12 



SOUND WAVES 



B. Particle Velocity in a Spherical Wave. — The particle velocity from 

 equations 1.11 and 1.36 is 



--(;«..) 



i± ^jk{ct-r) 



Retaining the real part of equation 1.41 the particle velocity is 



\_kr 



cos k{ct — r) — sin k{ct 



1.41 



1.42 



C. Phase Angle between the Pressure and the Particle Velocity in a Spheri- 

 cal Wave. — The particle velocity given by equation 1.42 may be written 



A 1 1 



« = — ^1— + k^s'm [k{ct - r) - d] 



1.43 



where tan d = 1/kr. 



Comparing equation 1.43 with equation 1.40 for the pressure it will be 

 seen that the phase angle between the pressure and velocity in a spherical 

 wave is given by 



= tan-i — 

 kr 



1.44 



For very large values of kr, that is, plane waves, the pressure and par- 

 ticle velocity are in phase. The phase angle frequency characteristics for 

 various distances from the center of a spherical wave system are shown in 

 Fig. 1.2. 



5 6 7 8 9|q2 2 3 4 5 6 7 » 9 |q3 



FREQUENCY IN CYCLES PER SECOND 



5 « 7 6 9 



icf 



Fig. 1.2. Phase angle between the pressure and particle velocity in a spherical sound wave 

 for distances of \, |, 1, 2 and 5 feet from the source. (Courtesy of The Blakiston Company 

 from Olson and Massa, Applied Acoustics.) 



