Fourier Analysis
Playing a musical instrument, such as a guitar or an organ, generates a set of standing waves that cause a
sympathetic oscillation in the surrounding air. Such an oscillation consists of a fundamental harmonic, whose
frequency determines the pitch of the musical note heard by the listener, accompanied by a set of overtone
harmonics that determine the timbre of the note. By definition, the oscillation frequencies of the overtone
harmonics are integer multiples of that of the fundamental. Thus, we expect the pressure perturbation generated
in a listener's ear to have the general form
(339)
where
is the angular frequency of the fundamental (i.e.,
) harmonic, and the
and
are the
amplitudes and phases of the various harmonics. The preceding expression can also be written
(340)
where
and
. The function
is periodic in time with period
. In other words,
for all
. This follows because of the mathematical
identities
and
, where
is an integer. [Moreover, there is
no
for which
for all
.] Can any periodic waveform be represented as a linear
superposition of sine and cosine waveforms, whose periods are integer subdivisions of that of the waveform,
such as that specified in Equation (
340
)? To put it another way, given an arbitrary periodic waveform
,
can we uniquely determine the constants
and
appearing in Equation (
340
)? It turns out that we can.
Incidentally, the decomposition of a periodic waveform into a linear superposition of sinusoidal waveforms is
commonly known as
Fourier analysis
.
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 Fall '08
 Staff
 Fourier Series, Waves And Optics, Waveform

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