Читать книгу Introduction to the Physics and Techniques of Remote Sensing - Jakob J. van Zyl - Страница 25

The phase velocity is the velocity at which a constant phase front progresses (see Fig. 2.5). It is equal to

(2.22)

If we have two waves characterized by (ω − Δω, k − Δk) and (ω + Δω, k + Δk), then the total wave is given by

(2.23)

Figure 2.5 Phase velocity.

In this case, the plane of constant amplitude moves at a velocity υ_g, called the group velocity:

(2.24)

As Δω and Δk are assumed to be small, then we can write

(2.25)

This is illustrated in Figure 2.6. It is important to note that υ_g represents the velocity of propagation of the wave energy. Thus, the group velocity υ_g must be equal to or smaller than the speed of light c. However, the phase velocity υ_p can be larger than c.

If the medium is nondispersive, then

(2.26)

This implies that

(2.27)

(2.28)

However, if the medium is dispersive (i.e., ω is a nonlinear function of k), such as in the case of ionospheres, then the two velocities are different.