Whitham equation - Alchetron, The Free Social Encyclopedia

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.

Water waves

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 K_ww(s) is:

The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of c_ww(k) for long waves with kh ≪ 1:

with δ(s) the Dirac delta function.

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

References

Whitham equation Wikipedia

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