which originally involved beta, and gamma is the dummy variable 
of integration for those expressions which involved alpha origin- 
ally. Let the last expression in equation (8.60) be a short hand 
notation for the, expression which precedes it. 
Equation (8.61) is the result when this short hand notation 
is substituted into equation (8.59). Each of the indicated inte- 
grals is a Fresnel Integral and its value is consequently known. 
The terms preceding the two trigonometric terms in (8.61) deter- 
mine the envelope of the traveling wave, and since an expression 
of the form G cos6 + H siné can be written in the form 
(a? + q2)V2 sin(e + tan7/c/d), the last expression in equation 
(8.61) shows the results of this transformation. 
The expression, FF(x,y,t),is equal to the sum of the squares 
of the two coefficients in equation (8.61). It will turn out 
to be the two dimensional equivalent of the one dimensional Fres- 
nel filter described in Chapter 7. When the process of Squaring 
and clearing terms is carried out, the final result is the last 
expression in equation (8.62). FF(x,y,t) will be referred to 
as the Fresnel filter for a storm of finite width. 
Interpretation of results 
The expression for FF(x,y,t), given in equation (8.62), is 
a product of two terms which involve Fresnel Integrals. The function 
will first be treated for a fixed value of x as a function of y 
and t. Each of the terms in the large bracket is essentially 
two inside of a certain range of t and y for a fixed value of 
xe Outside of this range at least one of the terms is nearly 
zero and the product is therefore nearly zero. Consider the first 
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