Whitham equation

Template:Short description In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves.

The equation is notated as follows:

$\frac{\partial \eta }{\partial t}+\alpha \eta \frac{\partial \eta }{\partial x}+\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}K(x-\xi )\frac{\partial \eta (\xi ,t)}{\partial \xi }\text{d}\xi =0.$

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. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.

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:

$c_{\text{ww}}(k)=\sqrt{\frac{g}{k}\tanh (kh)},$ Template:Pad while Template:Pad ${\alpha }_{\text{ww}}=\frac{3}{2}\sqrt{\frac{g}{h}},$

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

$K_{\text{ww}}(s)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{c}_{\text{ww}}(k){\text{e}}^{iks}\text{d}k=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{c}_{\text{ww}}(k)\cos (ks)\text{d}k,$

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 Template:Nowrap:

$c_{\text{kdv}}(k)=\sqrt{gh}(1-\frac{1}{6}{k}^2{h}^2),$ Template:Pad $K_{\text{kdv}}(s)=\sqrt{gh}(\delta (s)+\frac{1}{6}{h}^2{\delta }^{\prime \prime }(s)),$ Template:Pad ${\alpha }_{\text{kdv}}=\frac{3}{2}\sqrt{\frac{g}{h}},$

with δ(s) the Dirac delta function.

• Bengt Fornberg and Gerald Whitham studied the kernel K_fw(s) – non-dimensionalised using g and h:

$K_{\text{fw}}(s)=\frac{1}{2}\nu {\text{e}}^{-\nu |s|}$ Template:Pad and Template:Pad $c_{\text{fw}}=\frac{{\nu }^2}{{\nu }^2+k^2},$ Template:Pad with Template:Pad ${\alpha }_{\text{fw}}=\frac{3}{2}.$

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

$(\frac{{\partial }^2}{\partial x^2}-{\nu }^2)(\frac{\partial \eta }{\partial t}+\frac{3}{2}\eta \frac{\partial \eta }{\partial x})+\frac{\partial \eta }{\partial x}=0.$

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

References

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