BOTTOM REFLECTION. SHALLOW-WATER TRANSMISSION 



225 



hydrophone can be interpreted in terms of a simple 

 dispersion law, i.e., a propagation of different fre- 

 quencies with different velocities. 



The physical reasons underlying the dispersion 

 phenomena just mentioned can be seen by consider- 

 ing the simple case of a progressive wave of a single 

 frequency /. Let us try to construct such a wave by 

 assuming the pressure disturbance to be 



p = e^"<'/'-^"M(z), (12) 



where a: is a horizontal coordinate, and z is the depth 

 below the free surface of the ocean. This function jo 

 must satisfy the wave equation 



(322 + f?) = ^P (13) 



in the water; that is, when z is less than the depth h 

 and must satisfy the analogous wave equation"* 



Jb2v , ^\ _ 



(14) 



in the bottom, that is, when z is greater than h,. In 

 addition, p must satisfy boimdary conditions at the 

 free surface and at the interface between water and 

 bottom. These conditions are 



p = at z = 



Pwater ~ Pbottom ^t Z = h. 



(15) 

 (16) 



0?) =(-f) ^^^=^- (1^) 



\p OZ/,vater \Pl oZ/ bottom 



Assuming for simplicity that water and bottom are 

 uniform, so that c and Ci are independent of z, we 

 have, on inserting expression (12) into equation (13), 



— = 44--yMforz<;.. (18) 



To satisfy this equation and the condition (15), we 

 must take 



M = A sin 



(-^f4^) 



for z <h. (19) 



This equation is formally correct regardless of whether 

 the quantity under the radical is positive or negative. 

 However, it will be shown below that this quantity 

 must be positive if equation (12) is to represent a 

 physically possible disturbance. Similarly, to satisfy 

 the wave equation in the bottom we must have 



■■ The theory presented here ignores shearing stresses in the 

 bottom and thus treats the bottom as a fluid rather than as a 

 solid. This assumption, although reasonable for MUD bot- 

 toms, is of course not true for ROCK. However, many features 

 of the disturbance predicted by the present theory would 

 doubtless also be observed over a rock bottom. 



f = -[i-fl« '»->'• 



(20) 



Now, if [(1/X=) - (f /cf)] is negative, M will be a 

 periodic function of z in the bottom, and according 

 to equation (12) the pressure disturbance in the bot- 

 tom will consist of progressive waves going diagonally 

 up or down. The disturbance created by an explosion 

 will consist in part of a superpo.sition of progressive 

 waves of this type which travel diagonally downward 

 in the bottom; these waves are, however, a relatively 

 unimportant part of the signal received in the water 

 at a great distance, since their energy spreads out in a 

 downward direction and thus decreases fairly rapidly 

 with distance in the horizontal plane. The part of the 

 signal which is most important in the present applica- 

 tion consists, instead, of a superposition of waves of 

 the form (12), for values of X and / which make 

 [(l/X^) - (f /c?)] in equation (20) positive. The two 

 independent solutions of equation (20) for this case 

 will be exponential functions of z, one increasing to 

 infinity as z increases, the other decreasing to zero. 

 The former of these is physically inadmissible; so we 

 may conclude that if a pressure wave of the desired 

 form exists at all, it must be of the form 



M = 5 exp f -2ir]/^^ - ■? 3) for z > h, (21) 



and of course of the form (19) for z < h. However, it 

 is easily shown that no matter what values are given 

 to the constants A and B, it is not possible to satisfy 

 both of the boundary conditions (16) and (17) unless 

 X and / are related in a particular way. For, on in- 

 serting expressions (19) and (21) into these condi- 

 tions, and using the abbreviations 



we obtain 



A sin ixh = Be~ 



f^A 



cos fih = 



vB 



pi 



(22) 

 (23) 



Dividing the first of these equations by the second 

 ehminates A and B, and gives the following relation 

 which must be satisfied by / and X. 



- tan fih = 



(24) 



If this relation is satisfied, a suitable choice of the 

 ratio B/A will insure that both equations (22) and 

 (23) are satisfied. It is easily verified that if Ci > c 



