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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.
Other frequently encountered structures comprise pulse trains (also known as periodic travelling waves), spiral waves and target patterns. These three solution types are also generic features of two- (or more-) component reaction–diffusion equations in which the local dynamics have a stable limit cycle [13]
Periodic waves oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a travelling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave. In a standing wave, the amplitude of vibration has nulls at ...
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.
The traveling wave on a Leaky-Wave Antenna is a fast wave, with a phase velocity greater than the speed of light. This type of wave radiates continuously along its length, and hence the propagation wavenumber kz is complex, consisting of both a phase and an attenuation constant. Highly directive beams at an arbitrary specified angle can be ...
The waves are stable, and can travel over very large distances (normal waves would tend to either flatten out, or steepen and topple over) The speed depends on the size of the wave, and its width on the depth of water. Unlike normal waves they will never merge – so a small wave is overtaken by a large one, rather than the two combining.
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Tidal waves can be either internal (travelling waves) with positive eigenvalues (or equivalent depth) which have finite vertical wavelengths and can transport wave energy upward, or external (evanescent waves) with negative eigenvalues and infinitely large vertical wavelengths meaning that their phases remain constant with altitude.