Dispersion equation

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.

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

[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)
[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)