42 



RAY ACOUSTICS 



may be called sound rays, in analogy with light rays. 

 Sound energy may be regarded as traveling out along 

 these rays with the speed c. The wave fronts assume 

 in this description the secondary role of surfaces 

 everywhere normal to the rays. 



An individual sound ray cannot exist as a physical 

 phenomenon. An isolated sound ray would mean a 

 state of the fluid where the condensation was confined 

 to the immediate neighborhood of a particular 

 straight line. Beams of narrow cross section can be 

 produced by directing a wave front onto a very nar- 

 row slit ; but if the size of the slit becomes comparable 

 to the wavelength of the sound, the sound leaving the 

 slit is not a narrow beam, but a cone. This phenome- 

 non is called diffraction, and will be discussed in 

 Section 3.7. It is mentioned here only to indicate 

 that the concept of a sound ray refers not to the 

 propagation of a narrow beam with sharp edges, but 

 merely to the direction of propagation of actual wave 

 fronts. 



Spherical wave fronts represent only one particular 

 case of sound propagation, even in an infinite fluid. 

 First, the wave front can be spherical only if the 

 initial disturbance has no preferential direction (a 

 vibrating bubble satisfies this condition). Second, the 

 expanding surface remains a sphere concentric with 

 the origin only if the soimd velocity c is either con- 

 stant throughout the fluid, or has spherical symmetry 

 about the sound source. 



The wave factor f(t — r/c) in equation (1) is re- 

 sponsible for the conclusion that the disturbance is 

 propagated with the velocity c. The remaining factor, 

 1/r, is called the amplitude factor since it is responsi- 

 ble for the decrease in sound intensity as the distance 

 from the source increases. The rate at which sound 

 intensity is weakened with distance can be easily 

 computed by using the concept of rays as carriers of 

 sound energy, provided we assume that energy is 

 generated only at the source and then flows through 

 space without gain or loss. For reasons of symmetry, 

 the energy flow from the source must take place 

 along the radial sound rays. There will be a definite 

 number of rays inside a unit solid angle. These rays 

 will intercept an area of 1 sq ft on a sphere of 

 radius 1 ft whose center is at the source, an area 

 of 4 sq ft on a sphere of radius 2 ft, and generally 

 r^ sq ft on a sphere of radius r. Since the total energy 

 flow is the same for all these spherical surfaces, the 

 energy flow per unit area, or sound intensity, must 

 be inversely proportional to the square of the distance 

 of the unit area from the source. 



3.1.2 



General Waves 



The frontal attack on the wave equation was the 

 solution of the boundary problem by the method of 

 normal modes. This method was found to be too com- 

 plicated. It was shown in Section 3.1.1 that the 

 method of sound rays gave a simple and plausible ac- 

 count of sound propagation for the case of spherical 

 symmetry. A natural approach to the general prob- 

 lem would be to generalize the definition of sound 

 rays, and see if light could thereby be thrown on the 

 general case of variable sound velocity and arbitrary 

 initial distributions of p. 



First we must generalize the definition of wave 

 fronts. In what follows we shall restrict ourselves to 

 harmonic sound waves, that is, sound waves which 

 have been produced by a soimd source which under- 

 goes single-frequency harmonic vibrations. In ac- 

 cordance with Section 2.4.3, the pressure at any 

 point inside the fluid can be represented as the real 

 part of an expression having the form 



p = A{x,y,z)e'''^'y'''''> (3) 



in which the angle B at each point in space increases 

 linearly with time, 



e = 2Trf[t - t{x,y,z)'}. (4) 



We shall now call a wave front all those points at 

 which the phase angle d has a specified value, say So- 

 At any time t, this surface is defined by the equation 



<x,y,z) = t - — ■ 



(5) 



For later convenience, we shall replace t{x,y,z) by an 

 expression W{x,y,z)/c,i, in which Co is the velocity of 

 sound under certain designated standard conditions. 

 Equation (3) then takes the form 



p = A{x,y,z) e2"/['-^|^], (6) 



both A and W being real functions of the space co- 

 ordinates. The defining equation (5) of an individual 

 wave front assumes the form 



where 



W{x,y,z) = Co(< - to) 

 Bo 



(7) 



to = 



2ir/ 



The term U has different values for different wave 

 fronts, but is constant both in space and in time for a 

 given wave front. The function W clearly has the 

 dimension of a length. 



In order to make use of the concept of sound rays 

 to describe the energy propagated by such generalized 



