PROPERTIES OF SOUND WAVES 



19 



mates a point source at the origin, the following 

 solution is obtained. 



p(r,l) = cr 



0-D , /0+D 



+ (1 - Cl) 



(41) 



Physically, we can eliminate the solution 

 (1/r) p{t + r/c). This solution corresponds to wave 

 propagation in the negative r direction, with the 

 speed c; in other words, to a wave which starts out 

 at some negative time with a great radius and con- 

 tracts into the point x = y = z = Oa,t the time t = 0. 

 The first solution is physically valid since it resembles 

 actual propagation from a point source. It implies 

 that the spherical wave spreads out from the point 

 source into ever-increasing spheres with the speed c. 

 Therefore, an initial ping of duration t seconds will 

 cause the resulting sound energy to be contained 

 within an expanding spherical shell of thickness ct. 



If the source is harmonic (emits a pure tone of the 

 frequency/), the initial conditions are of the form 

 a cos 2irf(t — e) 



P = ; 



n 



and if Tq is nearly zero, the pressure at the distance r 

 from the source and time t is given by 



a cos 2x/( t 



V 



{'-;-) 



(42) 



The constants / and e have the same physical signifi- 

 cance as for the plane wave case; /is the frequency of 

 the vibration and e is the phase constant which 

 orients the vibration in time. There is a difference, 

 however, in the interpretation of a. In the plane wave 

 case, o represents the maximum pressure change in 

 the wave at all distances from the source; since a is a 

 constant, all these pressure changes are equal. For 

 spherical waves described by equation (42), however, 

 it is clear that the maximum pressure change at the 

 distance r is given by a/r, decreasing as r increases. 

 The constant a is no longer the amplitude at all 

 ranges, but merely the amphtude at the particular 

 range r = 1. 



2.4 PROPERTIES or SOUND WAVES 

 2.4.1 Pressure versus Fluid Velocity 



Plane Waves 



For a plane wave we have, from equation (4), 

 da du 



dt ~ ~'dx' 



where u is the particle velocity in the positive x 

 direction. Because of equation (18), this equation 

 can be transformed into 



dp 

 Tt 



du 



K 



dx 



(43) 



From equations (21) and (18), there results the 

 following expression for dp/ dx: 



dp 



dx 



= — Po 



du 

 dt' 



(44) 



Equations (43) and (44) will be used to derive the 

 general relationship between the excess pressure and 

 the particle velocity in a plane wave. Assume that 

 as initial conditions the plane x = has its excess 

 pressure given by piO,t) = pit) between t = and 

 t = <o, and p(0,i) = for all other values of /. Then 

 the general solution of the plane wave equation, if 

 we assume that the wave moves in the positive x 

 direction, is given by 



p{x 



,t) = v{t-^^. 



By differentiating both sides of this equation with 

 respect to t and also with respect to x, we obtain 



dt ^\ J' dx c^\ J 



This means that 



dp 1 dp 



dx c dt 



Combining equations (43) and (45) gives 



-i 

 dx\ 



and by combining equations (44) and (45) 



:(«-?)-- 



-(p - pocu) 



0. 



(45) 



(46a) 



(46b) 



Equation (46a) means that (m — cp/x) is a func- 

 tion of t alone; and equation (46b) implies that 

 (p — poCu) is a function only of x. But these two 

 parentheses are proportional, differing by the factor 

 ( — Poc) since poc^ = k. Therefore, each parenthesis 

 must be identically equal to some constant. This 

 constant turns out to be zero for both since at any 

 given point both p and u vanish before and after the 

 disturbance has passed. The following relation be- 

 tween particle velocity and pressure results: 



^ p- (47a) 



c 

 u = - p 



K 



PoC 



