E. J. Skudrzyk 219 
TABLE 12.11 
i i < [pee limes 
Correlation function R (p) Pscl 7 _ kT gts] 
or power spectrum E (k) |pol Qn 
The above factor ] 
cr 
Case 
—1/R 
R(p)=e 8nR*/[1 + (CR)7]? 
2 | R(p)= en (t/R)? 7/2 R3 - UR/2)? 
3 | R@M=l+@/R77? TRie-TR 
AC af 3 
4 | R@=G-1/R);rSR p= Be epaiP4 2 wc Pr] 
(rR) [TR 
R(p)=0;r>R 
1 
2 
_ a (m— 1) —m 
by E(k) ge (K/Ko) Gas) <a2> KB 
2 
for K= ksin(9/2) > Ko pat <a”> 
TK , 3 
ey k 
2r 
6 | e«)=Kx7 2 7 [2 sin(O9/2)) °° 
As the angle between the direction of incident sound and the direction of scat- 
tering becomes smaller, the scattered intensity is determined by larger patch 
sizes. 
The spectral intensity of the thermal fluctuations increases with decreasing 
k; scattering, therefore, increases toward the direction of the traveling wave. 
Scattering in the forward direction is described by the very large patches; the 
first-order approximation, Eq. (20), that was obtained by retaining only the 
linear powers in the Rayleigh integral, then breaks down. A better approximation 
(see Section 12.4) shows that the pressure scattered by very large patches is 
90° out of phase with the transmitted sound and affects the phase rather than 
the amplitude of the signal. The scattered pressure, then, is no longer coherent 
with the incident sound; the scattering integral degenerates to a mathematical 
correction that has no physical significance, and its square is no longer a meas- 
ure of the divergence of the energy. The scattering integral p.. primarily de- 
scribes the change in phase because of the variation of the sound velocity in 
the medium. 
If the power spectrum of the sound velocity fluctuations is of a Kolmogorov 
type, Eq. (41) leads to 
<p?> _ rk*K 2k sin(Oo/2)~”? _ 1K 4 ¥5 (963 =1/, 
oe TORE [ak sin (6,/DI2 SoH (2 sin (0o/2)] (43) 
Scattering, then, increases with the cube root of the frequency. For forward 
scattering, 0, =0, and the above solution breaks down because of the assumption 
L + ~ in the evaluation of theintegralin Eq. (40). If the p integration is performed 
for finite L, the solution becomes 
(coke (ioe) BniGeein yo itn au 
BS Seo EM9) SENS UNS 44 
“yoo mre | Tk ; K-T K+T ‘ ) 
