Whitham equation
From Infogalactic: the planetary knowledge core
In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves:[1][2][3]
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]
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:[4]
-
- while
- with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is:[4]
- 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:[4]
- with δ(s) the Dirac delta function.
- Bengt Fornberg and Gerald Whitham studied the kernel Kfw(s) – non-dimensionalised using g and h:[5]
-
- and with
- The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[5]
- 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).[5][3]
Notes and references
Notes
- ↑ Debnath (2005, p. 364)
- ↑ Naumkin & Shishmarev (1994, p. 1)
- ↑ 3.0 3.1 Whitham (1974, pp. 476–482)
- ↑ 4.0 4.1 4.2 4.3 Whitham (1967)
- ↑ 5.0 5.1 5.2 Fornberg & Whitham (1978)
References
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