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In dispersive wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a dispersion relation = (), where ω is the angular frequency, and k is the wavevector describing plane-wave solutions.
Numerical simulation of the Fisher–KPP equation. In colors: the solution u(t,x); in dots : slope corresponding to the theoretical velocity of the traveling wave.. In mathematics, KPP–Fisher equation (named after Andrey Kolmogorov, Ivan Petrovsky, Nikolai Piskunov [1] and Ronald Fisher [2]) also known as the KPP equation, Fisher equation or Fisher–KPP equation is the partial differential ...
Solitary wave in a laboratory wave channel. In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets.
The phase velocity is the rate at which the phase of the wave propagates in space. The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.
So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of H, solitary gravity waves of depression do not exist.
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]
N-wave type solutions of the Burgers equation, starting from the initial condition (,) = / (+) /. Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation [ 1 ] occurring in various areas of applied mathematics , such as fluid mechanics , [ 2 ] nonlinear acoustics , [ 3 ...
While periodic travelling waves have been known as solutions of the wave equation since the 18th century, their study in nonlinear systems began in the 1970s. A key early research paper was that of Nancy Kopell and Lou Howard [1] which proved several fundamental results on periodic travelling waves in reaction–diffusion equations.