214 Lecture 12 
where €=IR = 2kRsin(@)/2). The scattered pressure has decreased tohalf its max- 
imum value when ¢€= 2.50 rad. The corresponding scattering angle is given by 
Pos) Xv 
) 0 
(29 sos —— = OVS 30 
i nEOMNRER 2R (30) 
Outside this region, where sin ¢ « cos ¢, the scattered pressure becomes 
ps —jkr, yr 
Psc = a a (31) 
For small values of ¢, that is, for scattering in the forward direction or for low 
frequencies, the Taylor development leads to the classical solution [Eq. (28)]. 
For a large scatterer, the scattered pressure fluctuates rapidly with changing 
scattering angle, and the envelope of these fluctuations decreases inversely 
proportional to the square of the sine of half of the scattering angle and to the 
square of the frequency. The phase of the scattered pressure is given by that 
of the incident wave at the center of the scatterer and the distance of the point 
of observation from the center; therefore, it corresponds to the distance the 
sound has actually traveled. However, this result applies only if the scatterer 
is very small in comparison to the distance from the scatterer, so that the 
higher-order terms in the Rayleigh integral can be neglected. 
It is also interesting to study the effect of a continuous transition of the 
sound velocity from the undisturbed medium to the center of the scattering 
patch. Four cases of a gradual velocity transition are easily soluble [10,11]. 
They are represented in Table 12.1 as cases 3 to 6: 
ag(l1—r/R)? for r<R 
for r>R (32) 
ao 
Ore ayugetevereeet ene ———— 
ae) [1 + (r/R)7I? 
icons tokcrsvele shaw a@ysenen 
Figure 12.12 shows a graphical comparison of the scattered pressure for 
the cases represented in Table 12.1. The ordinate represents the scattered pres- 
sure; the abscissa, the quantity [r, whichis twice the product of the undisturbed- 
medium wave number with the effective radius R of the scatterer, as defined 
by Eqs. (82) and the sine of half the scattering angle. Forward scattering 
is, thus, described by points on the vertical axis, and backward scattering 
is described by points that, proportionally, are more to the right as the wave 
number and the radius of the scatterer become greater. Since the same scat- 
tering strength has been assumed [see Eq. (25)], forward scattering is the same 
in every case. For a given scattering power or a given volume of scatterers, 
the classical scatterer is by far the most effective. The scattered pressure is 
independent of the angle and has the maximum value possible. If the scatterer 
