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Whitham equation
In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves:
:
\frac
+ \alpha \eta \frac
+ \int_^ K(x-\xi)\, \frac\, \text\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.
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:〔
::
c_\text(k) = \sqrt\, \tanh(kh)},
while $\alpha_\text = \frac \sqrt\right\},$
:with ''g'' the gravitational acceleration and ''h'' the mean water depth. The associated kernel ''K''ww(''s'') is:〔
::
K_\text(s) = \frac \int_^ c_\text(k)\, \text^\, \textk.

* The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of ''c''ww(''k'') for long waves with :〔
::
c_\text(k) = \sqrt \left( 1 - \frac k^2 h^2 \right),

K_\text(s) = \sqrt \left( \delta(s) + \frac h^2\, \delta^(s) \right),

\alpha_\text = \frac \sqrt},

:with ''δ''(''s'') the Dirac delta function.
* Bengt Fornberg and Gerald Whitham studied the kernel ''K''fw(''s'') – non-dimensionalised using ''g'' and ''h'':
::$K_\text\left(s\right) = \frac12 \nu \text^$ and $c_\text = \frac,$ with $\alpha_\text=\frac32.$
:The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:〔
::
\left( \frac - \nu^2 \right)
\left(
\frac
+ \frac32\, \eta\, \frac
\right)
+ \frac
= 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).〔〔

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