954 
The method used in determining the time constant of the shock wave was to 
express n as a function of r in terms of the parameters of an assumed exponential 
shock wave (exponential with distance behind the front), the peak pressure and time 
constant. This may be done by writing p in Eq. (II-12) as a function of r, 
( IRI - r) 
Pinney : (12-14) 
where @,, is expressed in units of length. 
Substituting p from Eq. (II-14) into Eq. (II-12) gives 
( IRI - x) ae —5 
n(p) = 29 5 8Pnag ®@ Op - bp sex & = » (11-15) 
where n, (the index of refraction at zero pressure), a, and b are functions of the 
temperature for a given type of film, and is the pest pressure for the given 
shot as datermined by the method outlined in s Appendix I AE 
This function of r is then substituted for n in Eq. (II-13). In Eq. (II-13) 
Dg is obtained from Eq. (II-15) by putting r = (RI. 
Eq. (II-13) is a function of the given ray of light on a given film, that is, 
of the pair of intersections (apparent and actual) of the pair of grid lines studied, 
because of the explicit presence of the angle i,. This is obtained for each pair 
of intersections by the following equation: 
Bin to = % P (11-16) 
sin @ Dg 
where the angle @ (Fig. 134) is obtained for the given point (that is, for the 
given pair of intersections) in the calculation procedure for Pay for that point. 
Thua all the parameters of Eq. (II-13) are determined except @,. Some general 
discussion of this equation is felt to be necessary. A statement of Snell's Law 
for spherical symmetry, in which case the path of a given ray of light, as was dis- 
cussed previously, lies in a great circle, is given by the following: 
Dp (Risin i, = nr sin i = 6, (II-17) 
where i is the angle made by the path of light and the radius vector r to a given 
point on the path, and n is the index of refraction at ra for some given @, in 
Eq. (II-15). Then in Eq. (II-13) the denominator vanishes at the point of total 
reflection, that is, at the point where sin i = l. 
The right hand side of Eq. (II-13), from physical arguments, has a finite 
integral from r = {R/ to r = rpiy - € where, 
Trin.2(Tmin-%,) = Bp IRI sin ip . (II-18) 
By making € sufficiently small, the integration can be carried up to within an 
infinitesimal distance from ryjne Then since the path of light is symmetrical 
about the radius vector of length rmin, the whole path is know from the point 
of entry into the shock wave to the point of exit from the shock wave. 
The first step in the calculation of 9, is to determine Oy jn where 
(rc! - D,).n(r" - Dy,@pymin) = Dp IRI sin i). (II-19) 
Physically @, | is the lowest value of the exponential decay constant which will 
allow the wes of light studied to get as far into the shock wave as the point 
at which it is observed to strike the lucite grid, that is, atr =r! -D, (Fig. 134). 
13 15616 
