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What we can learn from the wave equation
1. How do you tune a stringed instrument? A wind instrument? A drum?
2. What other factors determine the frequency produced by an instrument?
3. The set of harmonics that an instrument can produce is called its spectrum (plural
spectra). Suppose the fundamental frequencies of a stringed instrument, a flute, a
clarinet, and a drum are all tuned to 440 Hz. Which frequencies belong in the
spectrum of each instrument? How do the spectra differ from each other?
4. Suppose one flute plays a tone whose fundamental is 440 Hz and the other plays a
tone whose frequency equals the third harmonic of the first flute. List the elements of
the spectrum of each flute.
Frequency, Periodic Functions, the Wave Equation, Fourier Analysis, and Spectra
Main points of this section

The human ear responds to sound waves that are modeled by sinusoidal
functions.

If time is measured in seconds, the frequency of the sinusoid Asin( 2t   ) is 
Hertz (cycles per second), and the loudness (measured in decibels, related to the
logarithm of power intensity) depends on A (amplitude).

 is
If you play two sinusoidal sounds at the same time, the
 resulting vibration
modeled by the sum of the functions for each sound. The loudness is related to
the logarithm of the sum—in other words, the resulting sound is not twice as
loud! If the two frequencies are almost—but not quite—equal, then you hear
beats.

Musical sounds are modeled by periodic functions.

Every periodic function can be represented as a sum of sinusoidal functions. This
sum, which may be infinite, is called the Fourier series of the function. The
Fourier series for a periodic function that has finitely many discontinuities and
finitely many places where the derivative is undefined converges to the function
except at the discontinuities.

We spent a good deal of time calculating Fourier series for periodic functions.
The formulas are summarized in Benson. The theory of even and odd functions
often simplifies the calculations.

The one-dimensional wave equation models how stringed and wind
instruments vibrate and produce sound. If you choose the boundary conditions
correctly, solutions to the wave equation for stringed and wind instruments
usually have frequencies  , 2 , 3 ,... (  is the fundamental frequency). There is
an exception for wind instruments: if one end of the instrument is open and the
other is closed, solutions have frequencies ,3,5,.... The spectrum of an
instrument is the collection of harmonics it produced. The higher harmonics are


quieter than the fundamental. The intensities of the harmonics are determined by
physical properties of the instrument—its shape, the materials it is make of, the

room you play it in, etc.

The fundamental frequency of sound produced by a stringed instrument depends
on only three things: the string’s length, tension and linear density. The
fundamental frequency of a wind instrument’s sound depends on its length and
the size of its bore (the cross-sectional area of the inside of the instrument).

The two-dimensional wave equation models how drums vibrate. If you choose
appropriate boundary conditions, you can solve the wave equation; however, the
frequencies produced are not rational multiples of the fundamental frequency.
The modes of vibration of a drum are represented by Chladni patterns, which
indicate the curves along which the displacement of the drumhead is zero. Drums
will not produce the same harmonics as stringed and wind instruments.
What’s next…

This was a lot of math! How does it relate to music? We will see that the
spectrum of an instrument determines which musical intervals sound good when
played on that instrument. Pythagoras discovered this fact in the sixth century
BC. This means that the same intervals sound good on stringed and wind
instruments—exactly the instruments used in most Western music.

A musical scale is a sequence of pitches used to construct melodies and chords.
The intervals between the frequencies of notes in the scales used in Western
music are usually the same intervals that appear in the spectrum of a stringed or
wind instrument. Coincidence? We don’t think so!

Does every culture use the same intervals in a scale? No. Indonesian music uses
bells that have spectra that are not rational multiples of the fundamental (this is
because the three-dimensional wave equation applies to the vibration of bells).
The scale played by a gamelan—an Indonesian bell orchestra—is quite difference
from the Western musical scale.