BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 



221 



these results cannot be compared with data for 

 sinusoidal sound, since tiie character of the bottom 

 in the locality of the experiments is at present un- 

 known, and since measurements with sinusoidal 

 sound at low frequencies are not very complete. 



Information can also be obtained from explosive 

 sound regarding the geological strata beneath the 

 bottom. Figure 23 shows a typical ray diagram for 



Figure 23. Ray paths in a stratified bottom. 



sound originating in the water over a stratified bot- 

 tom, in which each successive layer has a higher 

 sound velocity than the one above it. Without 

 bothering about the detailed form of the pressure 

 pulse received at a distant hydrophone, a subject 

 which will be discussed fully in the two following sub- 

 sections, we may study the way in which the time of 

 arrival of the first measurable disturbance varies with 

 the range from the explosive to the hydrophone. If 

 this range, EH in Figure 23, is sufficiently short, the 

 first disturbance will arrive by a pat^ which lies en- 

 tirely in the water. But if EH is greater than a certain 

 value To, whichdepends upon the depths of source and 

 hydrophone and upon the velocity of sound in the 

 top layer of the bottom, sound traveling along the 

 ray EABH will arrive before the direct sound wave 

 through the water; in such a case the fact that the 

 sound velocity over the path A Bis greater than that 

 in the water more than compensates for the fact that 

 EABH is longer than the direct path EH. To find 

 out when this occurs, let c be the velocity of sound in 

 the water (assumed uniform for simplicity), Ci the 

 velocity in the top layer of the bottom, r the hori- 

 zontal range, and z the height of explosive and hydro- 

 phone above the bottom, assuming for simplicity 

 that both are at the same level. We shall first show 

 that the positions of A and B, which minimize the 

 time of travel, are those for which the angles EAB 

 and ABH obey the refraction law of ray theory, and 



shall then derive an expression for the value of r at 

 which the time of travel via EABH becomes .shorter 

 than via the direct route EH. 



The time required for a pulse of .sound to travel 

 a path such as EABH in Figure 23 is 



3(csc de + CSC dh) r - z(cot de + cot 9/,) ^^^ 

 t - 1 (5) 



C Ci 



If this time has a minimum as the position of point A 



is varied, this minimum must occur when dt dde = 0, 



that is, when 



z z 



— CSC Be cot ^e + - csc^ 6^ = 0, 



C Ci 



which is equivalent to 



cos de = —• 



Ci 



This is the well-known expression for the angle at 

 which the transition from refraction to total reflec- 

 tion occurs. Similarly, the requirement that t be a 

 minimum with respect to displacements of point B 

 gives 



cos dk = COS de = 



Ci 



(6) 



Eliminating the angles from equation (5) by use of 

 this relation, we have for the time of arrival by the 

 shortest path through the bottom 



2z , r 2z c/ci 



r 2z 



t = . -I ._ 



cV 1 - e-l^^ C'i ci Vl 



c'/cl 



_ r 2zVcl 



Ci 



CCi 



(7) 



This equals the arrival time r/c of the direct pulse 

 through the water when 



-4. 



Ci + c 

 Ci — c 



(8) 



Now if the time interval between the explosion and 

 the first signal at the hydrophone is plotted against 

 the range r, the graph will start out as a straight line 

 passing through the origin and of slope 1/c; and at 

 the range given by equation (8) the slope will change 

 abruptly to 1/ci. Thus all the quantities c, ci, and h 

 could be determined from this plot. If the plot is 

 continued to larger values of r, another abrupt 

 change of slope may occur when the travel time via 

 a path EMNQRH lying partly in a denser stratum 

 (medium 2) becomes shorter than via EABH. If the 

 bottom contains still deeper strata with higher sound 

 velocities, further changes of slope will occur. By 

 methods similar to those outlined above, the depths 



