350 



PRINCIPLES 



or dS = 4x7-2 gju 2ede = Sirr^ sin 6 cos 0d0. 



(17) 



Since the intensity of the reflected sound equals the 

 energy reflected per unit time divided by the area 

 over which it is distributed, then 



J _dJ _ 2tA'^ sin 6 cos Odd _ A^ 

 " ~dS ~ Sirr^ sin 6 cos Odd " ~ 4^^ "' 



(1^ 



Thus Ir is independent of 6 and therefore is inde- 

 pendent of the direction of the reflected sound, and 

 equation (18) is identical with equation (7) derived 

 from a simpler analysis. Rigorously, then 



T = 10 log fc = 10 log (--) = 20 log (-) , (19) 



where T is the target strength and A the radius of 

 the sphere. 



Eriuation (19) applies only to target .strengths 

 measured far away from the sphere. Close to the 

 sphere, the target strength will also depend on both 

 the direction d and the range r. 



19.3 



EFFECT OF PULSE LENGTH 



So far it has been tacitly assumed that continuous 

 sound strikes the target and is reflected back to the 

 projector. Usually, however, sound pulses of finite 

 length are sent out, and most target strengths are 

 measured with such sound pulses. In general, target 

 strength will be a function of pulse length, and the 

 dependence of echo intensity on signal length must 

 be investigated. 



Consider a curved surface, such as a sphere or an 

 ellipsoid, each part of which reflects sound specularly 

 as a mirror would. This surface is normal to the inci- 

 dent beam at only one point, and only one ray is re- 

 flected back to the projector in the direction of the 

 incident ray. Therefore, the echo intensity — and 

 consequently the target strength — will be inde- 

 pendent of signal length, and the echo structure will 

 accurately reproduce the signal structure, unless mul- 

 tiple transmission paths which result from surface- 

 reflected or bottom-reflected sound, for example, 

 give rise to multiple echoes. This result, derived on 

 the basis of ray acoustics, is not valid if very short 

 pulses are used, since the wave character of sound 

 must then be considered. However, this result is 

 correct if the pulse is at least several wavelengths 

 long. 



On the other hand, consider an extended rough 

 surface, each part of which reflects sound in all direc- 

 tions. A pulse T seconds long is sent out from a 



projector a distance r from the target which has an 

 extension z in the direction of the incident beam, 

 as illustrated in Figure 4. Now the first part of the 

 signal will reach the nearest part of the target at a 

 time r/c after it was emitted where c is the velocity 

 of sound and will be returned to the projector at a 

 time 2r/c. The last part of the signal will leave the 

 projector at a time r, reaching the nearest part of 

 the target at r -f- r/c and the farthest part of the 

 target at a time t -|- r/c + z/c; it will return to the 

 projector at a time r -|- 2r/c -f 2z/c. The duration of 

 the echo will be the difference between the time when 

 t he first part of the signal reaches the nearest part of 

 the target and is retiu-ned, and the time when the end 

 of the signal is reflected from the farthest part of the 

 target and is received at the projector. Then if the 

 duration of the echo is o-. 



19.3.1 



= T + 2z/c. 



Long Pulses 



(20) 



First, let the signal be long compared to the ex- 

 tension of the target (Figure 4A). Then 



<7 « r (21) 



and the echo length will approximately equal the 

 signal length. Assume that the reflected energy is 

 always directly proportional to the incident energy, 

 and therefore to the product of the signal intensity 

 and the signal length. Then the echo intensity will 

 depend on the signal intensity but not on the signal 

 length. 



Now let the signal length equal the depth of the 

 target in the direction of the beam (Figure 4B). Then 



2z 

 0- = T -h - = 3r, (22) 



c 



and the echo length will be three times the signal 

 length. The echo will no longer resemble the signal, 

 as the echo intensity grows to a maximum when the 

 target is illuminated by the entire signal. 



19.3.2 



Short Pulses 



Lastly, let the signal be short compared to the 

 depth of the target (Figure 4C). Then 



.«^^ (23) 



c 



and the echo length will approximately equal twice 

 the extension of the target in the direction of the 

 beam. The echo intensity now will depend on the 



