Chapter 3 

 RAY ACOUSTICS 



CHAPTER 2 was devoted to the rigorous computa- 

 tion of the acoustic pressure p as a function of 

 position in the fluid and of time. In situations where 

 the acoustic pressure could be determined the sound 

 intensity at an arbitrary spot and at an arbitrary 

 time could be calculated. However, it was noted that, 

 in most situations involving initial and boundary 

 conditions similar to those met with in actual sound 

 transmission in the ocean, this computation was at 

 best very laborious and at worst completely impos- 

 sible to carry out. Ray acoustics provides a more con- 

 venient though less rigorous approach. 



In the study of sound the ray concept has not 

 played so great a role as in optics. The reason for this 

 is that the wavelengths of most audible sounds are 

 not small compared to the obstacles in the path of the 

 sound. Consequently, sounds audible to the ear do 

 not travel straight-line or nearly straight-line paths; 

 they bend around comers and fill almost all of any 

 space into which they are directed. However, for the 

 short wavelengths used in supersonic sound, the ray 

 methods have an important, if Umited, application. 

 This chapter elaborates the theory of sound rays, 

 describes the computation of soimd intensity from the 

 ray pattern, and finally, examines the conditions 

 under which ray intensities may be expected to ap- 

 proximate the intensities calculated according to the 

 rigorous methods of the second chapter. 



3.1 WAVE FRONTS AND RAYS 



3.1.1 Spherical Waves 



The wave equation was solved exphcitly for p in 

 one very important case: an impulse sent out by a 

 point source into a homogeneous medium under the 

 assumption of spherical symmetry. The pressure as a 

 fimction of time and space was found to be 



P 



=;K'-:)- 



In this expression, r is the distance from the source. 



and the function f{t — r/c) is determined by the out- 

 put of the source. Specifically, the source can be 

 characterized by the statement that, while the pres- 

 sure itself at the source is infinite, the product rp in 

 the immediate vicinity of the source has a finite value 

 at every instant, namely /(<)• 



Obviously, this fimction f{t — r/c) determines 

 when a pulse emitted by the source at a particular 

 instant will arrive at a given point of space. If the 

 pulse should, for instance, have started in time at 

 some instant < = « so that for all values of the argu- 

 ment less than « the function f{t) vanishes, then the 

 onset of the disturbance at a distance r from the 

 source will be observed at the time 



€ + 



Likewise, we find that the front of the pulse will have 

 reached at the time t a distance r, given by 



r = c(t- e). (2) 



What has just been stated about the front of the 

 pulse might just as well have been said about any 

 other specified part of the pulse; only, « would in that 

 event characterize the time at which the specified 

 part of the pulse was radiated by the source. What 

 has been called, vaguely, part of the pulse, is often 

 more concisely called phase, particularly in connec- 

 tion with harmonic pulses. The term e then char- 

 acterizes the phase of the pulse considered and is 

 ordinarily referred to as a phase constant. 



The surface defined by equation (2) is, of course, a 

 sphere of radius c{t — e) at the time t. As the time 

 increases, the radius of the sphere increases at the 

 rate of c imits per second. The surface of this sphere 

 of constant phase is called the wave front. The energy 

 represented by the disturbance of equilibrium condi- 

 tions clearly spreads out radially from the source. We 

 may focus attention on the direction of energy flow 

 by mentally drawing an infinite number of radii from 

 the source to the wave front. These radii may be re- 

 garded as representing the paths of energy flow and 



41 



