399 



In the formulas, values of W are in pounds and all values of X and 

 Z in feet; p^ is the total hydrostatic pressure in atmospheres. 



The case, 6=0, applies to a rigid horizontal bottom. The formu- 

 las for = n, on the other hand, are found to apply to a free surface, with 

 Z measured away from the surface and hence downward, provided changes are 

 made corresponding to the assumption that, for the free surface, 



M= - J, N^= -j^ [36a, b] 



in place of Equations [29a, b]. Hence for a rigid horizontal bottom and for 

 the free surface of the water, the formulas can conveniently be written 



together as follows: 



_ - 2.60 B2R2 



r ^ 1 + 4000 fi/ 

 B, = ±|o.0346[l ± 0.23:^] 1^-0. 223 [1 + 0.18^](^'| [38] 

 or, to a good approximation 



B^ = ±[o.0346 ^ - 0.223 (^f] [38a] 



The upper sign refers to the rigid bottom and the lower sign to the free sur- 

 face. The symbol IB^I denotes the numerical value of B^ taken without regard 

 to sign; and Z, - Z^ represents in each case the displacement measured posi- 

 tively away from the surface. 

 The first period is 



T, = r,o[i ±0-20^] [39] 



These formulas probably become unreliable when 



Z <2R, = ^(f) 



Pa 



Because of the negative sign between the two parts of B^, the grav- 

 itational effect and the effect of the surface oppose each other in the case 

 of a free surface or a rigid surface below the charge, whereas the two ef- 

 fects are in the same direction when a rigid surface lies above the charge. 

 The gas globe from a small charge near the surface of the water sinks Instead 

 of rising. 



