Whitham equation

Short description: Non-local model for non-linear dispersive waves

In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves.

The equation is notated as follows:

$\frac{\partial \eta }{\partial t}+\alpha \eta \frac{\partial \eta }{\partial x}+\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}K(x-\xi )\frac{\partial \eta (\xi ,t)}{\partial \xi }\text{d}\xi =0.$

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. 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:

$c_{\text{ww}}(k)=\sqrt{\frac{g}{k}\tanh (kh)},$   while   ${\alpha }_{\text{ww}}=\frac{3}{2}\sqrt{\frac{g}{h}},$

with g the gravitational acceleration and h the mean water depth. The associated kernel K_ww(s) is, using the inverse Fourier transform:

$K_{\text{ww}}(s)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{c}_{\text{ww}}(k){\text{e}}^{iks}\text{d}k=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{c}_{\text{ww}}(k)\cos (ks)\text{d}k,$

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 kh ≪ 1:

$c_{\text{kdv}}(k)=\sqrt{gh}(1-\frac{1}{6}{k}^2{h}^2),$   $K_{\text{kdv}}(s)=\sqrt{gh}(\delta (s)+\frac{1}{6}{h}^2{\delta }^{\prime \prime }(s)),$   ${\alpha }_{\text{kdv}}=\frac{3}{2}\sqrt{\frac{g}{h}},$

with δ(s) the Dirac delta function.

• Bengt Fornberg and Gerald Whitham studied the kernel K_fw(s) – non-dimensionalised using g and h:

$K_{\text{fw}}(s)=\frac{1}{2}\nu {\text{e}}^{-\nu |s|}$   and   $c_{\text{fw}}=\frac{{\nu }^2}{{\nu }^2+k^2},$   with   ${\alpha }_{\text{fw}}=\frac{3}{2}.$

The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:

$(\frac{{\partial }^2}{\partial x^2}-{\nu }^2)(\frac{\partial \eta }{\partial t}+\frac{3}{2}\eta \frac{\partial \eta }{\partial x})+\frac{\partial \eta }{\partial x}=0.$

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).

Notes and references

References

• Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, ISBN 9780817643232 
• Fetecau, R.; Levy, Doron (2005), "Approximate Model Equations for Water Waves", Communications in Mathematical Sciences 3 (2): 159–170, doi:10.4310/CMS.2005.v3.n2.a4 
• Fornberg, B.; Whitham, G.B. (1978), "A Numerical and Theoretical Study of Certain Nonlinear Wave Phenomena", Philosophical Transactions of the Royal Society A 289 (1361): 373–404, doi:10.1098/rsta.1978.0064, Bibcode: 1978RSPTA.289..373F 
• Hur, Vera Mikyoung (2017), "Wave breaking in the Whitham equation", Advances in Mathematics 317: 410–437, doi:10.1016/j.aim.2017.07.006 
• Moldabayev, D.; Kalisch, H.; Dutykh, D. (2015), "The Whitham Equation as a model for surface water waves", Physica D: Nonlinear Phenomena 309: 99–107, doi:10.1016/j.physd.2015.07.010, Bibcode: 2015PhyD..309...99M 
• Naumkin, P.I.; Shishmarev, I.A. (1994), Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society, ISBN 9780821845738 
• Whitham, G.B. (1967), "Variational methods and applications to water waves", Proceedings of the Royal Society A 299 (1456): 6–25, doi:10.1098/rspa.1967.0119, Bibcode: 1967RSPSA.299....6W 
• Whitham, G.B. (1974), Linear and nonlinear waves, Wiley-Interscience, doi:10.1002/9781118032954, ISBN 978-0-471-94090-6, Bibcode: 1974lnw..book.....W