174 
III. PRESSURE WAVE 14 
6. THEORY OF SECONDARY IMPULSES 
Let us start with a sphere of compressed gas of negligible density sur- 
rounded by incompressible water at rest, and neglect gravity. During the motion, 
the pressure p and particle velocity u of the water at a distance r from the center 
will be given by Equations [8] and [2] of Appendix II. 
in which r, is the radius of the sphere of gas, u, = dr,/dt, p, is the pressure of 
the gas, p, is the hydrostatic pressure, p is the density of water. At a great dis- 
tance the Bernouilli tern, 5 pu’, is negligible. Near the center, however, this term 
is not negligible; it causes the pressure transmitted to a distance to be determined, 
“not by the pressure p, of the gas alone, but by the quantity p,+ 5 Pur. As the gas 
expands, p, decreases but uz increases. The pressure impulses are thus made broader 
than would be expected from the variation with time of the gas pressure alone. 
It is evident that oscillations will now occur in the general manner des- 
cribed in Section 3 preceding. Since no energy is lost here, however, the gas must 
return at each collepse to its initial pressure; hence all secondary pressure waves 
Will be alike, and each one will be symmetrical about its center. The first pressure 
wave will be only a half-wave, arising from a single outstroke, whereas each subse- 
quent wave is due to instroke plus outstroke. The impulse at distant points due to 
each secondary wave will thus be twice that due to the primary wave. 
For the period of the oscillations an expression is readily obtained in the 
form of an integral (see Appendix II, Equations [17], [19], [22]). In two extreme 
cases the value of the integral is easily found. 
The period T, of small radial oscillation of a gas bubble about its equi- 
librium size, with its pressure oscillating slightly above and slightly below hydro- 
static pressure, is given by Minnaert's formula (8): 
V; p 
1 i 2779 37D 
in which 7, is the equilibrium radius of the bubble, p the density of the surrounding 
liquid, p, the hydrostatic pressure, and y the ratio of the specific heats of the gas 
at constant pressure and at constant volume, respectively. For air in sea water, at 
a depth h, roughly 
th = ar %% Vaaee seconds, 
where 7, and h are expressed in feet. That the oscillations can occur so rapidly 
becomes plausible when one recalls that the velocity of efflux of water under a 
pressure of only one atmosphere is 47 feet per second. 
As the amplitude of oscillation increases, the period increases. When the 
maximum radius becomes 6 times the minimum, if y= 1.4, a numerical integration 
