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# Whitham equation

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### Whitham equation

In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves:[1][2][3]

\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.

This integro-differential equation 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.[4]

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

## Contents

• Water waves 1
• Notes and references 2
• Notes 2.1
• References 2.2

## Water waves

c_\text{ww}(k) = \sqrt{ \frac{g}{k}\, \tanh(kh)},   while   \alpha_\text{ww} = \frac{3}{2} \sqrt{\frac{g}{h}},
with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is:[4]
K_\text{ww}(s) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} c_\text{ww}(k)\, \text{e}^{iks}\, \text{d}k.
c_\text{kdv}(k) = \sqrt{gh} \left( 1 - \frac{1}{6} k^2 h^2 \right),   K_\text{kdv}(s) = \sqrt{gh} \left( \delta(s) + \frac{1}{6} h^2\, \delta^{\prime\prime}(s) \right),   \alpha_\text{kdv} = \frac{3}{2} \sqrt{\frac{g}{h}},
with δ(s) the Dirac delta function.
K_\text{fw}(s) = \frac12 \nu \text{e}^{-\nu s}   and   c_\text{fw} = \frac{\nu^2}{\nu^2+k^2},   with   \alpha_\text{fw}=\frac32.
The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[5]
\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).[5][3]

## Notes and references

### Notes

1. ^ Debnath (2005, p. 364)
2. ^ Naumkin & Shishmarev (1994, p. 1)
3. ^ a b Whitham (1974, pp. 476–482)
4. ^ a b c d Whitham (1967)
5. ^ a b c Fornberg & Whitham (1978)

### References

• Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer,
• Fornberg, B.;
• Naumkin, P.I.; Shishmarev, I.A. (1994), Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society,
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