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1949 SCHLIEREN PHOTOGRAPHY 531
From the above figure
f(x) = 0. | x | > A
f(x} = e~ik4> -A<x<-a (18)
f(x) = e»* (*/<»)* —a<x<a e*+ a<x<A.
The integration of (16) then gives O for \x\ > A
h(z} _ I tisk(-<j> + (y/M -A < x< a sk(_d>
"7T "tisk(<t>/a + y/f)z -a<x<a
cis fc(* + (y/M a< x< A
X{[Cik(y/f)(-a-z) -Ci k(y/f)(-A -2)]+* [Si k(y/f)(-a 2) Si k(y/n(-A-z)]} + {tis k(4>/a+y/flz\ X{ ]Ci fc(0/a + y/f)(a-z)-C\ *(0/a + V//X-0-2)] +/fSifc(0/o +?///)(« -2) -Si A (0/a +?///)( -a -0)]} (19) + }cisfc(0 +(y/A)} X {[C\k(y/f](A -z)-C\ k(y/f)(a-z)] +i[8ik(y/f) (A -Z)-S\k(y/f)(a-z)}}}
where9
y = w + ?/i
cis (0) = cos (6) + f sin (6) Si (0) = sine integral of (0) Ci (0) = cosine integral of (0).
The light-intensity distribution I (z) incident on a photographic plate is given by the product of (19) with its compex conjugate.
I(z) = h(z) X A(i). (17)
It is evident from (19) that it will be most difficult to find the inverse of (17); i.e., to determine </> and a from measured values of light intensity in order to obtain quantitative information from a schlieren photograph.
POINT SOURCE
The contrast of the schlieren image of a disturbance will be defined as the logarithm of the ratio of the maximum light intensity in the image of the disturbance to the minimum light intensity in the background. For disturbances of the type illustrated in Fig. 5, the maximum light intensity occurs at the point z = 0 corresponding to the center of the gradient region 3. This quantity will be denoted by I (0). The minimum intensity occurs in the vicinity of z = 0.8A and will be denoted by 7(8). The contrast C is then given by C = log 7(0) log 7(8) or
(20) The density D, is given by log 7(0).