PROPERTIES OF SOUND WAVES 



21 



because of equation (18). By adding equations (49) 

 and (50), we obtain 



Total energy oi v = —~ — + ^P ■ (51) 



2 2k 



By dividing equation (51) by the volume Vo'. 

 Energy density at {x,y,z) 



~ 2 2/c 



= 'j{ul + ul + ul)+f^- (52) 



We are now in a position to give a general expres- 

 sion for the intensity of a progressive sound wave, the 

 characteristic which determines its loudness. Inten- 

 sity is defined for a general progressive wave as the 

 amount of energy which crosses a unit area normal to 

 the direction of propagation in unit time. Since the 

 energy travels at the same rate as the sound pulse, 

 the instantaneous rate of energy flow will be equal to 

 the energy density at the point in question times the 

 sound velocity at this point. The intensity will be the 

 time average of the instantaneous rate of energy flow, 

 or 



Intensity at (x,y,z) 



= Time average of c I — — h 



\ 2 



2k) 



~2~ 2^ 



(53) 



where the bar over a quantity denotes the time 

 average of that quantity. 



We shall now attempt to calculate the intensities 

 explicitly for various types of sound waves. We shall 

 first consider plane waves and spherical waves, and 

 then more general waves. 



Plane Waves 



A plane progressive wave satisfies equation (47); 

 and therefore equation (50) reduces, for that case, to 



Potential energy of y = ^p^v^v? (54) 



which is exactly equal to the expression (49) for the 

 kinetic energy of v. We therefore get the result that 

 for a plane progressive wave the kinetic and potential 

 energies possessed by any small volume element at 

 any time are equal. The kinetic and potential energies 

 attain their greatest values at the spots where the 

 particle velocity and excess pressure have their max- 

 ima or minima; and they vanish at the spots where 



the particle velocity and the excess pressure are both 

 zero. 



Because of this equahty of kinetic and potential 

 energies for a progressive plane wave, equation (53) 

 simplifies to 



Intensity = cpait^ = 



-,_P_ 



PaC 



(55) 



by equations (47) and (26). 



In a plane progressive wave that is also harmonic 

 the pressure is a sinusoidal function of the time with 

 maximum value a. Since the average value of sin^ d 

 over a complete period is l-^, -p^ = c^l2. 



Intensity = 



2poC 



(56) 



Also, from equation (55) and the fact that u is also 

 a sinusoidal function of the time. 



Intensity = t/JoCt^iax- (57) 



Spherical Waves 



For a spherical wave far from the source the 

 formulas derived for plane waves are approximately 

 true. We must be careful in applying them, however, 

 to remember that the amplitude of the pressure 

 vibration is no longer constant, but diminishes in- 

 versely with distance. 



Using equations (57) and (48), we obtain 



1 a? 



Intensity = (58) 



2poC r- 



for harmonic spherical waves where a is the maximum 

 pressure change at a distance one unit from the 

 source. 



Equation (58) is the familiar inverse square law of 

 intensity loss for a spherical wave spreading out from 

 a point source into an infinite homogeneous medium. 



Let F represent the amount of energy radiated by 

 the source into a unit solid angle"* in one second. 

 Then 



Total rate of emission = 45rF. 



(59) 



"'Solid angle is the three-dimensional analogue to the 

 ordinary, two-dimensional, plane angle. It measures the angu- 

 lar spread of such three-dimensional objects as a cone, a light 

 beam, or the beam of a radio transmitter. Its measure is de- 

 fined as follows. Construct a sphere of arbitrary size with the 

 apex of the solid angle as its center. The solid angle will then 

 cut out a certain area of the surface of the sphere. This area, 

 divided by the square of the radius of the sphere, is the meas- 

 ure of the solid angle. It is dimensionless and does not depend 

 on the sphere radius chosen. The unit solid angle is frequently 

 called the steradian. The full solid angle, comprising all direc- 

 tions pointing from the apex, has the value 4x. 



