two component fields at a certain location. Then the total field is
(2.20)
The average power is
(2.21)
If the two waves are incoherent relative to each other, then
This is the case of optical interference fringes generated by two overlapping coherent optical beams. The bright bands correspond to where the energy is above the mean and the dark bands correspond to where the energy is below the mean.
2.1.7 Group and Phase Velocity
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 υg, 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 υg represents the velocity of propagation of the wave energy. Thus, the group velocity υg must be equal to or smaller than the speed of light c. However, the phase velocity υp 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.
2.1.8 Doppler Effect
If the relative distance between a source radiating at a fixed frequency ν and an observer varies, the signal received by the observer will have a frequency ν′, which is different than ν. The difference, νd = ν ′ − ν, is called the Doppler shift. If the source–observer distance is decreasing, the frequency received is higher than the frequency transmitted, leading to a positive Doppler shift (νd > 0). If the source–observer distance is increasing, the reverse effect occurs (i.e., νd < 0) and the Doppler shift is negative.
Figure 2.6 Group velocity.
The relationship between νd and ν is
(2.29)
where υ is the relative speed between the source and the observer, c is the velocity of light, and θ is the angle between the direction of motion and the line connecting the source and the observer (see Fig. 2.7). The above expression assumes no relativistic effects (υ ≪ c), and it can be derived in the following simple way.
Referring to Figure 2.8, assume an observer is moving at a velocity υ with an angle θ relative to the line of propagation of the wave. The lines of constant wave amplitude are separated by the distance λ (i.e., wavelength) and are moving at velocity c. For the observer, the apparent frequency ν′ is equal to the inverse of the time period T′ that it takes the observer to cross two successive equiamplitude lines. This is given by the expression
(2.30)
which can be written as
(2.31)
The Doppler effect also occurs when the source and observer are fixed relative to each other but the scattering or reflecting object is moving (see Fig. 2.9). In this case, the Doppler shift is given by
(2.32)
Figure 2.7 Doppler geometry for a moving source, fixed observer.
Figure 2.8 Geometry illustrating wave fronts passing by a moving observer.
