Whitham equation

{{short description|Non-local model for non-linear dispersive waves}}

In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves. {{harvtxt|Debnath|2005|p=364}}{{harvtxt|Naumkin|Shishmarev|1994|p=1}}{{harvtxt|Whitham|1974|pp=476–482}}

The equation is notated as follows:{{Equation box 1|cellpadding|border|indent=:|equation= $\frac{\partial \eta}{\partial t}
+ \alpha \eta \frac{\partial \eta}{\partial x}
+ \int_{-\infty}^{+\infty} K(x-\xi)\, \frac{\partial \eta(\xi,t)}{\partial \xi}\, \text{d}\xi
= 0.$ |border colour=#0073CF|background colour=#F5FFFA}}This integro-differential equation for the oscillatory variable η(x,t) is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves in 1967.{{harvtxt|Whitham|1967}} Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.{{harvtxt|Hur|2017}}

For a certain choice of the kernel K(x − ξ) it becomes the Fornberg–Whitham equation.

Water waves

Using the Fourier transform (and its inverse), with respect to the space coordinate x and in terms of the wavenumber k:

For surface gravity waves, the phase speed c(k) as a function of wavenumber k is taken as:

:: $c_\text{ww}(k) = \sqrt{ \frac{g}{k}\, \tanh(kh)},$ {{pad|2em}} while {{pad|2em}} $\alpha_\text{ww} = \frac{3}{2} \sqrt{\frac{g}{h}},$

:with g the gravitational acceleration and h the mean water depth. The associated kernel K_ww(s) is, using the inverse Fourier transform:

:: $K_\text{ww}(s) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} c_\text{ww}(k)\, \text{e}^{iks}\, \text{d}k
= \frac{1}{2\pi} \int_{-\infty}^{+\infty} c_\text{ww}(k)\, \cos(ks)\, \text{d}k,$

:since c_ww is an even function of the wavenumber k.

The Korteweg–de Vries equation (KdV equation) emerges when retaining the first two terms of a series expansion of c_ww(k) for long waves with {{nowrap|kh ≪ 1}}:

:: $c_\text{kdv}(k) = \sqrt{gh} \left( 1 - \frac{1}{6} k^2 h^2 \right),$ {{pad|3em}} $K_\text{kdv}(s) = \sqrt{gh} \left( \delta(s) + \frac{1}{6} h^2\, \delta^{\prime\prime}(s) \right),$ {{pad|3em}} $\alpha_\text{kdv} = \frac{3}{2} \sqrt{\frac{g}{h}},$

:with δ(s) the Dirac delta function.

Bengt Fornberg and Gerald Whitham studied the kernel K_fw(s) – non-dimensionalised using g and h:{{harvtxt|Fornberg|Whitham|1978}}

:: $K_\text{fw}(s) = \frac12 \nu \text{e}^{-\nu |s|}$ {{pad|2em}} and {{pad|2em}} $c_\text{fw} = \frac{\nu^2}{\nu^2+k^2},$ {{pad|2em}} with {{pad|2em}} $\alpha_\text{fw}=\frac32.$

:The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:

:: $\left( \frac{\partial^2}{\partial x^2} - \nu^2 \right)
\left(
\frac{\partial \eta}{\partial t}
+ \frac32\, \eta\, \frac{\partial \eta}{\partial x}
\right)
+ \frac{\partial \eta}{\partial x}
= 0.$

:This equation is shown to allow for peakon solutions – as a model for waves of limiting height – as well as the occurrence of wave breaking (shock waves, absent in e.g. solutions of the Korteweg–de Vries equation).

Notes and references

=Notes=

=References=

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Category:Water waves

Category:Partial differential equations

Category:Equations of fluid dynamics