686 - 8 - 



Consider the txplcsion of 2000 tons oT H.E. pn the surface (such as, for example, the explosion 

 of a ship loafled with bomBs) . The .'.bov^ results on the 1 oz. charge now apply provided the distances 

 are multiplied by 115 times and wave heights by 20.4. 



Other examples may be easily constructed from the scaling laws and the figures just given for the 

 1 oz. charge. 



/V ie Have Heights and fotal Impulse : 



It will be noticed that the above numerical estimates are only concerned with wave velocities, and 

 with interference phenomena such as the position at which a certain wave is the greatest. To calculate 

 the wave heights requires a knowledge of the total positive impulse delivered to the surface. If I is 

 this impulse, then, for example, the height in inches of the second crtjst at 10 feet from a 1 oz. charge 

 exploded on the surface is approxirratoly 



H = 11 X 10"' I 



where I is in pounds/weight/seconds. 



An approximate value of I may be obtained from the knowledge that the explosion of a charge in 

 free air may be regarded as the release at the origin c' a volume of gas at atmospheric pressure 20,000 

 times the charge volume. (This factor is made up of 1000 for the gas products and 19,000 for the shock 

 wave heating of the air near the charge). Hence, the pressure wave from a 1 oz. charge on the surface 

 may be very roughly computed by the theory of the sound waves caused by the release at the origin of 

 25 cubic feet of gas at atmospheric pressure. Then from the theory of sound, the impulse over the water 

 surface should be 



I = Vyp, /2c = 32 lb. /weight/seconds. 



wh,;re V is volume of gas (25 feet ) y is ratio of specific heat of air (l.U) p is atmospheric pressure 

 (2100 lb. /feet ) and c Is the velocity of sound (ll80 ftet/second). Hence the heigHtof the second wave 

 at 10 feet is one third of an inch. This estimate is almost Certainly too low by a factor at least two, 

 but better agreement is hardly to be expected from the theory of sound. 



Underwater Explosion s. 



The wave system from an initial dome and crater of the form shown in Figure 1 is required. 

 However, before evaluating this, it is Tnstructive to repeat the discussion given for the surface impulse 

 to the present case. As before, the group velocity and wave velocities, position of the greatest crest 

 at any time tc, follow simply from the area of the disturbance, and net from the details. 



According to the Cauchy-Potsson theory (see Lamb page 430) the wave height at cj, t due to unit 

 volume placed on tne surface at 0,0 is 



1 2 



£ Ut) = 7 » (gt^/oj) 



2 n w 



H (f?) = -A, +.iLlie' - ^^ '^^ -^^ 0^ ♦ (32) 



•^' 61 101 



with the asymptotic formula when gt /R is large 



H> (5) = 'An cos 't'/'*) 

 a 2-^ 



gt^ 2 



I = — rT% Q cos (gt /4 oj) 



(33) 



Numerical values of the function H (f) are given !n Table 2 while % comparfson of H \9) and the 

 asymptotic expression H (5) Is shown in Figure 3. It is seen that the asymptotfc expression gives 

 numerical values within lU-15* for all 6 > Z, 



Assiming 



