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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).
The characteristics of the PDE are = (where sign states the two solutions to quadratic equation), so we can use the change of variables = + (for the positive solution) and = (for the negative solution) to transform the PDE to =.
The blue line is the real part of the solution, the red line is the imaginary part, the black line is the wave envelope (absolute value) and the green line is the centroid of the wave packet envelope. The Eckhaus equation can be linearized to the linear Schrödinger equation: [4] + =,
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.
In mathematical physics, the wave maps equation is a geometric wave equation that solves D α ∂ α u = 0 {\displaystyle D^{\alpha }\partial _{\alpha }u=0} where D {\displaystyle D} is a connection .
A one-way wave equation is a first-order partial differential equation describing one wave traveling in a direction defined by the vector wave velocity. It contrasts with the second-order two-way wave equation describing a standing wavefield resulting from superposition of two waves in opposite directions (using the squared scalar wave velocity).
the nonlinear Schrödinger equation (NLS equation) for the complex-valued amplitude of narrowband waves (slowly modulated waves). Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called cnoidal waves. These are approximate solutions of the Boussinesq equation.
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]