Wave propagation

Wave propagation is any of the ways in which waves travel through a waveguide.

With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.

For electromagnetic waves, propagation may occur in a vacuum as well as in a material medium. Most other wave types cannot propagate through vacuum and need a transmission medium to exist.

Another useful parameter for describing the propagation is the wave velocity that mostly depends on some kind of density of the medium.

Wave velocity


Wave velocity is a general concept, of various kinds of wave velocities, for an wave's phase and speed concerning energy (and information) propagation. The phase velocity is given as:


 * $$v_p = \frac{\omega}{k},$$

where:
 * vp is the phase velocity (in meter per second, m/s),
 * ω is the angular frequency (in radians per second, rad/s),
 * k is the wavenumber (in radians per meter, rad/m).

The phase speed gives you the speed at which a point of constant phase of the wave will travel for a discrete frequency. The angular frequency ω can not be chosen independently from the wavenumber k, but both are related through the dispersion relationship:


 * $$\omega = \Omega(k).\,$$

In the special case Ω(k)=ck, with c a constant, the waves are called non-dispersive, since all frequencies travel at the same phase speed c. For instance electromagnetic waves in vacuum are non-dispersive. In case of other forms of the dispersion relation, we have dispersive waves. The dispersion relationship depends on the medium through which the wave propagate and on the type of waves (for instance electromagnetic, sound or water waves). For sound waves, the more dense a medium is, the faster the waves will travel as particles will be closer together and thus energy can be transferred among them at a greater rate.

The speed at which a resultant wave packet from a narrow range of frequencies will travel is called the group velocity and is determined from the gradient of the dispersion relation:


 * $$v_g = \frac{\delta \omega}{\delta k}$$

In almost all cases, a wave is mainly a movement of energy through a medium. Most often, the group velocity is the velocity at which the energy moves through this medium.