291 
-3- 
The first approximation is to assume the shock wave reversible, so that in Figure 1, B and C 
are supposed to coincide. Then Y= NS and iy follows Immediately from (1). 
Now we obtain a first approximation to T,- T', from (7) by neglecting vt = V2 On LeHaSe 
For simplicity, assume that c, = 1cal./gm Then 
0.00266 = (v,- vo P, + 3+MB03) = 0.16009 0, v", (8) 
where @ Is 1, - tT, and the factor 0.0426 has arisen from expressing cal./gm. in mechanioal units 
pv, p being 10? Rome/eat cm. and v in c.c. 
Using the experimental value of the coefficient of expansion at pressure p, over the range 
-6 to tT we then get a second approximation to Yor Second approximations to U and u follow 
immediately. The Iteration process converges so rapidly that further approximations are not 
warranted by the accuracy of the experimental data. 
If p< 1, the heating @ In °C, above the adiabatic heating, is given by the formula 
O = 0,0342p? - 0.0122p" + 0.0038p” - 0,00110° + ..., (9) 
The velocity of sound in any thermodynamic state of water lying on the adiabatic through A 
is obtained by differentiation of (1). Expressing c in metres per second 
Sa = su6s (1 + p/3) 0434 (10) 
Thus, the velocity of sound in water at 20°C ana atmospheric pressure Is estimated at 
1461 m./second. This agrees exactly with the experimental value, within the limits of experimental 
error. 
The following table gives a selection of values of U, u, in = Urs Vy T, and °A for 
various Poe 
TABLE 1. 
2 in °C, U and 
Units; p in kgm. /sqecm. = 6,35 tons/square inch, WY infe.ce,, a oT 
u in m. fsecond. 
The values given in Table 1 agree well with those given In Report "A*%, except that the @ values 
have been improved, and with those quoted by Kistiakowsky and Wilson (Final report on the hydrodynamics 
of shock waves, N.D,R.C.) The temperatures given in this last report, where they differ from those 
above, are probably more accurate than those quoted here, because they allowed for the variation of 
C. with temperature. The difference never exceeds 1°C. The values of @ were not calculated by 
Kist iakowsky and Wilson, and thelr values extend only up to p= 12, 
In the step=by-step calculations described in following paragraphs, the entropy gradient 
terms in the hydrodynamic equations were omitted. Thus, the calculations were made on the assumpt ton 
that the temperature difference @ was insignificant as far as the propagation of oressure waves Is 
concerned, To be consistent throughout, we have therefore calculated U, u, ¥, ¢ behind the shock 
wave on the assumption that 8 and C in Figure 1 are consistent. Table 2 gives the numerical values. 
TABLE 2 oseee 
