18 



WAVE ACOUSTICS 



turbance. The quantity « is called the phase constant 

 because it fixes the position of the disturbance in 

 time. Two vibrations of the same frequency and the 

 same e have their zeros simultaneously, also, their 

 maxima. If they have different s's, one has its zeros a 

 fixed time interval ahead of the other, and we say 

 that there is a 'pliase difference between the vibrations. 



2.3.3 



Spherical Waves 



The sound at large distances from an actual source 

 resembles the sound from a point source more closely 

 than it does the sound from an infinite plane. Hence, 

 for some purposes it is more realistic to ab3,ndon the 

 assumption of plane waves, and assume instead a 

 point source at the origin which causes the pressure 

 in the surrounding medium to be a function only of 

 the distance r from the origin and of the time t. That 

 is, the pressure is given by some function 



V = Vir,t) (35) 



and is thus independent of the direction of the line 

 joining the source to the point in question. 



We shall now show that the wave equation (27) re- 

 duces, for the assumed case of spherical symmetry, to 

 the simple form (37). 



From simple analytic geometry, 



dr dr y dr 



^2 _ 3-2 _|_ 2^2 _|_ ^2; 



dx 



dy 



dz 



(3.6) 



In order to transform the wave equation, the vari- 

 ables x,y,z must be eliminated, and the variable r in- 

 serted. In order to do this, we must use the relations 

 (36) to calculate d^p/dx'^, d'^p/dy'^, and d'^p/dz'^ in 

 terms of r and the derivatives of p with respect to r. 

 This is done as follows. 



dp dp dr dpx 

 dx dr dx dr r 



because spatially p depends only on r. By differen- 

 tiating again, 



d^p dTdprl dp^ r^ ^ x^ l ^d/dp\ 



dx^ ~ dxLdrrJ ~ drL r^ J rdxXdr/ 

 dpVr^ - x^l x^ a^p 

 ~ ~dX. r^J r^ dr"^' 

 Similarly, we can show that 



d'^p dpTr'^ — 2/^1 2/^ d'^p 



dy^ ~ dn. 1^ J r^ df'- 



d'^p dpVr"^ — z^~\ ^ d'^p 



dz^ drL r^ J r^ dr^ 



Addition of these expressions for d'^p/dx^, d^p/dy^, 

 and d'^p/dz^, in order to obtain the right-hand side 

 of equation (27), gives 



d'^p d^p d^p d^p 2 dp 

 dx^ dy^ dz"^ dr^ r dr 



The latter expression is easily verified to be 

 (l/r)(3V3r^) (?■?), so we finally obtain 



d'^V d'^v d'^v 1 5^ 



-\ -\ = {rp), 



dx''- dy''- dz^ rdr^ 



and the general wave equation (27) reduces to 



d''(rp) d^ , , 



This equation has a form similar to that of equa- 

 tion (30) for plane waves with p replaced by rp, and 

 X by r. By using an argument similar to that in 

 Section 2.3.1, it can be shown that equation (37) is 

 satisfied by 



rp 



=<'^D. 



where / is an arbitrary function of one variable. By 

 dividing out the r, we get the following expression for 

 p(r,0 as the general solution of equation (37) : 



p{r,t) = 



/.(^-^)+4 + 



(38) 



Assume that the following initial conditions are 

 given. The initial disturbance is confined to a spheri- 

 cal shell of infinitesimal thickness at a distance r = ro 

 from the origin. We suppose that the excess pressure 

 in this spherical shell source is given by 



P(<) 



p(ro,0 = 



'•o 



(39) 



between the times i = and I = <o- We also suppose 

 that the excess pressure at points outside this shell 

 is zero at the time i = 0. The general solution of 

 equation (37) with these initial conditions is 



Vir,t) = \c.it - ^ -K ^») 



+ 



»-*('+:-?)]■ 



(40) 



because first, the right-hand expression is in the form 

 of equation (38), and therefore satisfies equation (37) ; 

 second, the right-hand expression satisfies the initial 

 conditions imposed. In particular, if the spherical- 

 shell source has a very small radius so that it approxi- 



