1509 
gives € = 6.71°)is shown in Figure 5. In this figure the origin 
of time is the instant of arrival of a wave which has traveled 
along the path ABC. The arrow indicates the expected arrival time 
of sound which has traveled along path AB'B"C. Since this is the 
earliest instant at which sound should begin to arrive at the 
receiver, we can see from the plot that the plane wave approximation 
is in considerable error. It is also qualitatively evident that 
the major errors occur in the treatment of the lower frequency 
components. 
If the range is increased from 150 ft. Govcoonm iter, 
the difference of time between the two paths increases to about 
0.4 milliseconds. This is due principally to the fact that 9-8, 
increases to .136 radians, that is by a factor of 5.1 over the 
previous value, and Mt varies as ( O-O.)®. This value of Jt 
is nearly three times as large as the time constant of the weve, 
so the plane wave approximation should be fairly good for ranges 
exceeding 200 ft. under the assumed conditions. 
5.4 It is possible to give a more exact treatment of the reflection 
of a spherical wave from a surface of discontinuity® »8. Here 
we shall merely quote some of the pertinent results from ref. 
(6) p. 49. Pekeris gives the following asymptotic expression for 
the "reflection coefficient" at a given angular frequency © : 
cos O-bS _ <C bv? 1? 8 C0278 + 25 (/+bS coe 8) 4 8LS sin 
oe = A / 5 coz O+ nes SiO coa® 
{cos 6. bg wRd*(coso + bd) [sim (7 a) 4 a co 8{ 
+O (G7/u? R*) 4 
(63) 
= 35) = 
