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

The equation is notated as follows :

\( {\frac {\partial \eta }{\partial t}}+\alpha \eta {\frac {\partial \eta }{\partial x}}+\int _{{-\infty }}^{{+\infty }}K(x-\xi )\,{\frac {\partial \eta (\xi ,t)}{\partial \xi }}\,{\text{d}}\xi =0. \)

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] Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.[5]

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:[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 Kww(s) is, using the inverse Fourier transform:[4]

\( {\displaystyle K_{\text{ww}}(s)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,{\text{e}}^{iks}\,{\text{d}}k={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,\cos(ks)\,{\text{d}}k,}\)

since cww 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 cww(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 Kfw(s) – non-dimensionalised using g and h:[6]

\( {\displaystyle K_{\text{fw}}(s)={\frac {1}{2}}\nu {\text{e}}^{-\nu |s|}} \) 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:[6]

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

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).[6][3]

Notes and references
Notes

Debnath (2005, p. 364)
Naumkin & Shishmarev (1994, p. 1)
Whitham (1974, pp. 476–482)
Whitham (1967)
Hur (2017)

Fornberg & Whitham (1978)

References

Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, ISBN 9780817643232
Fetecau, R.; Levy, Doron (2005), "Approximate Model Equations for Water Waves", Communications in Mathematical Sciences, 3 (2): 159–170, doi:10.4310/CMS.2005.v3.n2.a4
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, CiteSeerX 10.1.1.67.6331, doi:10.1098/rsta.1978.0064
Hur, V.M. (2017), "Wave breaking in the Whitham equation", Advances in Mathematics, 317: 410–437, arXiv:1506.04075, doi:10.1016/j.aim.2017.07.006
Moldabayev, D.; Kalisch, H.; Dutykh, D. (2015), "The Whitham Equation as a model for surface water waves", Physica D: Nonlinear Phenomena, 309: 99–107, arXiv:1410.8299, Bibcode:2015PhyD..309...99M, doi:10.1016/j.physd.2015.07.010
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, doi:10.1002/9781118032954, ISBN 978-0-471-94090-6

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