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SECTION V. EXTRAPOLATION OF THE STEP-BY-STEP 
RESULTS AND COMPARISON WITH EXPERIMENT, _ 
The next problem is to take the pressure and velocity distribution curves at the end of 
the 27th step, and extrapolate to very much greater radii, and later times. This can be done 
falrly accurately without much trouble, because the ordinary theory of sound is now not a bad 
approximation. Some modification at the shock wave front is however necessary, 
Let Pe be the pressure difference at the shock wave front when the radius is Roe as 
obtained by the last steo in the step-by-step caleulations. Then, If the theory of sound 
applied exactly to the pressurc discontInulty at the shock wave, the pressure difference P at 
the shock wave when the radius was R, R > Ro» would be 
P = PRO/R (16) 
The value of P given by this formula may be taken as a first approximation to the true value. 
Better approximations may be obtained by the following arguments. 
The speea at which a particular value ef the pressure at a point just behino the shock 
wave Is passing up into the shock wave, tnereby determining the pressure there, by the results 
of Section II1, is V+ u ~ U, or 74P metres per second, Hence by the time the shock wave has 
reached radius R, the pressure at the shock wave has come from a point which was distance a 
behind the shock wave when the latter was at Ros 
dir= (V+ u-U) at, 
the limits of t belny those corresponding with the passage of the shock wave from R, to R. 
Now, with jood approximation, we may replace Jt by dR/U, and take only the leading term of U, 
Mamely 1.55 X 10°, Similarly V + u— U may be replaced by 74Q0P, Moreover, P Is given 
approximately by (16). Thus 
5 
= 5 = 
d' PR (7400/1655 x 10°) GR/R 0,048 P_R Vog,R/R (17) 
Ry 
dis Ry and R are incm., and a is in 107 kymfcm. Wwe nave used the symbol d* instead of d to 
remind us of the approximate nature of the calculation by which it was obtained. Clearly d* is 
an upper limit to d oecause the value of P given by (16) is too large, and this error swamps al) 
the other errors, 
From the pressure-radius curve obtained in the last step of the step-by-step calculations, 
the pressure at distance d' behind the shock wave may be read off. Let this be n(d"). Thena 
second approximation for tne pressure at the shock wave front when the wave has reached r Is 
py = (Rj = 4") p (d")/r (18) 
To jet another approximation, we use this expression instead of (16) and repeat the 
calculations to get another approximation d" tod. Clearly 
a* = 0,048 (Ry - d) p (d") og R/R, 
The value d* is probably as much below d as gd’ is above. Thus the average of d* ana 
d" should be a very jood approximation to d. We take 
d = (d*+0')/2, 
Using this expression for 3, we use an equation similar to (18) for the shock wave 
pressure at radius R, namely 
Pp = (R,- 9) p (a)éR (19) 
This equatlon, together witn the results of tne last step in the step-by-steo calculations, 
enables us to calculate the pressure of the shock wave at any later time, 
In order seece 
