In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves:
∂ η ∂ t + α η ∂ η ∂ x + ∫ − ∞ + ∞ K ( x − ξ ) ∂ η ( ξ , t ) ∂ ξ d ξ = 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.
For surface gravity waves, the phase speed c(k) as a function of wavenumber k is taken as:with
g the gravitational acceleration and
h the mean water depth. The associated kernel
Kww(
s) is:
The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of cww(k) for long waves with kh ≪ 1:with
δ(
s) the
Dirac delta function.
Bengt Fornberg and Gerald Whitham studied the kernel Kfw(s) – non-dimensionalised using g and h: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).
