Ill- 15 



The coefficient Co can be found from (III- 11a) to any desired degree of accuracy, 

 since the spherical Bessel functions of order zero are given in terms of the simple 

 trigonometric functions: 



i (\cf>\ - ^^" ^^ „ /u^s _ cos ka 



We can therefore develop Co in a power series in ka, and if we keep all terms 

 up to and including (ka)^ , we find for the coefficient of (III- 18): 



(ka)= 



[ <^-^^^>^^f^^f-i-^3)(ka)-.o((ka)-)] 

 [gh^.(f^-ig-i)(ka)^-i(l-gh=)(M)%o((ka)^)] 



Co-i 3 r TZTi \ r\ :~T^ 7 — vi (111-19) 



For sufficiently small ka, all but the zero order terms may be ignored, and we 

 find that the scattered amplitude behaves indeed as the square of the frequency. 

 This is the approximation equivalent to (III- 15). We notice, however, that as 

 ka increases (though remaining much less than 1), the second order term in the 

 denominator approaches the zeroth order term when (ka)^ approaches 3 gh^. 



since the coefficient of the second term ^— - (1/6) g - 1/3 is essentially - 1/3 



for the combination of air and water. Let us therefore investigate (III- 19) more 

 carefully in the neighborhood of (ka)^ = 3 gh= =« l.SOxlO"*. In this neighbor- 

 hood, the value of the numerator remains essentially unity, since all other terms 

 are of the order of magnitude of gh^ . The denominator must be kept to third 

 order in ka since the zeroth and second order terms can be made to cancel. In 

 this way we obtain the approximation: 



(ka)^ 1 



i~ r \ w:^ ^°'' ^^^^^ ^^sh^ ("i"20) 



I.e. i ,. . -, . (ka) 



gh^ - -(ka)- 



If we examine the magnitude of the scattering strength, we find that the scattered 



wave passes through resonance when (ka)^ is 3 gh^ . In other words, I — ~ I 



I Co -i I 

 passes through a maximum. This may be seen most easily by observing that the 

 square modulus of the coefficient is bounded by unity and reaches its upper bound 

 at the point of resonance: 



artbur Aliitth.'Snc. 



S-7001-0307 



