Periodic Sound Waves

A periodic sound wave is a wave that has a
sinusoidal pattern of condensations and rarefactions.

 

    Now we look at this animation as a periodic sound wave.  The red bar on the left acts as the driving piston.  If it moves in a sinusoidal manner from left to right,  then the wave that is produced will be a sinusoidal wave.  Since the wave is sinusoidal, the wavelength, amplitude and frequency are constant. This is seen in nature as a tuning fork, which produces a periodic sound wave.  In a one dimensional tube as shown above, each particle undergoes simple harmonic motion.  The volume that is contained in one wavelength also undergoes this same motion.  We can represent the displacement of this volume as:

s(x,t) = s_max cos(kx - wt),

where s_max is the maximum displacement or displacement amplitude, k is the angular wave number, and w is the angular frequency of the piston.  
    We can also represent the change in pressure at any point in the same manner as we did for the volume displacement.  That is shown by:  

dP(x,t) = dP_max sin(kx - wt),

where dP_max is the maximum change in pressure, k is angular wave number, and w is angular frequency.

By combining the two equations we can receive the result:

dP_max = pvws_max

where the maximum change in pressure (dP_max), is equal to the density (p), times the velocity of the wave (v), times the angular frequency of the piston (w), times the maximum volume displacement (s_max).