The first integral is simply r1(x) = B a2/y2. The second integral can be broken into two parts:

Note that (a2 - y2) = -2ako. Thus, r(x) = B a2/y2 + A S(x) - [A ako/y]) exp(-y|x|) 5.3-12

Figure 5.3-6 illustrates the general form of this r(x). Clearly, r(x), because it is a frequency, is non-negative. For this to occur, it is possible to show that B must satisfy

B > A ko + 2ko/a . Note that the effect of the exponential, spatial distribution of

LI is to suppress the frequencies r(x) in the vicinity of the origin with an exponential shape. Curiously, the space constant of the exponential is y rather than a.

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