The trigonometric functions are useful for modeling periodic behavior. For example,
we may describe the motion of an object oscillating at the end of a spring (ignoring
any damping forces, such as friction, and assuming the object is at *x* = *0* at
time *t* = 0) with

*x*(*t*) = *a*sin(ω*t*),

where *a* is the amplitude of the motion and ω/2π is the frequency of
the motion.

For a more complicated example, consider the
motion of a molecule of air as a sound wave passes. The action of the sound
wave causes a particular molecule of air
to oscillate back and forth about some equilibrium position. If we let *x*(*t*) represent
the position of the air molecule at
time *t*, with *x* = 0 corresponding to the equilibrium position and *x* considered
to be positive in one direction from the
equilibrium position and negative in the other, then for many sounds *x* will be a
periodic function of *t*. In general, this
will be true for musical sounds, but not true for sounds we would normally classify
as noise. Moreover, even if *x* is a
periodic function, it need not be simply a sine or cosine function. The graph of *x* for a
musical sound, although periodic,
may be very complicated. However, many simple sounds, such as the sound of a
tuning fork, are represented by sine
curves. For example, if *x* is the displacement of an air molecule for a tuning fork which
vibrates at 440 cycles per second with a maximum displacement from equilibrium
of 0.002 centimeters, then

*x*(t) = 0.002sin(880π*t*).

In the early part of the 19th century, Joseph Fourier (1768-1830) showed that the
story does not end here.
Fourier demonstrated that any "nice" periodic curve (for example, one which is continuous)
can be approximated as
closely as desired by a sum of sine and cosine functions. In particular, this means that
for any musical sound the function *x* may be approximated well by a sum of sine and
cosine functions. For example, in his book *The Science of Musical Sounds*
(Macmillan, New York, 1926), Dayton Miller shows that, with an appropriate choice of units, the sequence of
functions

x_{1}(t) = 22.4sin(t) + 94.1cos(t) |

x_{2}(t) = x_{1}(t) + 49.8sin(2t) - 43.6cos(2t) |

x_{3}(t) = x_{2}(t) + 33.7sin(3t) - 14.2cos(3t) |

x_{4}(t) = x_{3}(t) + 19.0sin(4t) - 1.9cos(4t) |

x_{5}(t) = x_{4}(t) + 8.90sin(5t) - 5.22cos(5t) |

x_{6}(t) = x_{5}(t) - 8.18sin(6t) - 1.77cos(6t) |

x_{7}(t) = x_{6}(t) + 6.40sin(7t) - 0.54cos(7t) |

x_{8}(t) = x_{7}(t) + 3.11sin(8t) - 8.34cos(8t) |

x_{9}(t) = x_{8}(t) - 1.28sin(9t) - 4.10cos(9t) |

x_{10}(t) = x_{9}(t) - 0.71sin(10t) - 2.17cos(10t) |

give successively better approximations to the displacement curve of a sound wave generated
by the tone C_{3} of an organ pipe. Notice that the
terms in this expression for *x*(*t*) are written in pairs with frequencies which are
always integer multiples of the frequency of the first pair. This is a general fact which is
part of Fourier's theory; if we added more terms to obtain more
accuracy, the next terms would be of the form

*a*sin(11*t*) + *b*cos(11*t*)

for some
constants *a* and *b*. Notice also that
the amplitudes of the sine and cosine curves tend to decrease as the frequencies
are increasing. As a consequence, the
higher frequencies have less impact on the total curve.
Put another way, Fourier's theorem says that every musical sound is the sum of
simple tones which could be generated
by tuning forks. Hence in theory, although certainly not in practice, the instruments of
any orchestra could all be
replaced by tuning forks. On a more practical level, Fourier's analysis of periodic functions
has been fundamental for
the development of such modern conveniences as radios, televisions, stereos, and
compact disc players.

In the applet below,
clicking on the buttons will load an audio file to play the given tone for the functions
*x*_{1} through *x*_{10} described above. The units are scaled so that
the fundamental is played at a frequency of 261 cycles per second (middle C). Each audio
file is 41k, and so may take a moment to download the first time you play it. As overtones are added,
you can see the complexity of the motion increase, while remaining periodic.

The function *x*(*t*) which is 1 for 0 ≤ *t* < 0.5 and -1 for
0.5 ≤ *t* < 1, and then repeats these values over every interval of length 1, is an
example of a *square wave*. The following applet plots this square wave along with an
approximating trigonometric series. If *n* terms are requested, the approximating
sum is

*p _{n}*(

Note that although *p _{n}* is continuous, it approximates the discontinuous
square wave well for even small values of

According to Dayton Miller in

*The Science of Musical Sounds*, the function*x*(*t*) = 151sin(*t*) - 67cos(*t*) + 24sin(2*t*) + 55cos(2*t*) + 27sin(3*t*) + 5cos(3*t*)gives a good approximation to the shape of the displacement curve for the tone B

_{4}played on the E string of a violin.Graph each of the individual terms of

*x*on the interval [-15, 15]. Use a common scale for the vertical axis.Graph

*x*on [-15, 15].Graph

*x*and its individual terms (a total of 7 graphs) together on the interval [-15, 15].

Suppose we define a function

*f*by saying that it is periodic with period 1 and that*f*(*x*) = 1 - 2*x*for 0 ≤ x < 1.Sketch the graph of

*f*over the interval [-3, 3].Let

*g*(_{n}*x*) = 2(sin(2π*x*)/π + sin(4π*x*)/2π + sin(6π*x*)/3π + . . . + + sin(2nπ*x*)/nπ)What is the period of

*g*? Graph_{n}*g*,_{1}*g*,_{2}*g*,_{3}*g*,_{4}*g*, and_{5}*g*over the interval [-3, 3]._{10}What do you think happens to

*g*as_{n}*n*gets large?

For an interesting account of sound waves, Fourier's theorem, and related ideas in electromagnetism, read Chapters 19 ("The Sine of G Major") and 20 ("Mastery of the Ether Waves") in Morris Kline's

*Mathematics in Western Culture*(Oxford University Press, 1953).

Copyright © 2002 by Dan Sloughter.