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 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.

Water waves

• For surface gravity waves, the phase speed c(k) as a function of wavenumber k is taken as:^[4]

$c_\text{ww}(k) = \sqrt{ \frac{g}{k}\, \tanh(kh)},$   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 K_ww(s) is:^[4]

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

• The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of c_ww(k) for long waves with kh ≪ 1:^[4]

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

with δ(s) the Dirac delta function.

• Bengt Fornberg and Gerald Whitham studied the kernel K_fw(s) – non-dimensionalised using g and h:^[5]

$K_\text{fw}(s) = \frac12 \nu \text{e}^{-\nu s}$   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. Whitham (1974, pp. 476–482)
4. Whitham (1967)
5. Fornberg & Whitham (1978)

References

• Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, ISBN 9780817643232
• Fornberg, B.; Whitham, G.B. (1978), "A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena", Philosophical Transactions of the Royal Society A 289 (1361): 373–404, Bibcode:1978RSPTA.289..373F, doi:10.1098/rsta.1978.0064
• Naumkin, P.I.; Shishmarev, I.A. (1994), Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, ISBN 9780821845738
• Whitham, G.B. (1967), "Variational methods and applications to water waves", Proceedings of the Royal Society A 299 (1456): 6–25, Bibcode:1967RSPSA.299....6W, doi:10.1098/rspa.1967.0119
• Whitham, G.B. (1974), Linear and nonlinear waves, Wiley-Interscience, ISBN 0-471-94090-9