418 
4.1, we must make use of the theory of propagation of the shock and rare- 
faction waves emitted by the moving sphere surface. The advancing shock 
wave in waters we treat according to Sectim 3 as a wave in which rd 
or G is propagated outward with a velocity Cc e The receding rarefaction 
wave before the arrival of reflections from the center of the sphere we 
treat as a recessive wave in which re or Gis propagated from ac ») with 
velocity = run 
To illustrate the procedure, we outline the elimination of du ldr 
from the first of Eq. 4.1. From the theory of propagation, the equation of 
motion, and the definition of G, we have 
ot or 
pday ob Se ee (442) 
d@ = Qdrt rdp/p + rudku, 
Eqs. 4.2 lead to the following relation, 
eh Ce ae a pu = ee a 
uw (E-U) == + ae: (Cc - “eu? ; (4e3) 
Elimination of OuU/dI~ between Eq. 4.3 and the first of Eqs. 4.1 yields at 
the surface PF =Q, 
e Dpl Shale 2 Dar ye @(w-Sur/e)e2ue 
Q e¢2#-uertu* (Lot) 
In an exactly similar manner, we obtain from the propagation equation for the 
gas on the interior of the sphere 
Ma De? Sete a) Da 
ore Dt Cute - Dt 
or 
w* = a dp/p , 
Le 
x = gr 
ral Cw - gurl/e)+2 am 
a ere uc~*+ uX 
(4.5) 
31 
