398 SURFACE AND OTHER EFFECTS 



face of the dome as a function of distance r along the surface from the 

 point above the charge is easily obtained from Eq. (10.1) if the peak 

 pressure Pm is known as a function of distance R from the charge d feet 

 deep. Assuming a power law Pm = Pm(d) {d/RY where a '^ 1.15, 

 gives 



(10,2) .,0 . ?£=WY, + liV-f' 



u{r) ^ A , A 



u{o) \ dy 



2 



The outline of the dome is thus a smooth curve, the center of which 

 rises most rapidly, the contour at other points depending on the ratio 

 r/d. Measurements of the dome shape therefore permit a calculation 

 of the depth of explosion d ii sl scale for r is known. This and related 

 deductions have formed the basis of a number of methods for determin- 

 ing depths of underwater explosions. The first of these, proposed by 

 Shaw (101), consisted in determining the distance r' for which the rise 

 of the dome is one-half its central value. Shaw assumed the value 

 a = 1 for peak pressure variation with distance, which from Eq. (10.2) 

 gives r' = d, the depth of explosion. This result is modified slightly for 

 a > 1, but the method can still be applied. Shaw's method has the 

 disadvantage that a distance scale is needed, and there are practical 

 difficulties in accurate measurement of small initial heights, to which 

 the method strictly should be applied. Pekeris has for this reason out- 

 lined a procedure (82) based on extrapolating more accurate values at 

 later times, and has also analyzed a method in which the initial velocity 

 of the dome center is obtained by extrapolation of the measured velocity- 

 time curve. A critical study of these and other methods has been made 

 by Halverson and co-workers, which is discussed in section 10.3. 



The initial upward velocity of the dome of course decreases as it 

 rises, because of gravity, air resistance and differences in pressure be- 

 tween the upper surface and interior of the dome. If gravit}^ alone 

 acted, the deceleration would be 32 ft. /sec. 2, but Pekeris has found an 

 average deceleration for domes above depth charges of about 85 ft. /sec, 

 nearly 3 times this value. This increased deceleration could be at- 

 tributed either to drag resistance of drops of spray, or to the effect of 

 pressure excess on an unbroken surface. As Pekeris points out, the 

 observed deceleration is a strong argument against the existence of an 

 unbroken surface of the dome, as the deceleration of a sheet of thickness 

 z feet with pressure difference of one atmosphere (14.7 Ib./in.^) would be 

 14.7 X 32/0.433 z = (1100/z) ft./sec. The layer would on this basis 

 have to be at least 10 feet thick to give the observed deceleration. It 

 must therefore be concluded, both from calculation and from the ob- 



