Non-local model for non-linear dispersive waves
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:
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:
- while
- with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is, using the inverse Fourier transform:[4]
- since cww is an even function of the wavenumber k.
-
- with δ(s) the Dirac delta function.
- and with
- The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[6]
- 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]