1142 
524 WB 
The quantity poCy is commonly called the 
acoustic impedance of the medium. Equation (19) 
applies specifically to sea water having a salinity 
of 32 parts per thousand. 
10 
It is now possible to investigate the relative 
magnitude of the terms for internal and kinetic 
energy in Eq. (3). The part of the energy flux 
given by these terms is 
Fe= f (dout+ pn)udt. (20) 
0 
From Eqs. (14) and (15) 
gpu?=pAn=3(Ap)?/k, (14), (15) 
and the particle velocity is given by 
u=Ap/poCo, 
approximately. 
Using the exponential approximation for the 
pressure versus time relation, the part of energy 
flux given by Eq. (20) becomes 
ae (- --} fi 2d 
= — Ap) dt 
poCo 3 K 0 
1 t 
(2X 10-®P,,) it (ap)'dt, (21) 
poCo ry 
where pressures are in lb./in.? and the energy flux 
is in in. lb./in.2. The term (2X 10-§P,,) represents 
a fractional part of the total energy flux given by 
Eq. (17). The contribution of the kinetic and 
internal energy terms to the flux F; is, therefore, 
F, I> NGA 
OU eELCent: 
Pi NES 108 PS 
or approximately 
(2X10-4P,,) percent. (22) 
At the highest pressure levels so far investi- 
gated (ca. 30,000 to 40,000 Ib./in.?) this contri- 
bution is of the order of a few percent. For 
pressures below 10,000 Ib./in.? the contributions 
of the kinetic and internal energy terms are 
negligibly small. 
ARONS AND D. R. YENNIE 
IV. ENERGY DISSIPATION AT THE SHOCK 
FRONT 
11 
Acoustic theory, which does not admit dissi- 
pative effects, predicts that the pressure in a 
spherically divergent wave will decay as the 
inverse first power of the radial distance. It would 
naturally be expected that a finite amplitude 
wave should decay somewhat more rapidly, and 
this fact has been confirmed by experimental 
pressure-distance curves. 
As noted in Section 5, a decay of this type 
implies that some energy is being left behind as 
thermal energy in the water through which the 
wave has passed. Most of this dissipation can 
probably be ascribed to the irreversible thermo- 
dynamic process occurring at the shock front. 
Asan element of fluid passes through the shock 
front, it undergoes a sudden non-isentropic com- 
pression, the final state being determined by the 
Rankine-Hugoniot conditions.* When the pres- 
sure later drops to the hydrostatic level, it is 
found that the element of fluid has suffered a net 
increase of enthalpy (and entropy). This increase 
of enthalpy, which depends on the magnitude of 
the pressure at the shock front, is known as the 
dissipated enthalpy increment and will be desig- 
nated by the symbol h. 
The dissipated enthalpy increment is approxi- 
mately proportional to the cube of the shock- 
wave pressure for low and moderate pressures, 
the limiting law for low pressures being given by® 
h=(1/12)(d%/aP?),(AP.)*. (23) 
Using the Ekman equation of state* for sea 
water and applying Eq. (23), it is found that 
K=U52X10-“P,)*, (24) 
where kh: is in in.-lb./Ib. and AP, is in Ib./in.?. 
Equation (24) holds quite well for pressures up to 
5000 Ib. /in.?. 
12 
For higher pressures a modified adiabatic Tait 
equation of state has been used :4 
P=B(S)[(v/v)"—1], 
5 J. G. Kirkwood and H. Bethe; J. G. Kirkwood and E. 
Montroll, OSRD Reports No. 588 and 676. 
(2S) 
