32 



WAVE ACOUSTICS 



If it were not for the boundary condition p = at 

 y = 0, the problem would be solved by means of the 

 expression 



p(r,t) 



r \ c/ 



(121) 



We shall have to modify this solution in order to 

 satisfy the boundary condition as well. To this end, 

 we resort to a trick. We solve a fictitious problem, 

 one in which a source exactly like the first one is lo- 

 cated at a distance h on the other side of the inter- 

 face with the water extending through space and 

 with the initial conditions 



r'p'(r',t) = -F{t) (122) 



at points very close to the new source. This problem 

 has the solution 



p'{r',t) 



M'-9 



(123) 



where 



Vx^ -H z^ + (2/ - hy. 



Clearly, since r = r' at ?/ = 0, we have p + p' = 

 at a; = 0. Thus, the wave given by the sum of the 

 two disturbances described by equations (121) and 

 (123), or 



Fit - {r/c)l F\t - {r'/c)] 



V = 



(124) 



satisfies the imposed boundary conditions. Also, 

 equation (124) satisfies the initial conditions (120) 

 because the expression (124) can be rewritten as 



'^ = 4 - D - ^A - 9 



which reduces to equation (120) in the vicinity of the 

 actual source, where r » 0. Finally, equation (124) 

 satisfies the wave equation itself since the difference 

 of two solutions of that equation is itself a solution. 

 If the source S executes a harmonic vibration, the 

 solution (124) becomes 



/cos 27r/[i - {r/c)] cos 2Trf[t - {r'/c)S\ 



V = «| -. }■ 



(125) 



Formula (125) fully describes the effect of surface re- 

 flection on harmonic waves emitted by a single source 

 under the assumptions that air has negligible density 

 and elasticity and that the sea surface is a perfect 

 plane. 



From the method of construction of the solution 

 (124), it is possible to deduce that there should be a 



zone of low intensity near the surface. The reason is 

 that, at points near the surface, r and r' will be nearly 

 identical; and the two resulting fictional pressures 

 will almost balance each other. This type of destruc- 

 tive interference near the surface is called the Lloyd 

 mirror effect or image interference effect. The next few 

 paragraphs will discuss the width of this low intensity 

 zone and the intensity within this zone. 



Consider the intensity measured by a receiver at 

 the depth hi, located at a horizontal range R from 

 the source, as in Figure 12. We also assume that R 



r^-^piR.-h.o) 



(0,-h,0) 



FiGUBE 12. Fictitious scheme for solving wave equa- 

 tion and surface boundary conditions. 



is SO large compared with h and hi (the depths of 

 source and receiver) that second-order products of 

 h/R and hi/R may be neglected. 



By applying the Pjiihagorean theorem to Figure 

 12, then 



r = «|/i + *^ /.«|/i + * + ">■ 



R' 



Since k/R and hi/R are small, these equations may 

 be rewritten as 



and as a result 



_ i/ hi - fe V 



2\ R / 



(126) 



R 



1 2\ R J 



(127) 



r' R 



because 1/(1 — «) = l-|-tifeis small. Putting 

 these in equation (125), we obtain 



p = |[cos 2.f{t - ^) - cos 2.f{t - ^')] 



