Sources and Filters - The Physics of Birdsong - page 24

2.2.3 Resonances

Now, suppose that our vibrating membrane generates harmonic fluctuations
in a periodic way, with a period T . By this we mean that the membrane gen-
erates high and low densities in its surroundings (position x

0) alternately,
in a regular way ( P e = P e 0 cos( ωt ), with ω =2 π/T ), and these fluctuations
propagate along the tube. The other end of the tube reflects these fluctua-
tions, which return to the neighborhood of the membrane after traveling back
from the open end. Therefore, at every point x along the tube, the pressure
perturbation is a superposition of forward- and backward-traveling waves:

∼

P e ( x, t )= P e 0 cos( kx − ωt )+ P r cos( kx + ωt ) . (2.15)

What happens if, when the membrane is generating a high-pressure fluctu-
ation in its neighborhood, a reflected negative-pressure perturbation arrives?
The signal close to the membrane will be the sum of the two contributions,
and therefore will be strongly damped. This is known as destructive interfer-
ence . In contrast, if the pressure perturbation, after traveling to the open end
and returning back, arrives in phase with the perturbation being generated
by the membrane (for example, a high-pressure perturbation arrives as the
membrane is compressing its surroundings), the superposition of the signals
will be constructive . This helps to establish a signal of large amplitude. This
phenomenon is called resonance [Feynman et al. 1970].
The key quantities for establishing this strong signal in the neighborhood
of the membrane are the characteristic times of the problem. In order to
construct such a signal, the action of the membrane must be helped by the
reflected wave. For this to occur, there must exist a particular relationship
between the period of the signal generated by the membrane and the time it
takes the signal to propagate to the open end and back after its reflection.
When the signal returns, its shape is approximately equal to the shape that
it had when it was created, at a time 2 τ before (with τ being the time it
takes for the sound to travel a distance equal to the length of the tube L ).
The reflection changes the sign of the signal that returns. Therefore, this
perturbation has to be as similar as possible, considering the change of sign
that is produced in the reflection at the open end of the tube and the state of
the wave on its return, to the signal being created by the membrane, in order
to help constructively in the creation of a strong total signal. If the time taken
to travel from the membrane to the open end and back is 2 τ , a constructive
effect will occur if 2 τ = T/ 2. The reason is the following: two harmonic
oscillations whose phases differ by half a period will be in counterphase, as
illustrated in Fig. 2.5. In addition, the reflection of the wave produces an
inversion. Therefore, the delay of half a period accumulated in the trip plus
the inversion of the wave at its reflection implies that the wave that returns to
the membrane after reflection will be in phase with the signal being generated
by the membrane. Since τ = L/c (where c is the sound velocity), we have