414 
regard to the last term of Eq. 3.40, we remark that an alternative expres- 
sion forh is 
Ge = Lhe ay, 
heat o 
341 
c 
Ir/dt )e 5 
the 
where fo is propagation velocity of Y be: t) . From Eqe 3-22 we see that 
Cov C, in the acoustical approximation and thus, since (> 70.5 »h will 
Z = 
consist of terms of the order of o’* and & and higher. Since § or Co-ul/e 
is of order If r% we may consider the dominant term in h to be of the order 
r 2/ C, .- Using the Riemann equations 3.18 to compute the derivatives of 
re / Co 3 we get 
eee pene | 3 Ot eth), oO 
Thus the remainder term g is of order /y 2 In the pure acoustical approx- 
J 
imation, we note that T-¥¢-)-/C and 2, tends to unity, Therefore 
t 
‘ , 2 
(c+ u)g/G¢t) on SOM (ir es (3.43) 
A more refined analysis than the pure acoustical theory, based upon the ex=- 
pression » i a ; 
dt Mare oe oa (Bed) 
ess bk ine Ne Ay ; 
( ct et) a(t) . 
shows that if € contains terms of order }/)* then 
or —_ - 4og YY as wa b&b. (3.245) 
Ot 
The asymptotic behavior described by Eq,3.45 arises from the fact that there 
is some spreading of the wave profile even in the acoustical limit. 
