407 
with the solution, 
Y = pl(t-rfcs), 
where cy is the asymptotic sound velocity for waves of vanishing amplitude 
in the undisturbed fluid, the solution (t-1/¢,) being appropriate for 
an outgoing progressive wave. For Q andu » we obtain 
Q = Gle-r/ed/r , &ltd= gilt), 
u = plt-r/e)/r- ENO Cas. mae 
In this case, we note that rf is propagated outward with the finite and 
constant velocity C5 e The acoustical approximations to the Riemann func- 
tions r and 5 are of interest. For an outgoing spherical wave, they are 
pees ical) Phe iste 
Cai art (3.24) 
cece pit-1/e,) 
raat ele 
We note also that in the incompressible approximation, Lim (c> °0)@” =0, 
ete ey (3.25) 
ear 
we have 
Bt 
Thus the funce ton) h, defined as 
hs 54+ oler*, Best) 
vanishes both in the acoustical and in the incompressible domains of flow. 
It is also vanishingly small in the region behim the shock front in the 
initial stage of its formation, since § is always small just behind the 
shock front and 4 is equal to r [Powe At, 
Since in the i tatraseinietane acoustical limits r{2 or -G 
is propagated outward in the one case with infinite velocity and in the 
