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into the water. This is the Brewster effect.

      2.1.6 Coherency

In the case of a monochromatic wave of certain frequency ν₀, the instantaneous field at any point P is well defined. If the wave consists of a large number of monochromatic waves with frequencies over a bandwidth ranging from ν₀ to ν₀ + Δν, then the random addition of all the component waves will lead to irregular fluctuations of the resultant field.

      The coherency time Δt is defined as the period over which there is strong correlation of the field amplitude. More specifically, it is the time after which two waves at ν and ν + Δν are out of phase by one cycle; that is, it is given by:

      (2.18)

      The coherence length is defined as

      (2.19)

      Two waves or two sources are said to be coherent with each other if there is a systematic relationship between their instantaneous amplitudes. The amplitude of the resultant field varies between the sum and the difference of the two amplitudes. If the two waves are incoherent, then the power of the resultant wave is equal to the sum of the power of the two constituent waves. Mathematically, let E₁(t) and E₂(t) be the 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 and P = P₁ + P₂. If the waves are coherent, then . In the latter case, we have:

      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 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.

      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, ν_d = 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 (ν_d > 0). If the source–observer distance is increasing, the reverse effect occurs (i.e., ν_d < 0) and the Doppler shift is negative.

Figure 2.6 Group velocity.

      The relationship between ν_d 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 Скачать книгу