406 
Introduction of the new variables, 
in-= (ee 4/2), 
G =(7-wu)/2 ; (3.17) 
yields the Riemann equations 
or 4 (e#u) Ofome ott ) 
dt Pr 
Js : ( me u) os 5 ul : (3.18) 
ot dor . 
The kinetic enthalpy OQ is related to the Riemann functions and $ in the 
following manner 
dQ = (ct+uddr+ (C-WAS , (3.19) 
a relation we shall presently use. 
It is instructive to review the integration of Eqs. 3.13 in two 
limiting cases, the incompressible approximation and the acoustical approx- 
imation, The incompressible approximation is obtained in the limit C’—> 00, 
and Eq. 3.13 becomes 
2 
or 
which with the boundary condition (, ~~ O as M— 00, leads to the result 
y =plt) , and , 
WY 2 Glt)/r , Git) = lt), 
2 
us: op ( t/a 
In this case we note that f-(2. is propagated outward with infinite velocity. 
(3.21) 
The acoustical approximation is obtained by suppressing nonlinear 
terms in Eq. 3.13; 
2 % 
oF Weg 2 ¢ eo ee 
BS r 2 eS at? (3.22) 
