Wave Equations

The propagation phenomena of optical waves in random medium are described by a linear differential equation with random coefficients, instead of deterministic one (see Equation 2.17), used for description of deterministic models. Thus, a scalar wave Equation 2.17 can be presented in the following form:

where ^(r,t) is the wave field amplitude in the space and time domains, n(r, t) is the refractive index, which is a random function of space (r) and time (t), and c is the wave velocity in free space.

A compact scalar wave equation with a source g(r) (described, let’s say, with a Green “point source” function) can be presented as follows:

Here, Equation 2.32 is presented assuming a harmonic time dependence ^exp{ickt} = exp{ia>t} and a time-independent refractive index n, where k = (2n/A) is the wave number and A is the wavelength in medium under consideration. The source term g(r) is assumed to be given and not randomized (e.g., deterministic). In such an assumption, a usually used electromagnetic vector wave equation can be presented in the following form:

where E(r,t) is the vector presentation of the electromagnetic field.

We shall always treat the refractive index as a time-independent random function, which is equivalent to the assumption that the characteristic time of index fluctuations is much longer than the period of the propagating wave. The medium in such conditions will be taken statistically as homogeneous. This assumption excludes any medium where the atmospheric turbulence is concentrated in a small volume of space.

To conclude this subsection, let us show that the scalar wave equation 2.31 and the reduced scalar wave Equation 2.32 may be treated simultaneously. Equation 2.31 corresponds to an initial value problem well-known as the Cauchy problem, and must be given as T(r,0) and dT(r,0)/dt in order to find T(r,t) [32,39,44]. Equation 2.32 corresponds to a radiation problem [34]. Let us introduce the Laplace transform of the wave function T(r,t)

It satisfies the following equation, which is the Laplace transform of (2.32):

Equations 2.35 and 2.32 are identical if we take

We shall always choose these initial conditions for Equation 2.31 and treat it and Equation 2.32 simultaneously, interchanging q and c-k, whenever necessary.

 
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