E. J. Skudrzyk 213 
e 
wher 2x2pyaR3 
3 
Si 
is a measure of the strength of the scatterer; the magnitude s is equal to the 
amplitude of the scattered pressure at unit distance from the scatterer. Thus, 
the pressure scattered in the forward direction becomes proportional to the 
product ar, where a is the average deviation of the sound velocity from its mean 
value over the volume ; and ; is the volume of the scatterer. Polar coordinates 
p, 9, and ¢ have been introduced to replace €, 7, and ¢. 
12.3.2. Spherical Scatterers 
A great deal of information can be obtained by considering the simple case 
of a spherical scatterer and assuming that the sound velocity is a function of 
the distance from the center of the scatterer only. The angular integration can 
be easily performed as follows. The quantities a, B, and y- 1can be considered 
components of a vector A of magnitude 
A=l[a7+ B? + - 1)2]% =(2(0- yl” = (2(1-cos Oo)” =2 sin 22 (26) 
4 is the angle between the direction of the incident sound (that of the z or ¢ axis) 
and that of the scattered sound (@)= 0 for forward scattering). The magnitude 4 
and the direction (a, 8, and y-1) of the vector A depend upon the scattering angle 
only. In the integration over the scatterer, A is constant; therefore, the vector 
A may be assumed to be the axis of a system of polar coordinates p, 0, and ¢. 
The exponent in the integrand in Eq. (24) can then be written pAcos 6; and the 
integral becomes independent of ¢: 
-jkr 2 ith 4 = 
kpoe ieren0 : k2ppetkto 2 
Deg =—————_ a(p)el*4°C°S"a(-cos 6) p2dp Ae eS a(p) sin(I'p)pdp 
to h 0 Les 
To 
(27) 
where I= 2k sin(@)/2). The Rayleigh integral now becomes solvable for several 
different velocity distributions. 
In the classical theory of scattering, the dimensions of the scatterers are 
small compared to the wavelength of the radiation and the distance from the 
scatterer, ry. The phase change of the incident pressure, p;, and the scattered 
pressure, p.-, over the volume of the scatterer, 7 = 47R*/3 (where R is the radius 
of a sphere that has the same volume as the scatterer), may then be neglected; 
the scattered pressure 
Boke e7/K0 _ 87°Rpo R .-~ikn 
PE Pe ae Sree oe v (28) 
becomes independent of the scattering angle and is proportional to the square 
of the ratio of the diameter tothe wavelength. Particles that are small compared 
to the wavelength of the radiation are, therefore, very inefficient scatterers. 
If the scatterer is a finite sphere of constant sound velocity, the integral 
for scattered pressure in Eq. (27) simplifies to 
=, R as : 
cl Spc SETI i sin (Up) pdp Ss oig nS seens 
R°T'r0 0 uo ¢ 
(29) 
