REFLECTED PRESSURE 



355 



rigorously correct only for a plane surface, but results 

 in a good approximation for other surfaces as long as 

 the curvature is small over the distance of one wave- 

 length. 



Consequently, it will be assumed that in general 

 the gradient of p2 and the value of G are related to 

 each other at each point on the target surface by the 

 equation 



G = -— e 



2-K 



dz 



(14) 



If the boundary condition (6) is to be satisfied, 

 — dp^/dz in equation (14) may be replaced by 

 dpi/dz, and in terms of the incident-sound wave 



^ 1 -2«/(^Pl 



(j = — e — 



2ir dz 



(15) 



Since the incident soimd pressure is usually a har- 

 monic wave, it may be locally described by 



where h is the amplitude of the wave, / its frequency 

 and \ its wavelength, and q a coordinate parallel to 

 the direction of propagation. The derivative of pi in 

 the direction of propagation is then 



api 



dq X 



(17) 



The derivative of the amplitude h has been neglected 

 in this equation since this derivative is usually negli- 

 gible at distances from the source of many wave- 

 lengths. 



In any other direction, the derivative will equal 

 expression (17) multiplied by the cosine of the angle 

 between the direction chosen and the direction of 

 propagation q. If the angle between the direction of 

 propagation of the incident sound wave and a line 

 perpendicular to the target surface is d, then the 

 derivative of pi along a line perpendicular to the 

 target surface is 

 djpi 

 dz 



-— 6cos0e=^"^^'-«/^>■ 

 X 



(18) 



If this expression is substituted in equation (15), G 

 becomes 



G = — 6cosee-'"«^'- 



A 



(19) 



It is particularly interesting to evaluate the wave 

 amplitude h for the case where the incident wave is 

 caused by a point source of sound at a point P, a 

 distance r' from the point of the target surface con- 

 sidered. If at unit distance from P the amplitude of 



the incident spherical wave is B, then the local ampli- 

 tude 6 equals B/r'; the coordinate q may be replaced 

 by r', and equation (19) assumes the form 



iB 



G = ; cos 8 e 



X r 



■2Tir'/X 



(20) 



If this expression for G is substituted into equation 

 (9), the resulting integral for the reflected sound pres- 

 sure p2 becomes 



p, = —ti I ^e^"V'"~; dS (21) 



X J s rr 



where B is the amplitude of the original point source 

 at unit distance, r' is the range from the source to a 

 point on the surface of the target, r the range from 

 that point on the target surface to the point in space 

 where -pi is to be found, / the frequency and X the 

 wavelength of the sound. The integration is to be 

 carried out over the whole target surface S of which 

 dS is a surface element. 



20. 2. .3 Physical Interpretation 



So far the discussion has been wholly mathematical, 

 without the benefit of a physical argument to support 

 and justify the approximations made. Physically, the 

 analysis is based on the fundamental principle that 

 in the vicinity of a rigid surface the fluid motion in a 

 direction perpendicular to that surface must vanish. 

 If the incident pressure wave made the fluid move so 

 as to violate this condition, the rigid surface would 

 exert a force on the adjacent fluid elements just can- 

 celing this motion perpendicular to the surface. This 

 effect may be imagined by replacing each element of 

 area on the target surface by a small piston capable 

 of mo^dng in a direction perpendicular to the surface. 

 In the absence of the boundary condition, each of 

 these pistons would be moved back and forth in 

 rhythm with the motion of the adjacent fluid element. 

 In order to act as parts of a rigid surface, however, 

 these little pistons must each be pushed by a force 

 opposite to that of the motion of the fluid, just 

 sufficient to keep each piston permanently balanced 

 in its original position. This alternating force which 

 each piston exerts on the fluid has the same net effect 

 as the force which a transducer exerts on the sur- 

 rounding fluid, in other words, each acts as a sound 

 source with spherical wavelets emanating from each 

 individual piston. The appropriate amplitude and 

 phase of these wavelets has been calculated above. 

 The total reflected sound field then represents the 

 superposed effects of all these individual wavelets. 



