1532 
with @. to be expressed as a function of x by means of 
(93) and (97). 
We can now use equation (108) to determine the 
"equivalent viscous distance," AM, , by numerical integration 
for any qadué of. ~Asbindieatea previously, *- represents 
the propagation distance which would have led to the rise time 
existing at this value of R if only viscous effects had been 
present. Having , as a function of R, it is then possible 
to compute the effective a at any given R and from it in turn 
the rise time 7%, and the form of the wave from equation (91) 
or Figure 7. 
9.5 In order to effect the numerical integration of (108), 
it is necessary to select an appropriate starting point, i.e. 
fix the value of X, at some initial value of R. To do this, 
we first examine the general behavior of equation (108) in some 
detail. We write the eauation in more concise forn: 
dew y- 49 
AR 
ea (109) 
and examine its properties in the x-R plane, a schematic 
sketch of which is shown in Figure 9. 
The family of curves (shown as solid lines in Fig. 9) 
defined by: 
fo a 
ee 
Ae (110) 
- 56 = 
