EFFECT OF A BOUNDARY 



33 



plus negligible terms, which may be rewritten as 

 p = "-{-2.in[2./(.-4^)] . 



by the trigonometric identity for the difference of two 

 cosines. This equation reduces approximately, be- 

 cause of equation (126), to 



--l-WO"]-M'-!)]<-' 



At the point P, equation (128) tells us that the 

 amplitude of the pressure variation with time is ' 



Amplitude = -^ sin 2ir^ (129) 



where X = c/f is the wavelength of the sound. The 

 resulting sound intensity at P, which is proportional 

 to the square of the maximum acoustic pressure, will 

 be very small if the argument of the sine in equation 

 (129) is small. That is, the intensity will be low if 

 hih/X is very small compared with R, or, in other 

 words, if 



h<<'^- (130) 



h 



Assuming that equation (130) holds, the sine in 



equation (129) will be approximately equal to its 



argument, and we have 



, ,. , 4:Trahhi 

 Amphtude = ■ 



In terms of the intensity, this means 

 IGttV 

 X2 



Intensity oc 



■^aWhl / 1 \ 



x^ Vflv' 



(131) 



That is, in the layer of poor sound reception the sound 

 intensity falls off inversely as the fourth power of the 

 horizontal range at great ranges. 



For smaller values of horizontal range R, we find 

 that the amplitude vanishes wherever the argument 

 of the sine in equation (129) is an integral multiple of 

 T, or, in other words, where 



2hhi . . „ , „ „ 

 — =j,j = 0,l,2,3,-- 



while the amplitude will show greatest values in the 

 neighborhood of those points where the argument is 

 jr/2, 3ir/2 ■ • • ; in other words, where 



4hhi 

 R\ 



This sequence of interference minima and maxima is 

 called the image interference pattern. 



The image interference effect described here is only 

 occasionally observed in the sea for reasons which are 

 discussed in Section 5.2.1. 



Point Source Far from Sea Surface 



We assume now that the source is located so far 

 from the surface that the sound waves near the sur- 

 face can be regarded as plane waves. Only incident 

 waves propagated purely in the y direction are con- 

 sidered, that is, normal to the surface. Then equa- 

 tion (125) has to be replaced by 



p = afcos 2Tf(t - -) - cos 2ir/U -|- ^jj- (132) 



By applying to equation (132) the trigonometric 

 formula for the difference of two cosines, we obtain 



p = —2a sin 2ir- sin 2Trft. 

 A 



(133) 



— — = K, K = 1, 3, 5, • 



We notice a very curious thing about the disturb- 

 ance described by equation (133). The acoustic pres- 

 sure is zero over the entire fluid when ft is any in- 

 tegral multiple of }4- Further, the acoustic pressure 

 is zero for all time at points where y/\ is an integral 

 multiple of H- Thus we see that the interference be- 

 tween two plane waves of equal amplitude and of the 

 same frequency traveling in opposite directions pro- 

 duces, at least in this case, a disturbance of the 

 medium for which at any instant all points have 

 identical, or opposite phase. We no longer have pro- 

 gressive waves, but a phenomenon which we call 

 stationary or standing waves. The points where the 

 amphtude is zero for all time are called nodes; the 

 points where the amphtude term of equation (133) 

 is a maximum are called loops or antinodes. 



This state of affairs is permanent as long as the 

 source keeps vibrating. The nodes are permanent re- 

 gions of silence; and the loops are permanent regions 

 of maximum pressure amplitude. Such a state, in 

 which all points of the medium perform vibrations 

 of the form sin 2irft with an amphtude dependent on 

 position is called a stationary state of the medium. 



Reflection from Sea Bottom 



If water is separated by the plane 2/ = from a 

 medium with a density much greater than its own, 

 the boundary condition which must be fulfilled at 

 this plane is 



dp 



dy 



= 0- 



