Biomedical Engineering Reference
In-Depth Information
4 The Sources of Sound in Birdsong
In the previous chapters we discussed the fact that periodic fluctuations in
the airflow generate sound, and we have described some mechanisms through
which these fluctuations could be established. In this chapter, we shall focus
on those mechanisms that are present in the avian vocal organ. The rich-
ness of the physics involved in the operating syrinx will lead us to explore
several fields, from the physics of fluids to nonlinear dynamics. In order to
carry out this discussion, we begin with an analysis of one of the most thor-
oughly explored and often used models in the natural sciences: the harmonic
oscillator.
4.1 Linear Oscillators
4.1.1 A Spring and a Swing
Basically, any periodic motion can be described (at least when the move-
ments are small) in terms of a harmonic oscillator. The harmonic oscillator is
a system that performs oscillations as simple as the ones displayed in Fig. 4.1
[Kittel et al. 1965]. In this figure, we show a small mass, attached to a spring,
that leaves a record of its motion on a moving sheet of paper. In this example,
the motion of the spring is described in terms of the displacement of the mass
with respect to its rest position. It is natural to ask about the mechanism
used by the spring in order to induce a periodic motion in the mass. The key
is that the restitution force that the spring exerts on the mass is proportional
to its departure from its rest position (i.e., if the mass is displaced twice as
far, then the force applied to the mass by the spring is twice as large). If
initially we displace the mass by stretching the spring, the mass will expe-
rience a force proportional to the displacement. If we now release the mass,
it will be accelerated, increasing its velocity. As the mass passes through the
equilibrium position, it will experience no net force (since in this position the
spring is stretched by just the amount necessary to compensate the weight
of the mass). The inertia of the mass is responsible for the continuation of
its motion, and the mass passes through the equilibrium position. At this
instant, the mass begins to compress the spring, and a restitution force de-
celerates the mass until its motion momentarily stops, and it then begins its
