Digital Sound & Music: Concepts, Applications, & Science, Chapter 2, last updated 6/25/2013
frequencies, all of which give an instrument its characteristic sound. The
fundamental and harmonic frequencies are also referred to as the partials, since
together they make up the full sound of the resonating object. The harmonic
frequencies beyond the fundamental are called overtones. These terms can be
slightly confusing. The fundamental frequency is the first harmonic because
this frequency is one times itself. The frequency that is twice the fundamental is
called the second harmonic or, equivalently, the first overtone. The frequency
that is three times the fundamental is called the third harmonic or second
overtone, and so forth. The number of harmonic frequencies depends upon the
properties of the vibrating object.
One simple way to understand the sense in which a frequency might be natural to an
object is to picture pushing a child on a swing. If you push a swing when it is at the top of its
arc, you‟re pushing it at its resonant frequency, and you‟ll get the best effect with your push.
Imagine trying to push the swing at any other point in the arc. You would simply be fighting
against the natural flow. Another way to illustrate resonance is by means of a simple transverse
wave, as we‟ll show in the next section. Resonance of a Transverse Wave
We can observe resonance in the example of a simple transverse wave that
results from sending an impulse along a rope that is fixed at both ends. Imagine
that you‟re jerking the rope upward to create an impulse. The widest upward
bump you could create in the rope would be the entire length of the rope. Since
a wave consists of an upward movement followed by a downward movement,
this impulse would represent half the total wavelength of the wave you‟re
transmitting. The full wavelength, twice the length of the rope, is conceptualized
in Figure 2.9. This is the fundamental wavelength of the fixed-end transverse
wave. The fundamental wavelength (along with the speed at which the wave is
propagated down the rope) defines the fundamental frequency at which the
shaken rope resonates.
is the length of a rope fixed at both ends, then

is the fundamental wavelength of the
rope, given by
Equation 2.4
Figure 2.9 Full wavelength of impulse sent through fixed-end rope
Now imagine that you and a friend are holding a rope between you and shaking it up and
down. It‟s possible to get the rope into a state of vibration where there are stationary points and
other points between them where the rope vibrates up and down, as shown in Figure 2.10. This
Resonance as
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