# Wave propagation

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

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. Other wave types cannot propagate through a vacuum and need a transmission medium to exist[citation needed].

## Reflection of plane Waves in a half-space

The propagation and reflection of plane waves, i.e. Pressure waves (P-wave) or Shear waves (SH or SV-waves), in a homogeneous half-space have been of significant interest in the classical seismology. The analytical solution of this problem exists and has been used frequently in the literature for the verification purposes. The frequency domain solution pertaining to this problem can be obtained through the Helmholtz decomposition of the displacement field in the half-space and substitute into the Navier's equation.

### SV wave propagation

The analytical solution of SV-wave in a half-space indicates that the plane SV wave reflects back to the domain as a P and SV waves, leaving out special cases. The angle of reflected SV wave is identical to the incidence wave, while the angle of reflected P wave is greater than the SV wave. Note also that for the same wave frequency, the SV wavelength is smaller than the P wavelength. This fact has been depicted in this animated picture. [1]

The propagation of SV-wave in a homogeneous half-space (The horizontal displacement field)
The propagation of SV-wave in a homogeneous half-space (The vertical displacement field)

### P wave propagation

Similar to the SV wave, the P incidence, in general, reflects as the P and SV wave. There are some special cases where the regime is different.

## Wave velocity

Seismic wave propagation in 2D modelled using FDTD method in the presence of a landmine

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

${\displaystyle v_{p}={\frac {\omega }{k}},}$

where:

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 cannot be chosen independently from the wavenumber k, but both are related through the dispersion relationship:

${\displaystyle \omega =\Omega (k).\,}$

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 waves propagate and on the type of waves (for instance electromagnetic, sound or water waves).

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:

${\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}}$

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.