222 Lecture 12 
be reviewed here; this has been done elsewhere [12, 13,58]. However, it is pos- 
sible to derive practically the same results by elementary considerations 
exclusively. 
In Section 12.3, forward scattering was shown to be practically independent 
of the model assumed for the scatterers. Therefore, we can expect to obtain all 
the necessary information about the fluctuations of the amplitude and the phase 
of the transmitted signal by assuming scatterers that are as simple as possible, 
such as parallel discs having sound velocities slightly different from the mean 
value and thicknesses and radii of dx and R, respectively. The scattered pres- 
sure, then, is given by the integral 
k2 —jkx’ —jkr 
Pse = Poo oe dx 27 pdp (45) 
Integration can be performed in cylindrical coordinates setting 
2 
2 
r?=p?+x? 
r dr = pdp (46) 
If x’ is the distance between the sound source and the scatterer, and x the dis- 
tance of the point of observation, the pressure scattered by such a disc is given 
by 
mx = x!)2 +R2 
‘ —jkx td T2 2 
Psc = Po dxk*e—i** ae-ikt dr = po dxk” © =" eT = x) + RO ene Ge — x!) 
cd 
Nx = (47) 
The disc radius R can be considered as small in comparison to the range. Taylor 
development of the exponent then leads to the result 
Psc = JPo dx ke“ a |] — @VR2/2x = x! | (48) 
If R is very large, the second term in the brackets of Eq. (48) can be set 
equal to zero (as can be proved by the assumption of a very small attenuation). 
The scatterer pressure 
Op sc = — jka dxpo (49) 
then has a phase lagging 90° behind that of the incident sound velocity of the slab. 
The scattered pressure is a maximum if the second term in the bracket of 
Eq. (48) is -1; that is, if 
Rome Rea 
DG Te (50) 
or 
R?=)r or kR?=2nr - (51) 
where r has been written for x — x’. Ifthis condition is fulfilled, the disc will have 
exactly the same area as half the central zone in the Huygens zone construction. 
The average pressure that is scattered in the forward direction by such a 
disc is given by Dsc = 2jkaRpo 
