World Library  
Flag as Inappropriate
Email this Article

Whitham equation

Article Id: WHEBN0037072870
Reproduction Date:

Title: Whitham equation  
Author: World Heritage Encyclopedia
Language: English
Subject: Gerald B. Whitham, Water waves, List of nonlinear partial differential equations, Index of physics articles (W)
Collection: Equations of Fluid Dynamics, Partial Differential Equations, Water Waves
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Whitham equation

In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves:[1][2][3]

\frac{\partial \eta}{\partial t} + \alpha \eta \frac{\partial \eta}{\partial x} + \int_{-\infty}^{+\infty} K(x-\xi)\, \frac{\partial \eta(\xi,t)}{\partial \xi}\, \text{d}\xi = 0.

This integro-differential equation 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.

Contents

  • Water waves 1
  • Notes and references 2
    • Notes 2.1
    • References 2.2

Water waves

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 Kww(s) is:[4]
K_\text{ww}(s) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} c_\text{ww}(k)\, \text{e}^{iks}\, \text{d}k.
c_\text{kdv}(k) = \sqrt{gh} \left( 1 - \frac{1}{6} k^2 h^2 \right),   K_\text{kdv}(s) = \sqrt{gh} \left( \delta(s) + \frac{1}{6} h^2\, \delta^{\prime\prime}(s) \right),   \alpha_\text{kdv} = \frac{3}{2} \sqrt{\frac{g}{h}},
with δ(s) the Dirac delta function.
K_\text{fw}(s) = \frac12 \nu \text{e}^{-\nu s}   and   c_\text{fw} = \frac{\nu^2}{\nu^2+k^2},   with   \alpha_\text{fw}=\frac32.
The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:[5]
\left( \frac{\partial^2}{\partial x^2} - \nu^2 \right) \left( \frac{\partial \eta}{\partial t} + \frac32\, \eta\, \frac{\partial \eta}{\partial x} \right) + \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).[5][3]

Notes and references

Notes

  1. ^ Debnath (2005, p. 364)
  2. ^ Naumkin & Shishmarev (1994, p. 1)
  3. ^ a b Whitham (1974, pp. 476–482)
  4. ^ a b c d Whitham (1967)
  5. ^ a b c Fornberg & Whitham (1978)

References

  • Debnath, L. (2005), Nonlinear Partial Differential Equations for Scientists and Engineers, Springer,  
  • Fornberg, B.;  
  • Naumkin, P.I.; Shishmarev, I.A. (1994), Nonlinear Nonlocal Equations in the Theory of Waves, American Mathematical Society,  
  •  
  •  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.