II. PHASE AND AMPLITUDE FLUCTUATION SPECTRA, 
We shall assume the wavelength of the 
sound to be very much less than the dimensions 
of the smallest size component of the thermal 
variations. We shall also assume that the 
source is located at the origin of the co- 
ordinate system. Following Tatarski we may 
express the fluctuationgof the phase and the 
logarithmic amplitude ( hereinafter simply 
termed the amplitude fluctuation) at the plane 
x = L by means of the stochastic Fourier- 
Stieltjes integrals: 
S,(L,y,2) = S- S,= 
oe 5K ) 
+K, Zz 
If- a "s da(K,,K,,L) (1) 
and cal 
B(L,y, 2) = log(4/a, )= 
if it inet) da(K,,Kysb) (2) 
with random complex amplitudes ds and da. 
In particular, the correlations of the 
phase and amplitude fluctuations at the points 
(L,r/2,0) and (L,-r/2,0) using (1) and (2) are 
given by 
aoe S (L,r/2, 0)S¥(L,-r/2, 0) = 
i(K,+K}) (1/2) 
If de(K,,K, »L)ds*(K},K3,L) 
(3) 
and 
R,(r)= B(L,r/2,0)B*(L,-r/2,0) = 
Pp eAaRS) (/ 2) 
ie da(K,,K,,L)da*(K3,K3,L) 
(4) 
where ihe star (*) denotes the complex 
conjugate. 
Using equation (9.17) in Tatarski! and 
its analog for the correlation of the ampli- 
tudes of the random ere we get 
ee wi nis ff ips ar 
xp [= — z,| 
—, —, -_ 
N X) x) 1 2 
Lx, (L-x )\K? 
cos 2 2 
sin a 
1 aoe 
x exp[#ik,(1 + x,/x,)] (5) 
where Fy denotes the two dimensional spectral 
cos 
sin 
L(L=x, )K* 
kx 
32 
density of the index of refraction fluctuations 
and k denotes the wave number of the sound. We 
observe that the correlation functions are sym— 
metric abount the origin in the plane x = L. 
Using polar coordinates and employing the simpli- 
fying approximations used by Tatarski, equation 
(5) may be reduced to 
+00 
Ro(r) ad f dK K §,(K) 
m{r) b 7° 2 2 
os a a 
(2) 
where 9, (K) is the three dimensional spectrum 
of the index of refraction variations. 
If we let x = (LK')/K in the inner 
integral of (6) and interchange the order of 
integrations we get 
+00 
R(r) ary [ x dK J, (KL) 
R, (r) +09 0 2 
xz f By(K") O78, fxiar-x)-4 i, (7) 
K 
where K and K* have been interchanged. 
Now, since 
+09 
R(T) og if J, (Kr) FS(K) ax 
ar) g F,(®) 
(8) 
where F,(K) and F,(K) are the two dimensional 
spectra of the phase and amplitude fluctuations, 
then comparing (8) with (7) we have 
+00 2 
F.(K). ame fy (K') fete xray 3h] aK' 
P, (K) K N sin 2k 5 
K (9) 
Defining 
oTr 
x 27\/2k . 
P a OSB 
as the process wave number, and writing 
t. 
ne and n'e« ee 
P 
we may express (9) finally as 
24 2 
Fo(®), REEL fg cave n) &°85 [ante] at 
F,(n) Z sin (10) 
Given the three dimensional spectrum of 
the index of refraction variations, equation 
(10) may be used to determine the spectra of 
the transverse phase and amplitude fluctuations, 
If the results are inserted in equation (8) 
we may also obtain the correlation functions 
of the transverse phase and amplitude fluctua- 
tions. 
