REFLECTED PRESSURE 



353 



devoted to a qualitative discussion of the rcHectioii 

 from targets small compared to the wavelength, and 

 the echoes obtained with very short pulses. 



20.2 



REFLECTED PRESSURE 



Consider first a sound beam striking a surface ele- 

 ment dS of a perfectly reflecting, smooth and rigid 

 underwater target. Since this surface is rigid, the 

 primary effect of the target is to prevent the water 

 from moving perpendicularly to dS at the surface of 

 the target. In other words, at the surface of the tar- 

 get, the velocity u of the water, measured along a Una 

 perpendicular to the surface, must be zero, or 



Mz = 0, (1) 



where the z axis, at the point of incidence, is perpen- 

 dicular to the target surface. By differentiating equa- 

 tion (1) with respect to the time t, 



dt 



(2) 



But from equation (17) in Chapter 2 of this volume, 

 du^ dp 



-"1^ = 



dz 



(3) 



where p is the density of the medium, p the pressure 

 of the soimd wave, and z the coordinate perpendicular 

 to the surface. 



20.2.1 Boundary Condition 



Substitution of equation (3) into equation (2) gives 



dz ' 



which means that for a rigid target the component of 

 the pressure gradient perpendicular to the surface 

 must vanish at the surface. This is the boundary con- 

 dition which the solution of the wave equation must 

 satisfy. 



In the absence of the target, the sound source will 

 send out a wave whose resulting pressure at any par- 

 ticular point may be denoted by pi; then pi must be a 

 solution of the wave equation [equation (27) in Chap- 

 ter 2.] In the presence of the target, this pressure pi 

 does not satisfy the resulting boundary conditions at 

 the surface of the target. The actual sound pressure p, 

 which must satisfy both the wave equation and the 

 boundary conditions at the target surface, may be 

 written as 



where P2 constitutes the correction which must be 

 added to the undisturbed sound pressure pi in order 

 to satisfy the boundarj' conditions at the surface 

 of the target. 



By differentiating equation (5) with respect to z 

 and by substituting the result into equation (4) 



dpi I ^P2 _ Q 

 dz dz 



(0) 



which is another way of expressing the boundary 

 condition. 



Because the wave equation is a linear homogeneous 

 differential equation, the difference between the two 

 solutions p and pi is again a solution, and pz by itself 

 must therefore satisfy the wave equation. In other 

 words, the total sound field may be interpreted as the 

 combination of two sound fields. One of these, whose 

 pressure at any specified point is pi, is called the 

 incident sound; the other, whose pressure at the same 

 point is p2, is the reflected sound. Each of these 

 quantities satisfies the wave equation, but only their 

 sum satisfies the boundary conditions at the target. 

 In some places, the measured sound pressure may oc- 

 casionally consist wholly of one or the other of these 

 two soimd fields, depending on whether only the 

 sound projected from the source, or only the sound 

 reflected from the target is measured. The problem 

 tackled in this chapter is the evaluation of the re- 

 flected sound alone, since it is this quantity which is 

 most important in echo ranging. Therefore, an ex- 

 pression for p2 must be derived. 



(4) 20.2.2 Mathematical Formulation 



To obtain, rigorously, a general expression for p^ is 

 usually a very difficult problem. It is comparatively 

 easier to obtain an approximate solution by distrib- 

 uting, over the surface of the target, point sources of 

 sound. Then, if the distribution and strengths of 

 these point sources over the area are correctly chosen, 

 these point sources will emit sound in such a way as 

 to cancel the pressure gradient of the wave incident 

 on the surface, thus satisfying the condition (6). 



For a single point source, the solution of the wave 

 equation is 



~ _ :?.2.rt(/(-r/X) 



p — e 

 r 



(7) 



P = Pi + P2, 



(5) 



where p is the pressure of the sound field at a distance 

 r from the source, 5 is a constant which measures the 

 strength of the point source, iis-\/—l, t is the time. 



