In mathematical physics, the Whitham equation is a non-local model for

non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

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 A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

(and its inverse), with respect to the space coordinate ''x'' and in terms of the wavenumber ''k'': * For

surface gravity wave In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere ...

s, the

phase speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...

''c''(''k'') as a function of wavenumber ''k'' is taken as: ::

c_\text(k) = \sqrt,

while

\alpha_\text = \frac \sqrt,

: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(s) = \frac \int_^ c_\text(k)\, \text^\, \textk
                 = \frac \int_^ c_\text(k)\, \cos(ks)\, \textk,

: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 : ::

c_\text(k) = \sqrt \left( 1 - \frac k^2 h^2 \right),

K_\text(s) = \sqrt \left( \delta(s) + \frac h^2\, \delta^(s) \right),

\alpha_\text = \frac \sqrt,

:with ''δ''(''s'') the

Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

. * Bengt Fornberg and Gerald Whitham studied the kernel ''K''_fw(''s'') – non-dimensionalised using ''g'' and ''h'': ::

K_\text(s) = \frac12 \nu \text^

and

c_\text = \frac,

with

\alpha_\text=\frac32.

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

\left( \frac - \nu^2 \right)
  \left( 
    \frac
    + \frac32\, \eta\, \frac
  \right)
  + \frac
  = 0.

:This equation is shown to allow for

peakon In the theory of integrable systems, a peakon ("peaked soliton") is a soliton with discontinuous first derivative; the wave profile is shaped like the graph of the function e^. Some examples of non-linear partial differential equations with (multi-) ...

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

Notes

References

* * * * * * * * {{ref end Water waves Partial differential equations Equations of fluid dynamics