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) = smax cos(kx - wt),
where smax 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) = dPmax sin(kx -
wt),
where dPmax 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:
dPmax = pvwsmax
where the maximum change in pressure (dPmax),
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
(smax).