553 
- 5- 
19.7 tons/sq. in. On the other hand, when t* = 0,186 millisecond, the cylindrical surface of the bar 
is subjected to a normal pressure, which varies from 7.4 tons/sq.in, at the shock wave front to about 
2.1 tons/sq.in. at a distance of 3.2 cm. from the pressure end of the bar; the plane end surface of 
the bar is subjected to a normal pressure of about 1.6 tons/sq. in. 
Thus, after the arrival of the shock wave at the measuring end of the bar, a portion of the 
cylindrical surface of the bar is stressed normally, the length of the stressed portion increases 
non—linearly with time, and the value of the pressure at any point in this part of the bar decreases 
as time increases; in addition, the plane end surface of the bar is subjected to a normal pressure 
which is less than the shock-wave pressure and which decreases as time increases. in these 
circumstances, equation (3.4) giving the applied pressure in terms of the velocity of the measuring 
end of the bar, ceases to be true, and it is difficult to derive the relationship which wil? hold 
between the two quantities under the actual conditions of the experiment. |t is, however, possible 
to investigate the simple case of a cylindrical pressure bar, subjected to a constant, uniform 
hydrostatic pressure along a portion of its length, assuming that the length of the stressed portion 
increases uniformly with time. 
In Figure 5 let Ox represent the axis of a cylindrical pressure bar, the origin 0 being taken 
at the pressure end of the bar. we shall assume that the bar is sufficiently long to avoid overlapping 
of direct and reflected pulses, and that the radius of the bar is sufficiently smal} to enable dispersion 
effects to be neglected, Suppose that the pressure on the bar is initially zero and that a region of 
uniform hydrostatic pressure P, moves with a uniform velocity U in the pdsitive direction of the 
x — axis; let the boundary between this region and the region where the preesure is zero, i.e. the 
shock wave front in the experiment, arrive at 0 at time t' = 0. Exaggerating the changes in the 
radius of the bar, the state of affairs at time t' may be represented diagramatically by Figure 5, 
where the shock wave front has reached the point A (OA = Ut'), whilst the front of the stress pulse 
in the bar, which starts from 0 at time t" = 0, has reached the point D (0D = cot*). The bar thus 
consists of three regions, — the region of uniform hydrcstatic pressure, P,- region (1), say — the 
region of the stress pulse, in which the bar is strained in the direction ox, the cylindrica) surface 
being free from stress— region (2), say—and finally, the undisturbed region ~region (3), say, 
" 
cet u,, u = the particle velocities in regions (1) and (2) 
1 
ae) 
u 
tne pressure (in the direction ox) in region (2), 
Ay, A, A, = the area of cross-section of the bar in regions (1), (2), (3) respectively. 
Py Pr Po the density of the material of the bar in regions (1), (2), (3) respectively. 
It follows that: 
P = pc_u pons adic: sotGe “Gone idbo, toon (4, 2) 
|f we superpose a velocity U in the negative direction of Ox on the bar, the boundary BC is 
Drought to rest and the particle velocities in regions (1) and (2) become (u - uy) and (U = u) 
respectively. 
Under these conditions, conservation of mass at the boundary BC gives 
PyA, (U- u,) = A (U = u) oboe cd Yydoor acd (4.3); 
conservation of momentum at this boundary gives 
Pia, - PA = pA (U = u) (u, - 1) PsA, (U - u,) (u, - Wh) Gor. ond (4.4). 
In order to find the relationship Detween P, Pye U and coy it is necessary to eliminate 
Uys Pr Py and Ay from equations (U.3) and (4.4). For this purpose, consider a portion of the 
bar in the unstrained state (region (3) of Figure 5) of length Van radius fe and cross-sectional 
AFCA eaesee 
