Whitham equation explained

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 K_ww(s) is, using the inverse Fourier transform:

since c_ww 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 c_ww(k) for long waves with :

},

with δ(s) the Dirac delta function.

• Bengt Fornberg and Gerald Whitham studied the kernel K_fw(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).