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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 Stokes drift is the difference in end positions, after a predefined amount of time (usually one wave period), as derived from a description in the Lagrangian and Eulerian coordinates. The end position in the Lagrangian description is obtained by following a specific fluid parcel during the time interval.
The previous definition of phase velocity has been demonstrated for an isolated wave. However, such a definition can be extended to a beat of waves, or to a signal composed of multiple waves. For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier.
In some (unusual) cases both end points of a branch (family) of periodic travelling wave solutions are homoclinic solutions, [37] in which case one must use an external starting point, such as a numerical solution of the partial differential equations. Periodic travelling wave stability can also be calculated numerically, by computing the spectrum.
where ν is the frequency of the wave, λ is the wavelength, ω = 2πν is the angular frequency of the wave, and v p is the phase velocity of the wave. The dependence of the wavenumber on the frequency (or more commonly the frequency on the wavenumber) is known as a dispersion relation.
It might seem like a simple question. But the science behind a blue sky isn't that easy. For starters, it involves something called the Rayleigh effect, or Rayleigh scattering. But that same ...
The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real k, complex ω), or, allow group velocity to be a complex-valued quantity. [16] [17] Different considerations yield distinct velocities, yet all definitions agree for the case of a lossless, gainless medium.
Stokes's first definition of wave celerity has, for a pure wave motion, the mean value of the horizontal Eulerian flow-velocity Ū E at any location below trough level equal to zero. Due to the irrotationality of potential flow, together with the horizontal sea bed and periodicity the mean horizontal velocity, the mean horizontal velocity is a ...