PRESSURE 
e' 
pl teats 
DISTANCE 
v a! 
o! 
Fige3 - Curves showing the variation of pressure with distance 
at_two fixed times 
Pairs of ordinates such as PQ and P'Q' in fig. 3 which satisfy equation 6 
will then be a constant distance c(t, - t,) apart corresponding to a constant 
velocity of propagation c. The reduction ratio ri/ Ye », however, is not a 
constant for the two curves but will vary for each pair of ordinates. 
Unlike the pressure/time variation, the curves for pressure distribution 
with distance cannot be made strictly of the same shape by changing the 
pressure scale. However, if the effective length DF (= D'F') is small 
compared with r, the ratio r,/r Will be approximately constant for the two 
curves DEF and D'E'F' which can then be made approximately the same shape as 
the curves in fig. 2 for pressure/time variation by a suitable choice of 
scales. Thus, choosing the origin of time so that the pressure pulse starts 
from its centre at time t = 0, the point A in fig. 2 corresponds to time 
r,/c and the shape of the curve ABC is given by 
- 3 =F ¢ (ag) Rey! is, 08 Ss Rn eee tot er 
Similarly, the point D in fig. 3 corresponds to the extreme distance ct 
reached by the pulse in time t and the shape DEF is given by 
Boe eee eee See) ese cheek eee ce OG) 
If DF is now small compared with OD the factor P/r in equation 8 is sensibly 
constant over the curve DEF and equations 7 and 8 indicate that by a suitable 
choice of scales, the curves ABC and D&F would be approximately the same, 
although reversed with respect to the origins of time and distance. 
46. This similarity of the pressure/time and pressure/distance curves 
becomes increasingly more accurate.as the pulse travels outwards to great 
distances. The essential assumption is that the factor 1/r in equation 1 
may be treated as constant when phenomena over distances small compared with 
r are concerned. This assumption corresponds to neglecting curvature of 
the spherical wave front and treating the pulse as a plane wave. This 
approximation will be frequently used in the succeeding analysis for the 
effects due to the pressure pulse. 
