Whitham equation
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 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 it becomes the Fornberg–Whitham equation.
Water waves
Using the Fourier transform, with respect to the space coordinate x and in terms of the wavenumber k:- For surface gravity waves, the phase speed c as a function of wavenumber k is taken as:
- The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of cww for long waves with :
- Bengt Fornberg and Gerald Whitham studied the kernel Kfw – non-dimensionalised using g and h: