212 Lecture 12 
fe eTkor 
Psc=97 J 2Pi Sr (20) 
is represented by the Rayleigh integral [9] taken over the total scattering volume 
and the distance r is given by 
r= (x- €)?+(y - €)? +(2-&) (21) 
No assumptions other than Ac/co <1 have been required in the derivation of this 
formula. To simplify the notation, ko has been replaced by k, where k represents 
the average wave number in the inhomogeneous medium. As is standard practice 
in acoustics, the solutions here and in the following pages are represented in 
complex forms. The actual solution is given by the real part of the complex 
solutions. 
Physically, the Rayleigh integral denotes the sound pressure radiated by a 
source distribution that is excited by the incident sound. Every elementary vol- 
ume of the fluid turns into a sound generator that, at the expense of the energy 
of the incident sound, radiates (scatters) sound in all the different directions. 
The simplest and most efficient way to study scattering problems is to con- 
sider a parallel beam of sound p,; =poe~*? propagated in the direction of the 
positive £ axis. The scattering volume may be assumed to be small in compari- 
son to its distance, ro, from the receiver and to be centered at (€ 7, €)=0. Let 
the point of observation have the coordinates x, y, andz. The scattered pressure 
then is given by the Rayleigh integral 
pse (x,y, 2)= =F fa 1, 6) 5 pve" Jak dn dl (22) 
Tr 
where a = Ac/cy is the relative change in sound velocity and r [see Eq. (21)] is the 
length of the radius vector. It is standard practice to develop r, the exponent, 
into a Taylor series: 
xE yn 26 € 477+? Gene Ge 
ee ea Le ae 0 Se | (28 
=o Rear + Tp + /..=% —(a&+ Bn +yl)+ Ore + (23) 
In this equation ro is the distance of the receiver from the center of the scatter 
and a=x/ro, B=y/ro, and y=2z/rpg denote the direction cosines of the scattered 
sound. The Rayleigh integral then simplifies to 
Psc (x, y, z) = 
kp (2 - a- 7B - &y - 1)] dédnd€ (24) 
27 3 
To 
In the denominator, r has been replaced by rp on the assumption that the dimen- 
sions of the scatterers are small in comparison to the distance of the scatterer 
from the sound receiver. 
For forward scattering (@=f8=0, y=1, and @)=0), the scattered pressure 
becomes 
k2 —jkto a2 . ; 
Psc (0) a 2.a(p) p2dp = tt <a> ep (25) 
