378 SECONDARY PRESSURE WAVES 



ation, is capable of accounting for the dissipation indicated by period 

 measurements. 



One result which is roughly accounted for by theory is the fact that 

 the total energy loss near the first minimum changes very little regard- 

 less of migration, although the acoustic energy loss is greatty decreased 

 if the bubble has appreciable upward velocity. Under these conditions, 

 the bubble is considerably larger and photographs of such bubbles show 

 a flattening of the after surface (which may often be concave). A tur- 

 bulent wake is also formed as a result of water breaking away from the 

 bubble surface, so to speak, and forming eddies in which energy is dissi- 

 pated by work done against viscosity. 



The hydrod^^namic drag due to viscosity must, if the bubble moves 

 upward with a constant velocity U, be equal to the buoyant force of the 

 hollow, and can be expressed in terms of a drag coefficient Cd defined by 



Buoyant force = pogV = Cd[iPoU^A] 



where A is the projected area of the bubble normal to the direction of 

 motion and V its volume. Taylor and Davies (110) have measured 

 velocities of air bubbles released under liquids and found values of Cd 

 of the order 1.0 sec.~^ which had rather large scatter, for volumes 

 ranging from 1.5 to 30 cm.^ The time rate of energy dissipation in such 

 motion must he U X buoyant force, and hence is given by 



Rate of dissipation = ^poACdU^ 



Taylor and Davies were able to show that the loss so obtained was of 

 the same order as values calculated from a theory of distortion of a 

 bubble in an assumed turbulent field of viscous flow. 



The magnitude of energy losses from turbulence in upward bubble 

 motion has been estimated by the above equation for the Road Research 

 Laboratory measurements (95) of the bubble from a one ounce charge 

 detonated three feet below the surface. The rate of dissipation in- 

 creases as the cube of the velocity and hence is appreciable only near 

 the minimum radius, more than eighty per cent of the loss occurring in 

 the last millisecond of the eighty msec, period. The total loss up to the 

 first minimum for the estimated velocity at this time amounted to only 

 four per cent of the total bubble energy. The velocity during the short 

 time near the minimum is difficult to determine accurately, too low a 

 value being probable, and the drag coefficient might well be larger for 

 the irregular explosion bubble than for the air bubbles measured by 

 Taylor and Davies. It is therefore not unreasonable to suppose that 

 the energy losses in vertical motion can be explained, at least semi- 

 empirically, in this way. (It should be noted that, in the case con- 



