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An equation connecting the vibration frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333601.png" /> with the [[Wave vector|wave vector]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333602.png" /> of a planar wave. The wave evolves according to the exponential law
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333603.png" /></td> </tr></table>
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The dispersion equation is deduced from the equations describing the process under observation, and defines the dispersion of the wave (see, for example, the case of electrodynamic processes in [[#References|[1]]], [[#References|[2]]]). Depending on the nature of the problem, it may be used to find the vibration frequencies from the wave vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333604.png" /> or the magnitude of the wave vector from their direction and from the vibration frequency.
+
An equation connecting the vibration frequency  $  \omega $
 +
with the [[Wave vector|wave vector]] $  \mathbf k $
 +
of a planar wave. The wave evolves according to the exponential law
  
The former case is closely connected with solving Cauchy's problem and the study of the stability of the equilibrium position corresponding to the trivial solution of the equation of the wave process being studied. By expanding the initial conditions into a Fourier series the solution of Cauchy's problem may be written down as the superposition of planar waves over the frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333605.png" />. If, for some real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333606.png" />, these frequencies include at least one with a negative imaginary part, it indicates the existence of increasing solutions, i.e. instability.
+
$$
 +
\mathop{\rm exp} \{ i ( \omega t - \mathbf k\mathbf r ) \} .
 +
$$
 +
 
 +
The dispersion equation is deduced from the equations describing the process under observation, and defines the dispersion of the wave (see, for example, the case of electrodynamic processes in [[#References|[1]]], [[#References|[2]]]). Depending on the nature of the problem, it may be used to find the vibration frequencies from the wave vector  $  \omega _ {n} = \omega _ {n} ( \mathbf k ) $
 +
or the magnitude of the wave vector from their direction and from the vibration frequency.
 +
 
 +
The former case is closely connected with solving Cauchy's problem and the study of the stability of the equilibrium position corresponding to the trivial solution of the equation of the wave process being studied. By expanding the initial conditions into a Fourier series the solution of Cauchy's problem may be written down as the superposition of planar waves over the frequencies $  \omega _ {n} ( \mathbf k ) $.  
 +
If, for some real $  \mathbf k $,  
 +
these frequencies include at least one with a negative imaginary part, it indicates the existence of increasing solutions, i.e. instability.
  
 
The latter case of solving dispersion equations is connected with problems of excitation of monochromatic vibrations by external sources which harmonically vary with time.
 
The latter case of solving dispersion equations is connected with problems of excitation of monochromatic vibrations by external sources which harmonically vary with time.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "Electrodynamics of continous media" , Pergamon  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.P. Silin,  A.A. Rukhadze,  "Electromagnetic properies of plasma and plasma-like media" , Moscow  (1961)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Landau,  E.M. Lifshitz,  "Electrodynamics of continous media" , Pergamon  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.P. Silin,  A.A. Rukhadze,  "Electromagnetic properies of plasma and plasma-like media" , Moscow  (1961)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Some examples of dispersion relations (in the case of one space dimension, cf. also [[Dispersion relation|Dispersion relation]]) are afforded by the beam equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333607.png" /> with dispersion relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333608.png" />, and the linear Korteweg–de Vries equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d0333609.png" /> with dispersion relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d03336010.png" />.
+
Some examples of dispersion relations (in the case of one space dimension, cf. also [[Dispersion relation|Dispersion relation]]) are afforded by the beam equation $  \phi _ {tt} + \gamma  ^ {2} \phi _ {xxxx} $
 +
with dispersion relation $  \omega = \pm  \gamma k  ^ {2} $,  
 +
and the linear Korteweg–de Vries equation $  \phi _ {t} + c \phi _ {x} + \nu \phi _ {xxx} = 0 $
 +
with dispersion relation $  \omega = ck - \nu k  ^ {3} $.
  
For a linear equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d03336011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d03336012.png" /> is a polynomial, the corresponding dispersion relation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d03336013.png" />, so that the equation and the dispersion relation determine each other. The idea and usefulness of dispersion equations carry over to the non-linear case. If the dispersion relation is non-linear, waves with different wave numbers move with different phase velocities (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033360/d03336014.png" /> in the case of one space dimension), which accounts for the term dispersion.
+
For a linear equation $  P ( \partial  / {\partial  t } , \partial  / {\partial  x _ {1} } , \partial  / {\partial  x _ {2} } , \partial  / {\partial  x _ {3} } ) \phi = 0 $,  
 +
where $  P $
 +
is a polynomial, the corresponding dispersion relation is $  P ( - i \omega , ik _ {1} , ik _ {2} , ik _ {3} ) = 0 $,  
 +
so that the equation and the dispersion relation determine each other. The idea and usefulness of dispersion equations carry over to the non-linear case. If the dispersion relation is non-linear, waves with different wave numbers move with different phase velocities ( $  c = k  ^ {-} 1 \omega $
 +
in the case of one space dimension), which accounts for the term dispersion.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Brillouin,  "Wave propagation and group velocity" , Acad. Press  (1960)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Timman,  A.J. Hermans,  G.C. Hsian,  "Water waves and ship hydrodynamics" , M. Nijhoff  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.B. Whitham,  "Linear and non-linear waves" , Wiley  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Zanderer,  "Partial differential equations of applied mathematics" , Wiley (Interscience)  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Brillouin,  "Wave propagation and group velocity" , Acad. Press  (1960)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Timman,  A.J. Hermans,  G.C. Hsian,  "Water waves and ship hydrodynamics" , M. Nijhoff  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.B. Whitham,  "Linear and non-linear waves" , Wiley  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Zanderer,  "Partial differential equations of applied mathematics" , Wiley (Interscience)  (1983)</TD></TR></table>

Revision as of 19:36, 5 June 2020


An equation connecting the vibration frequency $ \omega $ with the wave vector $ \mathbf k $ of a planar wave. The wave evolves according to the exponential law

$$ \mathop{\rm exp} \{ i ( \omega t - \mathbf k\mathbf r ) \} . $$

The dispersion equation is deduced from the equations describing the process under observation, and defines the dispersion of the wave (see, for example, the case of electrodynamic processes in [1], [2]). Depending on the nature of the problem, it may be used to find the vibration frequencies from the wave vector $ \omega _ {n} = \omega _ {n} ( \mathbf k ) $ or the magnitude of the wave vector from their direction and from the vibration frequency.

The former case is closely connected with solving Cauchy's problem and the study of the stability of the equilibrium position corresponding to the trivial solution of the equation of the wave process being studied. By expanding the initial conditions into a Fourier series the solution of Cauchy's problem may be written down as the superposition of planar waves over the frequencies $ \omega _ {n} ( \mathbf k ) $. If, for some real $ \mathbf k $, these frequencies include at least one with a negative imaginary part, it indicates the existence of increasing solutions, i.e. instability.

The latter case of solving dispersion equations is connected with problems of excitation of monochromatic vibrations by external sources which harmonically vary with time.

References

[1] L.D. Landau, E.M. Lifshitz, "Electrodynamics of continous media" , Pergamon (1960) (Translated from Russian)
[2] V.P. Silin, A.A. Rukhadze, "Electromagnetic properies of plasma and plasma-like media" , Moscow (1961) (In Russian)

Comments

Some examples of dispersion relations (in the case of one space dimension, cf. also Dispersion relation) are afforded by the beam equation $ \phi _ {tt} + \gamma ^ {2} \phi _ {xxxx} $ with dispersion relation $ \omega = \pm \gamma k ^ {2} $, and the linear Korteweg–de Vries equation $ \phi _ {t} + c \phi _ {x} + \nu \phi _ {xxx} = 0 $ with dispersion relation $ \omega = ck - \nu k ^ {3} $.

For a linear equation $ P ( \partial / {\partial t } , \partial / {\partial x _ {1} } , \partial / {\partial x _ {2} } , \partial / {\partial x _ {3} } ) \phi = 0 $, where $ P $ is a polynomial, the corresponding dispersion relation is $ P ( - i \omega , ik _ {1} , ik _ {2} , ik _ {3} ) = 0 $, so that the equation and the dispersion relation determine each other. The idea and usefulness of dispersion equations carry over to the non-linear case. If the dispersion relation is non-linear, waves with different wave numbers move with different phase velocities ( $ c = k ^ {-} 1 \omega $ in the case of one space dimension), which accounts for the term dispersion.

References

[a1] L. Brillouin, "Wave propagation and group velocity" , Acad. Press (1960)
[a2] R. Timman, A.J. Hermans, G.C. Hsian, "Water waves and ship hydrodynamics" , M. Nijhoff (1985)
[a3] G.B. Whitham, "Linear and non-linear waves" , Wiley (1974)
[a4] E. Zanderer, "Partial differential equations of applied mathematics" , Wiley (Interscience) (1983)
How to Cite This Entry:
Dispersion equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dispersion_equation&oldid=46747
This article was adapted from an original article by D.P. Kostomarov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article