In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves.
The equation is notated as follows: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. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.
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:
while
with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is, using the inverse Fourier transform:
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 :
with δ(s) the Dirac delta function.
Bengt Fornberg and Gerald Whitham studied the kernel Kfw(s) – non-dimensionalised using g and h:
and with
The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:
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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