The size of a radio antenna is closely related to ... - Light and Matter

q / Graphs of loudness versus
frequency for the vowel “ah,”
sung as three different musical
notes. G is consonant with D,
since every overtone of G that
is close to an overtone of D
(marked “*”) is at exactly the
same frequency. G and C♯ are
dissonant together, since some
of the overtones of G (marked
“x”) are close to, but not right on
top of, those of C♯.
played loudly or softly. In reality, waves on a string obey the principle
of superposition approximately, but not exactly. The guitar,
like just about any acoustic instrument, is a little out of tune when
played loudly. (The effect is more pronounced for wind instruments
than for strings, but wind players are able to compensate for it.)
Now there is only one hole in our reasoning. Suppose we somehow
arrange to have an initial setup consisting of two identical pulses
heading toward each other, as in figure (g). They will pass through
each other, undergo a single inverting reflection, and come back to
a configuration in which their positions have been exactly interchanged.
This means that the period of vibration is half as long.
The frequency is twice as high.
This might seem like a purely academic possibility, since nobody
actually plays the guitar with two picks at once! But in fact it is an
example of a very general fact about waves that are bounded on both
sides. A mathematical theorem called Fourier’s theorem states that
any wave can be created by superposing sine waves. Figure p shows
how even by using only four sine waves with appropriately chosen
amplitudes, we can arrive at a sum which is a decent approximation
to the realistic triangular shape of a guitar string being plucked.
The one-hump wave, in which half a wavelength fits on the string,
will behave like the single pulse we originally discussed. We call
its frequency f o . The two-hump wave, with one whole wavelength,
is very much like the two-pulse example. For the reasons discussed
above, its frequency is 2f o . Similarly, the three-hump and four-hump
waves have frequencies of 3f o and 4f o .
Theoretically we would need to add together infinitely many
such wave patterns to describe the initial triangular shape of the
string exactly, although the amplitudes required for the very high
frequency parts would be very small, and an excellent approximation
could be achieved with as few as ten waves.
We thus arrive at the following very general conclusion. Whenever
a wave pattern exists in a medium bounded on both sides by
media in which the wave speed is very different, the motion can be
broken down into the motion of a (theoretically infinite) series of sine
waves, with frequencies f o , 2f o , 3f o , . . . Except for some technical
details, to be discussed below, this analysis applies to a vast range of
sound-producing systems, including the air column within the human
vocal tract. Because sounds composed of this kind of pattern
of frequencies are so common, our ear-brain system has evolved so
as to perceive them as a single, fused sensation of tone.
Musical applications
Many musicians claim to be able to pick out by ear several of the
frequencies 2f o , 3f o , ..., called overtones or harmonics of the fundamental
f o , but they are kidding themselves. In reality, the overtone
372 Chapter 6 Waves