396 SURFACE AND OTHER EFFECTS 



pressure has fallen from its peak value P^ by an amount {Po + Pb) and 

 hence P = P,n - {Po + Pb) = Pme-"''. If P^ is large, t'/d is small and 

 expanding the exponential gives t' = (Po + Pb)S/Pm. If Pb is only a 

 few atmospheres and Pm is large, f is a small fraction of the time constant, 

 and the depth A is only a small fraction of the length of the incident 

 wave. 



If the pressure is assumed to vary with distance as R~", where R is 

 the distance from the real source at a depth d, an approximate calcu- 

 lation setting the resultant pressure excess at a depth d equal to 

 — (Po + Pb) gives the result obtained by Pekeris (82) that 



^^_(Po + Pb)Rcod 



(^-^^^1) 



2P„d 



This is obtained by expanding the difference and ratio of the paths of 

 the two waves to point A in series and using only the first terms. The 

 distance A comes out to be very small if small values of Pb, of not more 

 than a few atmospheres, are assumed. For example taking Pb = 3Po, 

 Pm = 2250 lb./in.2, d = 50 feet (corresponding to 300 pounds of TNT 

 50 feet deep) gives A = .02 feet. On this picture then, a thin layer of 

 water will become detached and rise with virtually the peak velocity of 

 the surface. 



The initial velocity predicted by Eq. (10.1) should, if the foregoing 

 analysis is correct, agree with experimental values if the measured peak 

 pressure Pm is used. This comparison is simply made for the velocit}^ 

 and pressure at a point directly above the charge. A series of measure- 

 ments of this kind made by Pekeris from motion picture records for 

 large charges fired at 40 foot depth agreed with calculated values to 

 within 7 per cent in the average. In another experiment, streak pic- 

 tures of the rise of the dome above 5.1 and 5.5 pound cast TNT charges 

 fired at a depth of 4 feet gave initial velocities of 216 and 230 ft. /sec, 

 the values computed from gauge measurements of P,„ being 220 and 

 228 ft. /sec. It should be mentioned, however, that anomalously high 

 apparent initial velocities have been observed on motion picture records 

 of large charges, the cause for which is not known. 



As has been noted by Pekeris, several approximations are involved 

 in the use of Eq. (10.1) to relate peak pressure and initial particle 

 velocity at the upper surface of the dome. The first use is the fact that, 

 as written, Eq. (10.1) refers to acoustic waves of infinitesimal ampli- 

 tude travelling with the acoustic velocity c<,. For finite amplitude 

 shock waves, the Rankine-Hugoniot conditions at the shock front 

 should be used, which from Eq. (2.28) gives E(i. (10.1) if Co is replaced 

 by the shock front velocity U which is a known function of pressure. 



