274 
SS 
SECTION 1. THE HYDRODYNAMICAL EQUATIONS, 
Consider an invicid fluid in which pressure changes are occurring so fast that thermal 
conductivity may be nejlected, and such that cvery element of the fluid conforms to the same 
adiabatic equation, Thus, we are assuming that the fluid could be reduced by sultable local 
adlabatic pressure changes In different regions into a condition In which the pressure, density 
and temperature are all uniform. When thls Is so, the pressure may be taken to depend on 
density only. 
For a state of spnerical symmetry, the equation of continulty and the equation of motion 
are 
SP ge P=! Sip[ Quy |B 
ages Mate of Aa » | (4) 
ou Qu - _1 #3~ 
ot ae r p door ta 
Following the well—known solution of Riemann for the corresponding cne-dimensiona: case, 
we introduce two new functions P and Q, both of which Involve an arbitrary function f(0), whose 
precise definition we choose later, to suit our convenience, 
P= ti)+u, 0 = fhe)-u 
Multiply (1) by f'(o), and add to (2). 
ap OP - _4 OP OD_ ppg) PY ~ prr(gy 2 
are eye p dr Or CAM pti Ai 6) 
write 
[e(o) ]2 = -§ 
Deca 
thereby determining f(0), apart from an arbitrary constant. This canstant may be conveniently 
chosen by making the value of f() zero for some standard state, denoted by suffix zero, 
Vv 
te) = | + AB w = = dv (s) 
py de dv 
Po Yo 
Using the definition of f*(o) in (3), we obtain 
oP + Oke = -of' oP - 2y0 re 
aa Ce pt' (p) We : (2) (5) 
Now P is a function of two independent variables only, rf and t. ence 
op OP 
— dr+ —— dt 
3 
dP = 
or t 
Making use of (5), this becomes 
ap = 2° great (ud? + prey 22 + 2 try) 
or a r 
7 
= OPT ar -{ prio) + u} dt] — BL f#() at 
r 
NOW weeee 
See Lamb's Hydrodynamics, 
