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:
c ww ( k ) = g k tanh ( k h ) , {\displaystyle c_{\text{ww}}(k)={\sqrt {{\frac {g}{k}}\,\tanh(kh)}},} while α ww = 3 2 g h , {\displaystyle \alpha _{\text{ww}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}
with g the gravitational acceleration and h the mean water depth. The associated kernel Kww(s) is, using the inverse Fourier transform:
K ww ( s ) = 1 2 π ∫ − ∞ + ∞ c ww ( k ) e i k s d k = 1 2 π ∫ − ∞ + ∞ c ww ( k ) cos ( k s ) d k , {\displaystyle K_{\text{ww}}(s)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,{\text{e}}^{iks}\,{\text{d}}k={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }c_{\text{ww}}(k)\,\cos(ks)\,{\text{d}}k,}
since cww 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 cww(k) for long waves with kh ≪ 1:
c kdv ( k ) = g h ( 1 − 1 6 k 2 h 2 ) , {\displaystyle c_{\text{kdv}}(k)={\sqrt {gh}}\left(1-{\frac {1}{6}}k^{2}h^{2}\right),} K kdv ( s ) = g h ( δ ( s ) + 1 6 h 2 δ ′ ′ ( s ) ) , {\displaystyle K_{\text{kdv}}(s)={\sqrt {gh}}\left(\delta (s)+{\frac {1}{6}}h^{2}\,\delta ^{\prime \prime }(s)\right),} α kdv = 3 2 g h , {\displaystyle \alpha _{\text{kdv}}={\frac {3}{2}}{\sqrt {\frac {g}{h}}},}
with δ(s) the Dirac delta function. Bengt Fornberg and Gerald Whitham studied the kernel Kfw(s) – non-dimensionalised using g and h:
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