346 



PRINCIPLES 



Figure 1. Reflected rays at short and long ranges. 



Therefore, the way in which the sound intensity near 

 the target varies from point to point is complicated. 

 Rays reaching a point far away from the target all 

 come from essentially the same direction, no matter 

 from what part of the target they are reflected. Thus 

 the target "looks like" a point source, and the in- 

 verse square law of intensity will hold. This conclu- 

 sion, based solely on ray acoustics, is reinforced by 

 considerations of wave acoustics, mentioned in Sec- 

 tion 19.4 and described in more detail in Chapter 20. 

 Sufficiently far away from the target, then, Ir will 

 be not only directly proportional to /o but also in- 

 versely proportional to the square of the distance r, or 



/. = A- 



(1) 



Here k is a constant which in general depends on the 

 size, shape, and orientation of the target. It does not 

 depend on the strength of the somid striking the 

 target, or on the distance from the target, provided 



Ir is measured far enough away from the target to 

 make certain that the intensity of the reflected sound 

 will follow the inverse square law. Incidentally, this 

 relation is not valid for explosive sotmd, which is 

 treated in Chapters 8 and 9. 



Now according to equation (89) in Chapter 2, the 

 intensity la of the incident sound striking the target 

 is equal to the intensity F of the projected sound 1 yd 

 away from the source, divided by the square of the 

 distance r from the source to the target, provided 

 that r is much larger than the dimensions of the 

 source. Then 



F 



^0 = -,• (2) 



Substitute equation (2) into equation (1), and 



F 



Ir = k-- 



(3) 



Equation (3) is particularly interesting because it 

 shows that, for an ideal medium, the intensity of an 

 echo is inversely proportional to the fourth power of 

 the range, as long as the echo is measured at the 

 source and the range is much larger than the dimen- 

 sions of the target or source. If logarithms are taken 

 and equation (3) expressed in decibels, 



10 log /. = 10 log k + lOlogF - 40 log r. (4) 



19.1.2 General Transmission Loss 



All these equations are derived on the assumption 

 that the medium through which the sound travels is 

 ideal, that all the sound is transmitted freely without 

 refraction, absorption, or scattering, and that the 

 boundaries of the medium are so far away that their 

 effects on the propagation of sound waves may be 

 neglected. In other words, as the sound travels each 

 way, its intensity falls off according to the inverse 

 square law alone. The drop in intensity each way, in 

 decibels, is the transmission loss H, which for this 

 ideal case is simply 20 log r. The total transmission 

 loss 2H to the target and back again is then 40 log r. 



Generally, however, the intensity of transmitted 

 sound under water does not fall off according to the 

 inverse square law alone. Sound is absorbed and 

 scattered in sea water. It may be bent by tempera- 

 ture gradients and consequently focused or spread 

 out. Often the surface and bottom of the ocean sig- 

 nificantly affect both transmitted and reflected sound. 

 Therefore, H will seldom exactly equal 20 log r, and 



