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]

$K_{{\text{fw}}}(s)={\frac 12}\nu {\text{e}}^{{-\nu s}}$   and   $c_{{\text{fw}}}={\frac {\nu ^{2}}{\nu ^{2}+k^{2}}},$   with   $\alpha _{{\text{fw}}}={\frac 32}.$

The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:^[5]

$\left({\frac {\partial ^{2}}{\partial x^{2}}}-\nu ^{2}\right)\left({\frac {\partial \eta }{\partial t}}+{\frac 32}\,\eta \,{\frac {\partial \eta }{\partial x}}\right)+{\frac {\partial \eta }{\partial x}}=0.$

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