360 SECONDARY PRESSURE WAVES 



The first term is easily shown to be simply the increase in kinetic energy 

 of the fluid exterior to the surface of radius r (see, for example, section 

 8.2), and the second term is the increase in potential energy from work 

 done against hydrostatic pressure. The total afterflow energy of the 

 noncompressive flow thus goes, as of course it must, into mechanical 

 energy of the water. 



The kinetic energy of unit volume of water is given by ^poU^ = 

 \po{a/ry-{da/dty, and thus falls off inversely as the fourth power of dis- 

 tance from the origin. The maximum value at a given point occurs 

 when d/dt (a^ da/dt) = 0, and from Eq. (9.4) 



The pressure at this time is therefore less than hydrostatic, and the 

 maximum value of the kinetic energy of outward flow occurs when the 

 bubble is expanded beyond its equilibrium radius. The negative gauge 

 pressure would of course be measured only by a device offering no op- 

 position to the outward (or inward) flow. 



The total kinetic energy in the water when at a maximum repre- 

 sents a great part of the energy remaining after emission of the shock 

 wave, as the gas pressure in the products is less than hydrostatic and 

 the work done against gravity in expansion to the radius at this time is 

 of the order of thirty per cent of the total. This energy is concentrated 

 in a region near the gas sphere, and the large values of energy density 

 in this region have led many writers to assign great importance to it as 

 a factor in damage from near contact explosions. This viewpoint, 

 however, is certainly erroneous, insofar as it implies the existence of a 

 "kinetic wave" as separate and distinct from the pressure wave. As 

 Kennard (57) has emphasized, the effect of underwater explosions on 

 targets is determined by the pressure field; the pressure is physically 

 inseparable from motion of the water and must include any effects of 

 this motion. This does not mean, of course, that the pressure on a sur- 

 face is the same as that in free water, because the water and target to- 

 gether constitute a single dynamical system, each part of which affects 

 the other. 



The significance to be attached to afterflow velocity and afterflow 

 energy in a spherical pressure wave can perhaps be made more clear by 

 the simple example suggested by Kennard of an infinite, plane, rigid 

 plate in the field of an explosion. A spherical wave striking this plate 

 must be modified in such a way that the resultant normal velocity of the 

 water in contact witli the plate, including the afterflow velocity, is zero. 

 This condition is, however, exactly satisfied for acoustic waves by a 

 reflected spherical wave of the same intensity, proceeding as if it origi- 

 nated at the same distance behind the plate as the source of the original 



