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The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small wave packet will move, i.e. the direction of the group velocity. For light waves in vacuum, this is also the direction of the Poynting vector. On the other hand, the wave vector points in the direction of phase velocity.
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by =, where is a position in real space, is the wave vector in units of inverse meters, ω is the angular frequency with units of inverse time and is time.
Particle velocity (denoted v or SVL) is the velocity of a particle (real or imagined) in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound , but it can also be a transverse wave as with the vibration of a taut string.
As a wave, light is characterized by a velocity (the speed of light), wavelength, and frequency. As particles, light is a stream of photons. Each has an energy related to the frequency of the wave given by Planck's relation E = hf, where E is the energy of the photon, h is the Planck constant, 6.626 × 10 −34 J·s, and f is the frequency of ...
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.
Frequency dispersion of surface gravity waves on deep water. The red square moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red square traverses the figure in the time it takes the green dot to traverse half.
Ray tracing of a beam of light passing through a medium with changing refractive index.The ray is advanced by a small amount, and then the direction is re-calculated. Ray tracing works by assuming that the particle or wave can be modeled as a large number of very narrow beams (), and that there exists some distance, possibly very small, over which such a ray is locally straight.
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). It arises in fields like acoustics, electromagnetism, and fluid dynamics.