158 MAURICE EWING AND A. P. CRARY 
where » is the velocity at the depth y, is obtained. Eq. (2) yields 
dy/dx = + (6? csc? 5 — v?)1/2/y, (6) 
which may be combined with (5) to give the relation: 
x 
+ (1/27) f ole — b?)/(b? csc? 09 — v?) |*/2dy 
b 
(b? cot? 0o/4ra) [a — sin a cos al, (7) 
where sin a= + tan 4(v?/b?—1)/?. The positive sign should be taken 
if the wave has not reached its maximum depth. 

0.012, Observed © 



Observed ° 
- Computed — 
3 0.008 S ‘ 
e “ 
<4 ~ 
: 2 
iE 0.004 - 
10 ft 20 40 60 ft. 
Vertical Distance Horizontal Distance 
Fic. 6.—Travel-time curve for waves travelling vertically. 
Fic. 7.—Travel-time curve for waves travelling from surface to depth of 18 feet 6 inches 
The travel-time = fds/v becomes, upon substitution from (5) and 
(6) 
t= + (b/27a sin wo) f (1/v) X [(v? — b2)/(b? csc? 0p — v?) |!/2dv 
b 
= (b/2ma sin 09) X [a — sin 0 tan~! (tan a/sin 0) |. (8) 
For the limiting case, 9>=0, the path is vertical and (7) and (8) 
become 
x= 0 (7’) 
(b/2ma) [(v2/b? — 1)1/2 — tan! (v2/b? — 1)1/2], (8’) 
Formulas (7) and (8) make it possible to calculate the time required 
to traverse any part of any path. Two sets of data were taken to 
which these formulae may be applied. In the first case records were 
taken with the geophone buried at a series of depths up to about 20 
ft., the shot in each case being at the surface directly above. In the 
second case the geophone remained at a depth of 18.5 ft. while shots 
were fired on the surface at various distances from the mouth of the 
hole. Figs. 6 and 7 compare the travel-time obtained from these 
observations with those computed from the above equations using 
the values of a and 6 as computed from the time-distance curves. 
818 
