PROPAGATION OF SEISMIC WAVES 159 
WAVES REFLECTED OR REFRACTED AT DISCONTINUITIES 
By use of the formulas developed in the preceding section it is 
possible to calculate time-distance curves for waves reflected or re- 
fracted at the surface of a layer of rocks covered by material in which 
the velocity increases with depth in the manner discussed above. For 
a given depth the velocity v is given by Eq. (4). From Eqs. (7) and 
(8) it is possible to calculate time-distance curves for waves reflected 
at this depth by assuming a series of values for 0. If the reflecting 
surface is horizontal the time and the distance will be just double 
that calculated from these equations. If the reflecting surface is slop- 
ing it is necessary to make separate calculations for the incident and 
the reflected waves. 
In the case of waves which are refracted along the surface of a 
buried rock the angle of emergence is known from the velocity of 
the refracting layer. If, then, a depth is assumed the horizontal 
distance and the time of the wave in the upper layer can be computed 
from the value of the emergence angle and the velocity in the upper 
layer corresponding to the depth assumed, as in the case of reflected 
waves. The remainder of the distance traversed is in the higher speed 
material and the total time is the sum of the time spent there and the 
time of the incident and emergent rays in the upper layer.® For waves 
refracted along a sloping layer, the computations become increasingly 
more difficult as the emergence angle depends not only on the velocity 
of the high speed layer but also upon its depth. 
Although the primary purpose of the present work was to study 
waves propagated in the upper layer, the data represented in Figure 1 
afford a good illustration of refracted waves. The straight lines drawn 
in Profiles 3 and 4 represent an assumed horizontal layer which is 
100 ft. beneath the surface and has a velocity of 17,000 ft./sec. Be- 
cause of irregularities in the surface of the rock the observed points 
deviate considerably from the lines, but the depth assumed is certainly 
accurate to 5 or 10 percent. Profiles 1 and 2 indicate a limestone sur- 
face sloping up sharply to the east. By the aid of Figure 6 the depth 
at the east end may be estimated at approximately 20 ft. In the region 
between the 500 and 600 ft. stations there are systematic deviations 
marked by dotted lines on the time-distance curve. These indicate a 
fairly sharp depression in the surface of the limestone between these 
stations. The deviations of the points in general are an indication of 
the roughness of the limestone surface. An approximate cross section 
of this region is shown in Figure 1. 
5 Rutherford, Amer. Geophys. Union. Trans. (1933), P- 292- 
819 
