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Cnoidal wave solution to the Korteweg–De Vries equation, in terms of the square of the Jacobi elliptic function cn (and with value of the parameter m = 0.9). Numerical solution of the KdV equation u t + uu x + δ 2 u xxx = 0 (δ = 0.022) with an initial condition u(x, 0) = cos(πx). Time evolution was done by the Zabusky–Kruskal scheme. [1]
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.
As the relation is linear, the wave equation is said to be non-dispersive. To simplify, consider the one-dimensional wave equation with ω(k) = ±kc . Then the general solution is u ( x , t ) = A e i k ( x − c t ) + B e i k ( x + c t ) , {\displaystyle u(x,t)=Ae^{ik(x-ct)}+Be^{ik(x+ct)},} where the first and second term represent a wave ...
Two-frequency beats of a non-dispersive transverse wave. Since the wave is non-dispersive, phase and group velocities are equal. For an ideal string, the dispersion relation can be written as =, where T is the tension force in the string, and μ is the string's mass per unit length. As for the case of electromagnetic waves in vacuum, ideal ...
They happen to be equal in non-dispersive media. In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency. The phase velocity is the rate at which the phase of the wave propagates in space.
Dispersion of gravity waves on a fluid surface. Phase and group velocity divided by shallow-water phase velocity √ gh as a function of relative depth h / λ. Blue lines (A): phase velocity; Red lines (B): group velocity; Black dashed line (C): phase and group velocity √ gh valid in shallow water.
The second-order derivative represents the dispersion, while the κ term represents the nonlinearity. The equation models many nonlinearity effects in a fiber, including but not limited to self-phase modulation, four-wave mixing, second-harmonic generation, stimulated Raman scattering, optical solitons, ultrashort pulses, etc.
The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation. The linear frequency dispersion characteristics for the above set A of equations are: [ 5 ]