Ill- 90 



and 9 = tan (cos'^y)- The quantities a^i and bj^ are functions only of the paranie- 

 ter ph8. These functions are somewhat complicated and are given by Parker (Ref. 

 III-33, pg. 680) . If we assume ph9 ^ 1, we need consider only the term for n=0, 1, 2 

 in (III- 11 7). Therefore 



ku(x) = c + Cj sin(px + bi) + Cg sin (2px + bg) (III-118) 



Using (III-71), e can be expanded as the product of two series of Bessel functions. 



Substituting this in (III- 113b), B is given by 



o 



n= 



00 



E-i(2bi-b3) 

 j3n(ci)Jn(c3) e (III-119) 



Recall that in terms of our usual notation, B = - A . Furthermore, for ph0 ^ 1, 



' f ' 



it turns out that 2bi - bg « -2 n, and thus we have 



A^ = -^ h^{c,)]^(c^) (III-120) 



Since c^^ is of the order of kh, it follows that for kh small enough (kh«l), we may 

 ignore Bessel functions of order 2 or greater and therefore 



A^«- J„(c,)J^(c2) (III-121) 



This first coefficient is the only one for which Parker obtains even an approximate 

 expression. 



As mentioned earlier, no work seems to have been done in relating the 

 above theories to scattering observed in the sea. However, laboratory experiments, 

 with artificially constructed wave surfaces give at least qualitative support to the 

 theories. La Casce and Tamarkin (Ref. III-18) used a corrugated cork surface 

 floating on water to give the sinusoidal pressure release . Several surfaces were 

 used, with the incident sound at 10 kc intervals over the range 80 kc - 300 kc and 

 various angles of incidence. Leporskii (Ref. III-21) made similar measurements 

 using a totally submerged thin glass foil as a reflecting surface. 



artbur Sl.HtttkHnt. 



S-7001-0307 



