947 
THEORY OF PEAK PRESSURE AND TIME CONSTANT DETERMINATION FOR SHOCK WAVES 
BY THE METHOD OF OPTICAL DISTORTION 
R. R. Halverson 
A theory has been developed for several sets of experimental conditions designed to 
measure the peak pressure of shock waves by the method of optical distortion. For one of 
these experimental arrangements (1 below), a method has also been developed for calculating 
the time constant. Since this arrangement has had considerable success experimentally, it 
will be discussed in detail. 
ve 
Grid e enter of charge; ch off to the side; herical shock wave 
To study the distortion of light rays passing through the high pressure region 
behind the shock, an experimental arrangement as diagrammed in Figure 133 was used and 
the theory developed is based on these experimental conditions. E is the charge producing 
the shock wave S to be studied, C is the camera lens, F is a flash charge and G is a 
lucite sheet marked off in a wniform grid of 1/4 in. spacing. The grid is so placed 
that the grid lines intersect the shock wave diagonally in the portion of the shock 
studied. Two typical intersecting lines are shown in the front view in Figure 133. The 
charge was placed in the plane of the grid, and the line perpendicular to the grid 
passing through the center of the camera lens intereected the grid a short distance 
behind the shock, approximately 5% of the radius. 
In the system of Cartesian coordinates shown in Figure 133 the position of the 
center of the camera lens is defined at (Xo5 Yo Zo). On the assumption that the grid 
acts as a source of diffuse illumination, a ray whose reverse path is the vector 
M from the camera to the shock front is considered. The intersection of this ray with 
the shock front is defined as (x,y,z). If there were no distortion, the ray would con- 
tinue along the line £ M and strike the grid at (x", 0, 2"), but actually the ray follows 
some curved path P determined by the decay characteristics of the shock and strikes the 
grid at the point (x', 0,z'). Knowing the actual distance from the charge to a uniquely 
defined point on the grid, one obtains the radius of the shock wave from the photographic 
print by reference to this point after the scale factor of the print is determined from 
the undistorted part of the grid. The coordinates of the camera are also obtained by 
reference to this point. On the photographic print, the points (x", 0, 2"), and 
(x', 0, 2") may be located, the latter being obtained by extending lines from the un- 
distorted part of the grid until they intersect behind the shock, 
(a) Peak pressure determination. -- As a first approximation, an average pressure may 
be calculated by assuming a step shock wave, that is, a constant pressure behind 
the front. With this approximation the curve P is replaced by the vector N 
(Figure 133) from Gye) to (x',0,2'), and the step pressure calculated is, to 
this approximation, the pressure in the decaying wave on the spherical surface 
centered at E and passing through a point on P at which the tangent of P is parallel 
to N. We define this index of refraction or pressure for a given intergection of 
grid lines as Ray OF Pay» respectively. 
A very simple derivation of the index of rafraction_corresponding to this 
average pressure as a function of the distortion vector D, (the vector from 
(x",0,z") to (x!,0,z") for the given ray M,) and the geometry of the experiment 
can be given:in the system of coordimates discussed above. We are given 
¥e Zo)» (x",0,2"), (x',0,z') and R , the radius of the shock wave, The 
point txey,2) is readily obtained as the intersection of the line joining 
(X5,Yox%0) and (x",0,s") with the sphere of radius /RI . 
The following vectors are defined. 
p=xi+yjt zk, 
H = (x-x9)4+(y-y,) Jt (2-29)k, 
i = (x®=x)4+t (O~y) j+(2"-z)k, 
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