175 
15 III. PRESSURE WAVE 
6. THEORY OF SECONDARY IMPULSES 
indicates that T = = T, » approximately. Finally at very large amplitudes the 
period is given approximately by Willis' formula: 
1 5 
T = 1.88 % Ver = 1.149", ‘we 
where 7,, is the maximum radius of the bubble or W, is the maximum energy of the gas, 
moving adiabatically, during an oscillation. The gas need not be assumed to behave 
as an ideal gas. In all cases the theory indicates a decrease in the period with 
increasing hydrostatic pressure, as is actually observed for the intervals between 
the secondary impulses. For a gas expanding adiabatically from 4 given state, r, is 
proportional top, 8” where y is the ratio of the specific heats of, the gas. Hence, 
according to the small-amplitude formula, 7, is proportional to n, 2 37! » whereas 
according to the large-amplitude formula T is proportional to Be 
The total impulse, [pdt » is very simply related to the particle velocity 
in non-compressive radial motion (e.g., in Equation [3] of Appendix I, Section 1, let 
the velocity of sound c become infinite). When the amplitude of oscillation is large, 
the total positive impulse at a distance r from the center, ive, [pat taken over the 
part of the cycle during which the pressure p exceeds the hydrostatic pressure p,, is 
given by the formula (Appendix II, Equation [25]): 
Jc — Po) at = 048 5t pt wi 
When compressibility of the water is taken into account, all of these re- 
sults require modification and, unfortunately, the theory can be worked out only by 
methods of numerical integration. 
Whereas the motion of an incompressible liquid is all afterflow, in a com- 
pressible liquid the particle velocity contains an additional component that is pro- 
portional to the pressure and hence in phase with it (the term p/pc in Equation [3] 
in Appendix I, Section 1). The effect is both to modify the motion of the gas and to 
cause a radiation of energy. Such effects should become appreciable in water at 
pressures exceeding 1000 pounds per square inch. 
Because of the loss of energy, the gas will collapse less completely in 
each successive oscillation, and the maximum pressure and the total impulse will de- 
crease from one secondary wave to the next. It may even happen that the first second- 
ary impulse is smaller than the primary impulse. Furthermore, the interval between 
oscillations will decrease slowly, as is actually observed for the intervals between 
secondary impulses. Exact calculations for the secondary waves are needed. 
There is ample reason to believe that the loss of energy will be relatively 
large. It can be shown that a pressure curve such as that obtained when the water is 
treated as incompressible would involve, in actual water, a loss of energy in each 
oscillation comparable in magnitude with the energy of the gas. From his observa- 
tions, Hilliar (1) concluded that the part of the primary pressure wave which was 
