522 HERSEY AND BACKUS [CHAP. 13 



So far as we are aware the integral of (30) has not been evaluated or approxi- 

 mated for a piston transducer, while for that of (31) numerical integration has 

 been performed (NDRC, 1946a). They report measurements of w showing peak 

 values close to 3 x 10"'^ m~i at depths of about 400 m in the frequency range 

 from 10-80 kc/s. 



Machlup (see Machlup and Hersey, 1955) has treated the problem of back- 

 scattering of the omnidirectional shock wave from a nearby explosion to both 

 omnidirectional and piston or searchlight receivers. F{r) represents the energy 

 flux per unit solid angle of the shock wave ; b-{d) reduces to b{d) since the 

 explosion is considered omnidirectional; and, finally, if we assume r-^t so that 

 r = ctl'2 is a sufficient approximation, then the r-integration can be performed. 

 The total energy in the shock wave can be written 



/■*oo 



E = 4:77 \ F{t) dr. (33) 



Then equation (27) reduces to 



Ec r"/2 

 I{t) = — m{r cos d) b{9) sin 9 dd. (34) 



Treating the receiving transducer as a freely vibrating circular piston in an 

 infinite rigid baffle (Morse, 1948), 



m = 



2Ji{k sin 6) 



k sin 6 



(35) 



where Ji = Bessel function of order one, k = irdjA, d — diameter of the piston, and 

 A = c//= wavelength of the sound in the transmitting medium. Machlup (Mach- 

 lup and Hersey, 1955), seeking a simpler analytic function than (35), found 



b{e) = cos« {6), (36) 



where n= {k^l2) — 1, to be an adequate approximation to (35), where side lobes 

 are not important. 



Machlup chose b{6) = cos" 6 dehberately so that equation (34) could be put 

 in the form 



I{t) = ^ - \ ' m(r cos d) oos^ d d cos d (37) 



4 r^ Jo 



(a Fredholm equation of the first kind), which has a solution 



(38) 



Id 



m(z) — r- 



^ ' z^ dz 



Yc 2"+'^(2) 



where 



Using the relationship 



dy ^y d (log y) 

 dx X d (log x) 



