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

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

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