) } \) under the action of external force \( { \mathbf{F} } \). More precisely, \( { \frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\frac{1}{\rho}\,\nabla P} \allowbreak {+\nu^{2}\mathbf{u}+\mathbf{F}} \), where ρ is the density and \( { \nu=(\mu/\rho) } \) is the kinematic viscosity.

Nonlinear dispersion relation:

A mathematical relation between the wavenumber k, frequency ω and the amplitude a that is \( { D(\omega,k,a)=0 } \).

Nonlinear dispersive waves:

Waves with the given dispersion relation between the wavenumber, frequency and the amplitude.

Non‐linear Schrödinger (NLS) equation:

This is a nonlinear partial differential equation for the nonlinear modulation of a monochromatic wave. The amplitude, \( { a(x,t) } \) of the modulation satisfies the equation \( \smash{ i\big(\frac{\partial a}{\partial t}+\omega^{\prime}_{0}\,\frac{\partial a}{\partial x}\big)+\frac{1}{2}\,\omega^{\prime\prime}_{0}\,\frac{\partial^{2}a}{\partial x^{2}}+\gamma|a|^{2}a=0 } \), where \( { \omega_{0}} \allowbreak {=\omega_{0}(k) } \), and γ is a constant.

Ocean waves:

Waves observed on the surface or inside the ocean.

Phase velocity:

The velocity defined by the ratio of the frequency ω and the wavenumber k, \( \smash{ (c_{p}=\frac{\omega}{k}) } \).

Resonant or critical phenomenon:

Waves with unbounded amplitude.

Sinusoidal (or exponential) wave:

A wave is of the form \( u(x,t)=a\, \operatorname{Re}\exp[i(kx-\omega t)]=a\cos(kx-\omega t) \), where a is amplitude, k is the wavenumber \( \smash{ (k=\frac{2\pi}{\lambda}) } \), λ is the wavelength and ω is the frequency.

Solitary waves (or soliton):

Waves describing a single hump of given height travel in a medium without change of shape.

Stokes expansion:

This is an expansion of the frequency in terms of the wavenumber k and the amplitude a, that is, \( \smash{ \omega(k)=\omega_{0}(k)+\omega_{2}(k)a^{2}+\ldots\, } \).

Stokes wave:

Water waves with dispersion relation involving the wavenumber, frequency and amplitude.

Surface‐capillary gravity waves:

Waves under the joint action of the gravitational field and surface tension.

Surface gravity waves:

Water waves under the action of the gravitational field.

Variational principle:

For three‐dimensional water waves, it is of the form \( \smash{ \delta I=\delta\iint_{D}L\, \mskip2mu\mathrm{d}\mathbf{x}\, \text{d} t=0} \), where L is called the Lagrangian.

Velocity potential:

A single valued function \( { \phi=\phi(\mathbf{x},t) } \) defined by \( { \mathbf{u}=\nabla\phi } \).

Water waves:

Waves observed on the surface or inside of a body of water.

Waves on a running stream:

Waves observed on the surface or inside of a body of fluid which is moving with a given velocity.

Whitham averaged variational principle:

This can be formulated in the form \( \smash{ \delta\iint\mathcal{L}\, \mskip2mu\mathrm{d}\mathbf{x}\, \text{d} t=0 } \), where \( \smash{ \mathcal{L} } \) is called the Whitham average Lagrangian over the phase of the integral of the Lagrangian L defined by \( \smash{ \mathcal{L}(\omega,\mathbf{k},a,\mathbf{x},t)=\frac{1}{2\pi}\int_{0}^{2\pi}L\, \mskip2mu\mathrm{d}\theta } \), where L is the Lagrangian.

Whitham's conservation equations:

These are first order nonlinear partial differential equations in the form \( \frac{\partial k}{\partial t}+\frac{\partial\omega}{\partial x}=0 \), \( \frac{\partial}{\partial t}\left\{ f(k)A^{2}\right\} +\frac{\partial}{\partial x}\left\{ f(k)C(k)A^{2}\right\} =0 \), where \( { k=k(x,t) } \) is the density of waves, \( { \omega=\omega(x,t) } \) is the flux of waves, \( { A=A(x,t) } \) is the amplitude and \( \smash{ f(k) } \) is an arbitrary function.

Whitham's equation:

This first order nonlinear partial differential equation represents the conservation of waves. Mathematically, \( { (\partial k/\partial t)+(\partial\omega/\partial x)=0 } \) where \( { k=k(x,t) } \) is the density of waves and \( { \omega=\omega(x,t) } \) is the flux of waves.

Whitham's equation for slowly varying wavetrain:

This is written in the form \( { \frac{\partial}{\partial t}\,\mathcal{L}_{\omega}-\frac{\partial}{\partial x_{i}}\,\mathcal{L}_{k_{i}}=0 } \), where \( { \mathcal{L} } \) is the Whitham averaged Lagrangian.

Whitham's nonlinear nonlocal equations:

It is in the form \( \smash{u_{t}+ duu_{x}+\!\int_{-\infty}^{\infty} K(x\!-\!s) \, u_{s}(s,t) \, \mathrm{d} s=0} \), where \( { K(x)=\mathcal{F}^{-1}\left\{ c(k)=\omega/k\right\} } \) and \( { \mathcal{F}^{-1} } \) is the inverse Fourier transformation.

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  1. Department of Mathematics, University of Texas – Pan American, Edinburg, USA

    Lokenath Debnath

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Debnath, L. (2009). Water Waves and the Korteweg–de Vries Equation. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_586

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