SIMPLE TYPES OF SOUND WAVES 



15 



negative x direction. If, however, an expression of the 

 form (31) is the function describing the given physical 

 situation, the situation is more compUcated. The re- 

 sulting distribution of pressure will be the mathe- 

 matical resultant of the pressure distributions calcu- 

 lated for /i and fi; and a given value of the excess 

 pressure will no longer be propagated in a tingle 

 direction with the speed c. Discussion of this more 

 complicated type of wave propagation will be de- 

 ferred until Section 2.7. 



Now we consider two specific examples of the prop- 

 agation of plane waves in an infinite homogeneous 

 medium (no boundary conditions). In the first ex- 

 ample, we assume as an initial condition that the 

 exact pressure distribution is specified at the time 

 instant t = between the planes x = and x = Xo, 

 by p{x,0) = p(x); and also that the excess pressure 

 is zero at i = for all values of x less than and 

 greater than xq. We assume that this initial disturb- 

 ance gives rise to progressive waves traveling in the 

 positive X direction, that is, that the solution is of the 

 form p = f{t — x/c). Then the solution of the wave 

 equation with these conditions must be 



p{x,t) = p{x - d) (32) 



since first, it satisfies the initial conditions p(x,0) = 

 p{x); second, it is a function of (t — x/c) and there- 

 fore satisfies the wave equation (30) ; and third, there 

 can be only one solution to this physical problem. 



By the results of preceding paragraphs, we know 

 that a given value of the excess pressure will be 

 propagated in the x direction with the speed c. Thus, 

 at the time t the initial disturbance will be duplicated 

 between the planes x = ct and x = Xo -{- ct; and the 

 excess pressure will be zero for x < ct and x > xq + 

 ct. The disturbance of the fluid remains of width Xo, 

 remains unchanged in "shape," and is propagated 

 with the speed c. 



As another example, we suppose as an initial con- 

 dition that the values of the excess pressure are 

 specified only for the plane x = 0, but for the total 

 time interval between t = and t = to, by the equa- 

 tion p(0,i) = p{t) ard that the excess pressure at the 

 plane x = is zero for t < and t > to- Here, also, 

 we assume that this disturbance causes progressive 

 waves to be propagated in the positive x direction. 

 Arguing as in the preceding example, the solution of 

 the wave equation (30) with these imposed conditions 

 is 



The expression (33) differs somewhat in form from 

 equation (32) because the initial conditions are ex- 

 tended in time instead of in space. 



In this example, it is known that the excess pres- 

 sure will be the same at all combinations of space and 

 time where x — ct = constant. Since x — ct equals 

 zero when x = 0, t = 0, the value of the excess pres- 

 sure corresponding to x = 0,t = 0, must be assumed 

 by the plane x = ct a.t the time t. Further, since 

 x — ct equals — do a,t t = to, x = 0, the value of the 

 excess pressure corresponding to t = to, x = 0, will 

 be assumed by the plane x = ct ~ cto at the time 

 instant t. Thus, at time t the disturbance is confined 

 between the planes x = ct — cto and x = ct; that is, 

 the region of disturbance is always of width cto and 

 is propagated along the positive x axis with the 

 speed c. 



Sound Velocity 



We have seen that in some simple situations the 

 quantity c may be regarded as the velocity with 

 which the disturbance is propagated in the medium, 

 or more simply, the velocity of sound in the medium. 

 It will be recalled that c was defined in equation (26) 

 by 



-f.- 



where k is the bulk modulus and po is the density of 

 the fluid at equilibrium. 



If the medium is a perfect gas, the relation of the 

 sound velocity to the temperature and pressure can 

 be expressed in a simple formula. The pressure 

 changes produced by sound in a fluid are usually so 

 rapid that they are accomplished without appreciable 

 heat transfer, that is, they are practically adiabatic. 

 For a perfect gas suffering adiabatic pressure changes 

 the bulk modulus is yP, where P is the total pressure 

 and 7 is the ratio between the specific heat at con- 

 stant pressure and the specific heat at constant 

 volume. For a perfect gas suffering any kind of pres- 

 sure change P = pRT. Thus, the simple result fol- 

 lows that 



p{x 



,t) = p{i--} 



(33) 



c = VyRT. 



Hence, in air at normal pressure, which is not far from 

 a perfect gas, the sound velocity increases with the 

 square root of the absolute temperature. 



No such relationship can be derived for the velocity 

 of sound in sea water since the pressure, density, and 

 temperature of sea water are not related by any 



