Chapter 20 

 THEORY 



IN CHAPTER 19 the concept of target strength was 

 introduced and its meaning defined quantita- 

 tively; then the target strength of a perfectly reflect- 

 ing smooth sphere was derived in terms of ray 

 acoustics. The theoretical background will be pre- 

 sented in this chapter in terms of wave phenomena 

 with a mathematical discussion of the reflection of a 

 sound wave from a target of any shape, and a review 

 of the early theoretical calculations of the target 

 strength of a submarine. 



In principle, the reflection of sound from a target 

 can be exactly determined by solving the wave equa- 

 tion derived in Chapter 2 of Part I, as long as the 

 proper boundary conditions at the surface of the 

 target are satisfied. In practice, an exact computation 

 along these lines is mathematically very difficult ; the 

 difficulties are most marked for targets large com- 

 pared to the wavelength of the incident sound. Even 

 for a sphere the rigorous analysis which has been 

 worked out '■ ^ is rather complicated. Numerical ap- 

 plications of these precise formulas have been pub- 

 lished ^ for a rigid sphere, whose circumference is 

 from 1 to 10 times the wavelength; the results pro- 

 vide an interesting example of the exact behavior of 

 reflected sound in one simple case. However, even 

 for such relatively small targets the mathematical 

 analysis becomes tedious. 



20.1 



APPROXIMATIONS 



To obtain more general results, various approxima- 

 tions must be made, physical as well as mathematical. 

 The mathematical assumptions made in this chapter 

 are fairly standard and are believed to give essentially 

 correct results. The physical assumptions about the 

 nature of the reflecting surface are more important, 

 however, and require some justification. 



In the first place, most of this chapter applies only 

 to targets which are large compared to the wave- 

 length of the sound, and whose surface is smooth; in 

 other words, the radius of curvature of the surface is 



also large compared to the wavelength. These re- 

 strictions seem legit imate for most targets of practical 

 interest in echo ranging. 



In the second place, the material of which the tar- 

 get is composed is assumed to be rigid. In terms of 

 sound reflection, a target is said to be rigid if piCi/pid 

 is negligibly small, where p\ and Ci are the density and 

 sound velocity in the surrounding medium, and pz and 

 C2 are the density and sound velocity in the target. 

 When this condition is not fulfilled the problem be- 

 comes much more complicated. In most cases of in- 

 terest to subsurface warfare, the target is bounded 

 by thin metal plates, inside which there may be air 

 or water. The reflection of sound from such plates has 

 been studied,^' ^ and the results obtained show that 

 even for plates only 3^^ in. thick, such as generally 

 constitute submarine superstructures, the reflection 

 is practically perfect; transmission and absorption 

 are negligible. Thus, the assumption of perfect re- 

 flection from practical targets seems justified. 



Some of the additional assumptions which may be 

 made in the discussion of the reflection from targets 

 are discussed in an early British report." This work is 

 particularly interesting because it presents the most 

 complete available application of theory to the target 

 strength of underwater objects. 



The essential elements of the theory of target 

 strength, restricted by the physical assumptions 

 which have been made here, are presented in the 

 following sections. First, an approximate but general 

 formula is derived for the pressure of the sound re- 

 flected from a target. In Section 20.3, this result is 

 further simplified to give an equation for the target 

 strength of a reflecting surface in terms of the so- 

 called Fresnel zone theory. This latter equation is 

 then used to find practical formulas for the target 

 strengths of simple geometrical shapes, which are 

 applicable to the major reflecting properties of sub- 

 marines; the application to an actual submarine is 

 described in Section 20.5. All this latter analysis ap- 

 plies only to long pulses. The last two sections are 



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