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The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable. Envelope for a modulated sine wave.
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.
Consequently, the wave equation is approximated in the SVEA as: + = . It is convenient to choose k 0 and ω 0 such that they satisfy the dispersion relation: = . This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation:
In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope. A wave packet can be analyzed into, or can be synthesized from, a potentially-infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that ...
Huygens's principle asserts that the wave front set Φ q 0 (s + t) is the envelope of the family of wave fronts Φ q (s) for q ∈ Φ q 0 (t). More generally, the point q 0 could be replaced by any curve, surface or closed set in space.
The group velocity is positive (i.e., the envelope of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward). The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.
Animation of the wave-packet solution of the linear Schrödinger equation – corresponding with the above animation for the Eckhaus equation. 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.
A hyperbolic secant (sech) envelope soliton for surface waves on deep water. Blue line: water waves. Red line: envelope soliton. For water waves, the nonlinear Schrödinger equation describes the evolution of the envelope of modulated wave groups. In a paper in 1968, Vladimir E. Zakharov describes the Hamiltonian structure of water waves.