## 50Harmonics

### 50–1Musical tones

Pythagoras is said
to have discovered the fact that two similar strings under the same
tension and differing only in length, when sounded together give an
effect that is pleasant to the ear *if* the lengths of the strings
are in the ratio of two small integers. If the lengths are as one is to
two, they then correspond to the octave in music. If the lengths are as
two is to three, they correspond to the interval between $C$ and $G$,
which is called a fifth. These intervals are generally accepted as
“pleasant” sounding chords.

Pythagoras was so impressed by
this discovery that he made it the basis of a school—Pythagoreans they
were called—which held mystic beliefs in the great powers of numbers.
It was believed that something similar would be found out about the
planets—or “spheres.” We sometimes hear the expression: “the music
of the spheres.” The idea was that there would be some numerical
relationships between the orbits of the planets or between other things
in nature. People usually think that this is just a kind of superstition
held by the Greeks. But is it so different from our own scientific
interest in quantitative relationships?
Pythagoras’ discovery was
the first example, outside geometry,
of any numerical relationship in nature. It must have been very
surprising to suddenly discover that there was a *fact* of nature
that involved a simple numerical relationship. Simple measurements of
lengths gave a prediction about something which had no apparent
connection to geometry—the production of pleasant sounds. This
discovery led to the extension that perhaps a good tool for
understanding nature would be arithmetic and mathematical analysis. The
results of modern science justify that point of view.

Pythagoras could only have made
his discovery by making an experimental observation. Yet this important
aspect does not seem to have impressed him. If it had, physics might
have had a much earlier start. (It is always easy to look back at what
someone else has done and to decide what he *should* have done!)

We might remark on a third aspect of this very interesting discovery:
that the discovery had to do with two notes that *sound pleasant*
to the ear. We may question whether *we* are any better off than
Pythagoras in understanding
*why* only certain sounds are pleasant to our ear. The general
theory of aesthetics is probably no further advanced now than in the
time of Pythagoras. In this one
discovery of the Greeks, there are the three aspects: experiment,
mathematical relationships, and aesthetics. Physics has made great
progress on only the first two parts. This chapter will deal with our
present-day understanding of the discovery of
Pythagoras.

Among the sounds that we hear, there is one kind that we call
*noise*. Noise corresponds to a sort of irregular vibration of
the eardrum that is produced by the irregular vibration of some object
in the neighborhood. If we make a diagram to indicate the pressure of
the air on the eardrum (and, therefore, the displacement of the drum)
as a function of time, the graph which corresponds to a noise might
look like that shown in Fig. 50–1(a). (Such a noise
might correspond roughly to the sound of a stamped foot.) The sound of
*music* has a different character. Music is characterized by
the
presence of more-or-less *sustained tones*—or musical
“notes.” (Musical instruments may make noises as well!) The tone may
last for a relatively short time, as when a key is pressed on a piano,
or it may be sustained almost indefinitely, as when a flute player
holds a long note.

What is the special character of a musical note from the point of view of the pressure in the air? A musical note differs from a noise in that there is a periodicity in its graph. There is some uneven shape to the variation of the air pressure with time, and the shape repeats itself over and over again. An example of a pressure-time function that would correspond to a musical note is shown in Fig. 50–1(b).

Musicians will usually speak of a musical tone in terms of three characteristics: the loudness, the pitch, and the “quality.” The “loudness” is found to correspond to the magnitude of the pressure changes. The “pitch” corresponds to the period of time for one repetition of the basic pressure function. (“Low” notes have longer periods than “high” notes.) The “quality” of a tone has to do with the differences we may still be able to hear between two notes of the same loudness and pitch. An oboe, a violin, or a soprano are still distinguishable even when they sound notes of the same pitch. The quality has to do with the structure of the repeating pattern.

Let us consider, for a moment, the sound produced by a vibrating string. If we pluck the string, by pulling it to one side and releasing it, the subsequent motion will be determined by the motions of the waves we have produced. We know that these waves will travel in both directions, and will be reflected at the ends. They will slosh back and forth for a long time. No matter how complicated the wave is, however, it will repeat itself. The period of repetition is just the time $T$ required for the wave to travel two full lengths of the string. For that is just the time required for any wave, once started, to reflect off each end and return to its starting position, and be proceeding in the original direction. The time is the same for waves which start out in either direction. Each point on the string will, then, return to its starting position after one period, and again one period later, etc. The sound wave produced must also have the same repetition. We see why a plucked string produces a musical tone.

### 50–2The Fourier series

We have discussed in the preceding chapter another way of looking at
the motion of a vibrating system. We have seen that a string has
various natural modes of oscillation, and that any particular kind of
vibration that may be set up by the starting conditions can be thought
of as a combination—in suitable proportions—of several of the
natural modes, oscillating together. For a string we found that the
normal modes of oscillation had the frequencies $\omega_0$, $2\omega_0$,
$3\omega_0$, … The most general motion of a
plucked string, therefore, is composed of the sum of a sinusoidal
oscillation at the fundamental frequency $\omega_0$, another at the
second harmonic frequency $2\omega_0$, another at the third
harmonic $3\omega_0$, etc. Now the fundamental mode repeats itself every
period $T_1 = 2\pi/\omega_0$. The second harmonic mode repeats itself
every $T_2 = 2\pi/2\omega_0$. It *also* repeats itself every $T_1 =
2T_2$, after *two* of its periods. Similarly, the third harmonic
mode repeats itself after a time $T_1$ which is $3$ of its periods. We
see again why a plucked string repeats its whole pattern with a
periodicity of $T_1$. It produces a musical tone.

We have been talking about the motion of the string. But the
*sound*, which is the motion of the air, is produced by the
motion of the string, so its vibrations too must be composed of the
same harmonics—though we are no longer thinking about the normal
modes of the air. Also, the relative strength of the harmonics may be
different in the air than in the string, particularly if the string is
“coupled” to the air via a sounding board. The efficiency of the
coupling to the air is different for different harmonics.

If we let $f(t)$ represent the air pressure as a function of time for a musical tone [such as that in Fig. 50–1(b)], then we expect that $f(t)$ can be written as the sum of a number of simple harmonic functions of time—like $\cos\omega t$—for each of the various harmonic frequencies. If the period of the vibration is $T$, the fundamental angular frequency will be $\omega = 2\pi/T$, and the harmonics will be $2\omega$, $3\omega$, etc.

There is one slight complication. For each frequency we may expect
that the starting phases will not necessarily be the same for all
frequencies. We should, therefore, use functions like $\cos\,(\omega t
+ \phi)$. It is, however, simpler to use instead both the sine and
cosine functions for *each* frequency. We recall that
\begin{equation}
\label{Eq:I:50:1}
\cos\,(\omega t + \phi) = (\cos\phi\cos\omega t -
\sin\phi\sin\omega t)
\end{equation}
and since $\phi$ is a constant, *any* sinusoidal oscillation at
the frequency $\omega$ can be written as the sum of a term
with $\cos\omega t$ and another term with $\sin\omega t$.

We conclude, then, that *any* function $f(t)$ that is periodic
with the period $T$ can be written mathematically as
\begin{alignat}{4}
f(t) &= a_0\notag\\[.5ex]
&\quad\;+\;a_1\cos&&\omega t &&\;+\;b_1\sin&&\omega t\notag\\[.65ex]
&\quad\;+\;a_2\cos2&&\omega t &&\;+\;b_2\sin2&&\omega t\notag\\[.65ex]
&\quad\;+\;a_3\cos3&&\omega t &&\;+\;b_3\sin3&&\omega t\notag\\[.5ex]
\label{Eq:I:50:2}
&\quad\;+\;\dotsb && &&\;+\;\dotsb
\end{alignat}
where $\omega = 2\pi/T$ and the $a$’s and $b$’s are numerical
constants which tell us how much of each component oscillation is
present in the oscillation $f(t)$. We have added the
“zero-frequency” term $a_0$ so that our formula will be completely
general, although it is usually zero for a musical tone. It represents
a shift of the average value (that is, the “zero” level) of the
sound pressure. With it our formula can take care of any case. The
equality of Eq. (50.2) is represented schematically in
Fig. 50–2. (The amplitudes, $a_n$ and $b_n$, of the harmonic
functions must be suitably chosen. They are shown schematically and
without any particular scale in the figure.) The
series (50.2) is called the *Fourier series* for $f(t)$.

We have said that *any* periodic function can be made up in this
way. We should correct that and say that any sound wave, or any
function we ordinarily encounter in physics, can be made up of such a
sum. The mathematicians can invent functions which cannot be made up
of simple harmonic functions—for instance, a function that has a
“reverse twist” so that it has two values for some values of $t$! We
need not worry about such functions here.

### 50–3Quality and consonance

Now we are able to describe what it is that determines the “quality” of a musical tone. It is the relative amounts of the various harmonics—the values of the $a$’s and $b$’s. A tone with only the first harmonic is a “pure” tone. A tone with many strong harmonics is a “rich” tone. A violin produces a different proportion of harmonics than does an oboe.

We can “manufacture” various musical tones if we connect several “oscillators” to a loudspeaker. (An oscillator usually produces a nearly pure simple harmonic function.) We should choose the frequencies of the oscillators to be $\omega$, $2\omega$, $3\omega$, etc. Then by adjusting the volume control on each oscillator, we can add in any amount we wish of each harmonic—thereby producing tones of different quality. An electric organ works in much this way. The “keys” select the frequency of the fundamental oscillator and the “stops” are switches that control the relative proportions of the harmonics. By throwing these switches, the organ can be made to sound like a flute, or an oboe, or a violin.

It is interesting that to produce such “artificial” tones we need
only one oscillator for each frequency—we do not need separate
oscillators for the sine and cosine components. The ear is not very
sensitive to the relative phases of the harmonics. It pays attention
mainly to the *total* of the sine and cosine parts of each
frequency. Our analysis is more accurate than is necessary to explain
the *subjective* aspect of music. The response of a microphone or
other physical instrument does depend on the phases, however, and our
complete analysis may be needed to treat such cases.

The “quality” of a spoken sound also determines the vowel sounds that we recognize in speech. The shape of the mouth determines the frequencies of the natural modes of vibration of the air in the mouth. Some of these modes are set into vibration by the sound waves from the vocal chords. In this way, the amplitudes of some of the harmonics of the sound are increased with respect to others. When we change the shape of our mouth, harmonics of different frequencies are given preference. These effects account for the difference between an “e–e–e” sound and an “a–a–a” sound.

We all know that a particular vowel sound—say “e–e–e”—still
“sounds like” the same vowel whether we say (or sing) it at a high
or a low pitch. From the mechanism we describe, we would expect that
*particular* frequencies are emphasized when we shape our mouth
for an “e–e–e,” and that they *do not* change as we change
the pitch of our voice. So the relation of the important harmonics to
the fundamental—that is, the “quality”—changes as we change
pitch. Apparently the mechanism by which we recognize speech is not
based on specific harmonic relationships.

What should we say now about Pythagoras’ discovery? We understand that
two similar strings with lengths in the ratio of $2$ to $3$ will have
fundamental frequencies in the ratio $3$ to $2$. But why should they
“sound pleasant” together? Perhaps we should take our clue from the
frequencies of the harmonics. The second harmonic of the lower shorter
string will have the *same* frequency as the third harmonic of the
longer string. (It is easy to show—or to believe—that a plucked
string produces strongly the several lowest harmonics.)

Perhaps we should make the following rules. Notes sound consonant when
they have harmonics with the same frequency. Notes sound dissonant if
their upper harmonics have frequencies near to each other but far
enough apart that there are rapid beats between the two. Why beats do
not sound pleasant, and why unison of the upper harmonics does sound
pleasant, is something that we do not know how to define or
describe. We cannot say from this knowledge of what *sounds*
good, what ought, for example, to *smell* good. In other words,
our understanding of it is not anything more general than the
statement that when they are in unison they sound good. It does not
permit us to deduce anything more than the properties of concordance
in music.

It is easy to check on the harmonic relationships we have described by
some simple experiments with a piano. Let us label the $3$ successive
C’s near the middle of the keyboard by C, C$'$, and C$''$, and the G’s
just above by G, G$'$, and G$''$. Then the fundamentals will have
relative frequencies as follows:
\begin{alignat*}{4}
&\text{C}&&–2&&\quad
\text{G}&&–\phantom{1}3\\[1ex]
&\text{C}'&&–4&&\quad
\text{G}'&&–\phantom{1}6\\[1ex]
&\text{C}''&&–8&&\quad
\text{G}''&&–12
\end{alignat*}
These harmonic relationships can be demonstrated in the following way:
Suppose we press C$'$ *slowly*—so that it does not sound but we
cause the damper to be lifted. If we then sound C, it will produce its
own fundamental *and* some second harmonic. The second harmonic
will set the strings of C$'$ into vibration. If we now release C
(keeping C$'$ pressed) the damper will stop the vibration of the C
strings, and we can hear (softly) the note C$'$ as it dies away. In a
similar way, the third harmonic of C can cause a vibration of G$'$. Or
the sixth of C (now getting much weaker) can set up a vibration in the
fundamental of G$''$.

A somewhat different result is obtained if we press G quietly and then
sound C$'$. The third harmonic of C$'$ will correspond to the fourth
harmonic of G, so *only* the fourth harmonic of G will be
excited. We can hear (if we listen closely) the sound of G$''$, which
is two octaves above the G we have pressed! It is easy to think up
many more combinations for this game.

We may remark in passing that the major scale can be defined just by
the condition that the three major chords (F–A–C); (C–E–G); and
(G–B–D) *each* represent tone sequences with the frequency
ratio $(4:5:6)$. These ratios—plus the fact that an octave (C–C$'$,
B–B$'$, etc.) has the ratio $1:2$—determine the whole scale for
the “ideal” case, or for what is called “just intonation.”
Keyboard instruments like the piano are *not* usually tuned in
this manner, but a little “fudging” is done so that the frequencies
are *approximately* correct for all possible starting tones. For
this tuning, which is called “tempered,” the octave (still $1:2$) is
divided into $12$ equal intervals for which the frequency ratio
is $(2)^{1/12}$. A fifth no longer has the frequency ratio $3/2$, but
$2^{7/12} = 1.499$, which is apparently close enough for most ears.

We have stated a rule for consonance in terms of the coincidence of
harmonics. Is this coincidence perhaps the *reason* that two
notes are consonant? One worker has claimed that two *pure*
tones—tones carefully manufactured to be free of harmonics—do not
give the *sensations* of consonance or dissonance as the relative
frequencies are placed at or near the expected ratios. (Such
experiments are difficult because it is difficult to manufacture pure
tones, for reasons that we shall see later.) We cannot still be
certain whether the ear is matching harmonics or doing arithmetic when
we decide that we like a sound.

### 50–4The Fourier coefficients

Let us return now to the idea that any note—that is, a
*periodic* sound—can be represented by a suitable combination
of harmonics. We would like to show how we can find out what amount of
each harmonic is required. It is, of course, easy to compute $f(t)$,
using Eq. (50.2), if we are *given* all the
coefficients $a$ and $b$. The question now is, if we are given $f(t)$
how can we know what the coefficients of the various harmonic terms
should be? (It is easy to make a cake from a recipe; but can we write
down the recipe if we are given a cake?)

Fourier discovered that it was not really very difficult. The term $a_0$ is certainly easy. We have already said that it is just the average value of $f(t)$ over one period (from $t = 0$ to $t = T$). We can easily see that this is indeed so. The average value of a sine or cosine function over one period is zero. Over two, or three, or any whole number of periods, it is also zero. So the average value of all of the terms on the right-hand side of Eq. (50.2) is zero, except for $a_0$. (Recall that we must choose $\omega = 2\pi/T$.)

Now the average of a sum is the sum of the averages. So the average
of $f(t)$ is just the average of $a_0$. But $a_0$ is a *constant*, so
its average is just the same as its value. Recalling the definition of
an average, we have
\begin{equation}
\label{Eq:I:50:3}
a_0 = \frac{1}{T}\int_0^Tf(t)\,dt.
\end{equation}

The other coefficients are only a little more difficult. To find them we
can use a trick discovered by
Fourier.
Suppose we multiply both sides of Eq. (50.2)
by some harmonic function—say by $\cos7\omega t$. We have then
\begin{alignat}{2}
f(t)\cdot\cos7\omega t &= a_0\cdot\cos7\omega t\notag\\[.5ex]
&\quad+\;a_1\cos\hphantom{1}\omega t\cdot\cos7\omega t &&\;+\;
b_1\sin\hphantom{1}\omega t\cdot\cos7\omega t\notag\\[.65ex]
&\quad+\;a_2\cos2\omega t\cdot\cos7\omega t &&\;+\;
b_2\sin2\omega t\cdot\cos7\omega t\notag\\[.65ex]
&\quad+\;\dotsb &&\;+\; \dotsb\notag\\[.65ex]
&\quad+\;a_7\cos7\omega t\cdot\cos7\omega t &&\;+\;
b_7\sin7\omega t\cdot\cos7\omega t\notag\\[.5ex]
\label{Eq:I:50:4}
&\quad+\;\dotsb &&\;+\; \dotsb
\end{alignat}
\begin{alignat}{2}
f(t)\cdot\cos7\omega t &= a_0\cdot\cos7\omega t\notag\\[.75ex]
&\quad+\;a_1\cos\omega t \cdot\cos7\omega t\notag\\
&\qquad\qquad+\;b_1\sin\omega t\cdot\cos7\omega t&\notag\\[.75ex]
&\quad+\;a_2\cos2\omega t\cdot\cos7\omega t \notag\\
&\qquad\qquad+\;b_2\sin2\omega t\cdot\cos7\omega t\notag\\[1ex]
&\quad+\quad\dotsb\notag\\[2ex]
&\quad+\;a_7\cos7\omega t\cdot\cos7\omega t\notag\\
&\qquad\qquad+\;b_7\sin7\omega t\cdot\cos7\omega t\notag\\[.75ex]
\label{Eq:I:50:4}
&\quad+\quad\dotsb
\end{alignat}
*Now* let us average both sides. The average of $a_0\cos7\omega
t$ over the time $T$ is proportional to the average of a cosine over
$7$ whole periods. But that is just zero. The average of *almost
all* of the rest of the terms is *also* zero. Let us look at the
$a_1$ term. We know, in general, that
\begin{equation}
\label{Eq:I:50:5}
\cos A\cos B = \tfrac{1}{2}\cos\,(A + B) + \tfrac{1}{2}\cos\,(A - B).
\end{equation}
The $a_1$ term becomes
\begin{equation}
\label{Eq:I:50:6}
\tfrac{1}{2}a_1(\cos8\omega t + \cos6\omega t).
\end{equation}
We thus have two cosine terms, one with $8$ full periods in $T$ and
the other with $6$. *They both average to zero*. The average of
the $a_1$ term is therefore zero.

For the $a_2$ term, we would find $a_2\cos9\omega t$
and $a_2\cos5\omega t$, each of which also averages to zero. For the
$a_9$ term, we would find $\cos16\omega t$ and $\cos\,(-2\omega t)$. But
$\cos\,(-2\omega t)$ is the same as $\cos2\omega t$, so both of these
have zero averages. It is clear that *all* of the $a$ terms will
have a zero average *except* one. And that one is the
$a_7$ term. For this one we have
\begin{equation}
\label{Eq:I:50:7}
\tfrac{1}{2}a_7(\cos14\omega t + \cos0).
\end{equation}
The cosine of zero is one, and its average, of course, is one. So we
have the result that the average of all of the $a$ terms of
Eq. (50.4) equals $\tfrac{1}{2}a_7$.

The $b$ terms are even easier. When we multiply by any cosine term
like $\cos n\omega t$, we can show by the same method that *all*
of the $b$ terms have the average value zero.

We see that Fourier’s “trick” has acted like a sieve. When we multiply by $\cos7\omega t$ and average, all terms drop out except $a_7$, and we find that \begin{equation} \label{Eq:I:50:8} \operatorname{Average}\,[f(t)\cdot\cos7\omega t]=a_7/2, \end{equation} or \begin{equation} \label{Eq:I:50:9} a_7 = \frac{2}{T}\int_0^Tf(t)\cdot\cos7\omega t\,dt. \end{equation}

We shall leave it for the reader to show that the coefficient $b_7$ can be obtained by multiplying Eq. (50.2) by $\sin7\omega t$ and averaging both sides. The result is \begin{equation} \label{Eq:I:50:10} b_7 = \frac{2}{T}\int_0^Tf(t)\cdot\sin7\omega t\,dt. \end{equation}

Now what is true for $7$ we expect is true for any integer. So we can summarize our proof and result in the following more elegant mathematical form. If $m$ and $n$ are integers other than zero, and if $\omega = 2\pi/T$, then \begin{align} \label{Eq:I:50:11} &\text{I.}\quad \int_0^T\sin n\omega t\cos m\omega t\,dt = 0.\\[1ex] % ebook break &\left.\hspace{-2mm} \begin{alignedat}{3} &\text{II.}\quad \int_0^T\cos n\omega t \cos m\omega t\,dt ={}\\[1ex] &\text{III.}\quad \int_0^T\sin n\omega t \sin m\omega t\,dt ={} \end{alignedat} \label{Eq:I:50:12} \right\}\; \begin{cases} 0 & \kern{-1ex}\text{if $n \neq m$}.\\[1ex] T/2 & \kern{-1ex}\text{if $n = m$}. \end{cases}\\[1ex] % ebook break \label{Eq:I:50:13} &\text{IV.}\quad f(t) = a_0 + \sum_{n = 1}^\infty a_n\cos n\omega t + \sum_{n = 1}^\infty b_n\sin n\omega t.\\[1ex] \label{Eq:I:50:14} &\text{V.}\quad a_0 = \frac{1}{T}\int_0^Tf(t)\,dt.\\[1.5ex] \label{Eq:I:50:15} &\text{}\qquad a_n = \frac{2}{T}\int_0^Tf(t)\cdot\cos n\omega t\,dt.\\[1.5ex] \label{Eq:I:50:16} &\text{}\qquad b_n = \frac{2}{T}\int_0^Tf(t)\cdot\sin n\omega t\,dt. \end{align} \begin{align} \label{Eq:I:50:11} &\text{I.}\; \int_0^T\!\sin n\omega t\cos m\omega t\,dt = 0.\\[1ex] % ebook break &\left.\hspace{-2mm} \begin{alignedat}{3} &\text{II.}\; \int_0^T\!\cos n\omega t \cos m\omega t\,dt ={}\\[1ex] &\text{III.}\; \int_0^T\!\sin n\omega t \sin m\omega t\,dt ={} \end{alignedat} \right\}\notag\\ \label{Eq:I:50:12} &\kern{6em}\begin{cases} 0 & \kern{-1ex}\text{if $n \neq m$}.\\[1ex] T/2 & \kern{-1ex}\text{if $n = m$}. \end{cases}\\[1ex] % ebook break &\text{IV.}\; f(t) = a_0 + \sum_{n = 1}^\infty a_n\cos n\omega t\notag\\ \label{Eq:I:50:13} &\kern{5.5em}+\sum_{n = 1}^\infty b_n\sin n\omega t.\\[1ex] \label{Eq:I:50:14} &\text{V.}\; a_0 = \frac{1}{T}\int_0^T\!f(t)\,dt.\\[1.5ex] \label{Eq:I:50:15} &\text{}\quad a_n = \frac{2}{T}\int_0^T\!f(t)\cdot\cos n\omega t\,dt.\\[1.5ex] \label{Eq:I:50:16} &\text{}\quad b_n = \frac{2}{T}\int_0^T\!f(t)\cdot\sin n\omega t\,dt. \end{align}

In earlier chapters it was convenient to use the exponential notation for representing simple harmonic motion. Instead of $\cos\omega t$ we used $\FLPRe e^{i\omega t}$, the real part of the exponential function. We have used cosine and sine functions in this chapter because it made the derivations perhaps a little clearer. Our final result of Eq. (50.13) can, however, be written in the compact form \begin{equation} \label{Eq:I:50:17} f(t) = \FLPRe\sum_{n = 0}^\infty\hat{a}_ne^{in\omega t}, \end{equation} where $\hat{a}_n$ is the complex number $a_n - ib_n$ (with $b_0 = 0$). If we wish to use the same notation throughout, we can write also \begin{equation} \label{Eq:I:50:18} \hat{a}_n = \frac{2}{T}\int_0^Tf(t)e^{-in\omega t}\,dt\quad (n \geq 1). \end{equation}

We now know how to “analyze” a periodic wave into its harmonic
components. The procedure is called *Fourier
analysis*, and the separate terms are called
Fourier components. We have *not*
shown, however, that once we find all of the Fourier
components and add them together, we do
indeed get back our $f(t)$. The mathematicians have shown, for a wide
class of functions, in fact for all that are of interest to
physicists, that if we can do the integrals we will get back
$f(t)$. There is one minor exception. If the function $f(t)$ is
discontinuous, i.e., if it jumps suddenly from one value to another,
the Fourier sum will give a value at the
breakpoint halfway between the upper and lower values at the
discontinuity. So if we have the strange function $f(t) = 0$, $0 \leq
t < t_0$, and $f(t) = 1$ for $t_0 \leq t \leq T$, the Fourier
sum will give the right value everywhere
*except at* $t_0$, where it will have the value $\tfrac{1}{2}$
instead of $1$. It is rather unphysical anyway to insist that a
function should be zero *up to* $t_0$, but $1$ *right at*
$t_0$. So perhaps we should make the “rule” for physicists that any
discontinuous function (which can only be a simplification of a
*real* physical function) should be defined with halfway values
at the discontinuities. Then any such function—with any finite
number of such jumps—as well as all other physically interesting
functions, are given correctly by the Fourier sum.

As an exercise, we suggest that the reader determine the Fourier series for the function shown in Fig. 50–3. Since the function cannot be written in an explicit algebraic form, you will not be able to do the integrals from zero to $T$ in the usual way. The integrals are easy, however, if we separate them into two parts: the integral from zero to $T/2$ (over which $f(t) = 1$) and the integral from $T/2$ to $T$ (over which $f(t) = -1$). The result should be \begin{equation} \label{Eq:I:50:19} f(t) = \frac{4}{\pi}(\sin\omega t + \tfrac{1}{3}\sin3\omega t + \tfrac{1}{5}\sin5\omega t + \dotsb), \end{equation} where $\omega = 2\pi/T$. We thus find that our square wave (with the particular phase chosen) has only odd harmonics, and their amplitudes are in inverse proportion to their frequencies.

Let us check that Eq. (50.19) does indeed give us back
$f(t)$ for some value of $t$. Let us choose $t = T/4$, or $\omega t =
\pi/2$. We have
\begin{align}
\label{Eq:I:50:20}
f(t) &= \frac{4}{\pi}\biggl(\sin\frac{\pi}{2} +
\frac{1}{3}\sin\frac{3\pi}{2} + \frac{1}{5}\sin\frac{5\pi}{2} +
\dotsb\biggr)\\[1.5ex]
\label{Eq:I:50:21}
&= \frac{4}{\pi}\biggl(1 - \frac{1}{3} + \frac{1}{5} -
\frac{1}{7}\pm\dotsb\biggr).
\end{align}
The series^{1} has the value $\pi/4$, and we find that $f(t) = 1$.

### 50–5The energy theorem

The energy in a wave is proportional to the square of its amplitude. For a wave of complex shape, the energy in one period will be proportional to $\int_0^Tf^2(t)\,dt$. We can also relate this energy to the Fourier coefficients. We write \begin{equation} \label{Eq:I:50:22} \int_0^Tf^2(t)\,dt = \int_0^T\biggl[a_0 + \sum_{n = 1}^\infty a_n\cos n\omega t + \sum_{n = 1}^\infty b_n\sin n\omega t\biggr]^2\,dt. \end{equation} \begin{gather} \label{Eq:I:50:22} \int_0^Tf^2(t)\,dt =\\ \int_0^T\biggl[a_0 + \sum_{n = 1}^\infty a_n\cos n\omega t + \sum_{n = 1}^\infty b_n\sin n\omega t\biggr]^2\,dt.\notag \end{gather} When we expand the square of the bracketed term we will get all possible cross terms, such as $a_5\cos5\omega t\cdot a_7\cos7\omega t$ and $a_5\cos5\omega t\cdot b_7\sin7\omega t$. We have shown above, however, [Eqs. (50.11) and (50.12)] that the integrals of all such terms over one period is zero. We have left only the square terms like $a_5^2\cos^2 5\omega t$. The integral of any cosine squared or sine squared over one period is equal to $T/2$, so we get \begin{align} \int_0^Tf^2(t)\,dt &= Ta_0^2 + \frac{T}{2}\, (a_1^2 + a_2^2 + \dotsb + b_1^2 + b_2^2 + \dotsb)\notag\\[.5ex] \label{Eq:I:50:23} &= Ta_0^2 + \frac{T}{2}\sum_{n = 1}^\infty(a_n^2 + b_n^2). \end{align} \begin{gather} \label{Eq:I:50:23} \int_0^Tf^2(t)\,dt =\\[.5ex] Ta_0^2 + \frac{T}{2}\,(a_1^2 + a_2^2 + \dotsb + b_1^2 + b_2^2 + \dotsb)= \notag\\[.5ex] Ta_0^2 + \frac{T}{2}\sum_{n = 1}^\infty(a_n^2 + b_n^2).\notag \end{gather} This equation is called the “energy theorem,” and says that the total energy in a wave is just the sum of the energies in all of the Fourier components. For example, applying this theorem to the series (50.19), since $[f(t)]^2 = 1$ we get \begin{equation*} T = \frac{T}{2}\cdot\biggl(\frac{4}{\pi}\biggr)^2\biggl( 1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \dotsb\biggr), \end{equation*} so we learn that the sum of the squares of the reciprocals of the odd integers is $\pi^2/8$. In a similar way, by first obtaining the Fourier series for the function $f(t)=(t-T/2)^2$ and using the energy theorem, we can prove that $1 + 1/2^4 + 1/3^4 + \dotsb$ is $\pi^4/90$, a result we needed in Chapter 45.

### 50–6Nonlinear responses

Finally, in the theory of harmonics there is an important phenomenon which should be remarked upon because of its practical importance—that of nonlinear effects. In all the systems that we have been considering so far, we have supposed that everything was linear, that the responses to forces, say the displacements or the accelerations, were always proportional to the forces. Or that the currents in the circuits were proportional to the voltages, and so on. We now wish to consider cases where there is not a strict proportionality. We think, at the moment, of some device in which the response, which we will call $x_{\text{out}}$ at the time $t$, is determined by the input $x_{\text{in}}$ at the time $t$. For example, $x_{\text{in}}$ might be the force and $x_{\text{out}}$ might be the displacement. Or $x_{\text{in}}$ might be the current and $x_{\text{out}}$ the voltage. If the device is linear, we would have \begin{equation} \label{Eq:I:50:24} x_{\text{out}}(t) = Kx_{\text{in}}(t), \end{equation} where $K$ is a constant independent of $t$ and of $x_{\text{in}}$. Suppose, however, that the device is nearly, but not exactly, linear, so that we can write \begin{equation} \label{Eq:I:50:25} x_{\text{out}}(t) = K[x_{\text{in}}(t) + \epsilon x_{\text{in}}^2(t)], \end{equation} where $\epsilon$ is small in comparison with unity. Such linear and nonlinear responses are shown in the graphs of Fig. 50–4.

Nonlinear responses have several important practical consequences. We
shall discuss some of them now. First we consider what happens if we
apply a pure tone at the input. We let $x_{\text{in}} = \cos\omega
t$. If we plot $x_{\text{out}}$ as a function of time we get the solid
curve shown in Fig. 50–5. The dashed curve gives, for
comparison, the response of a linear system. We see that the output is
no longer a cosine function. It is more peaked at the top and flatter
at the bottom. We say that the output is *distorted*. We know,
however, that such a wave is no longer a pure tone, that it will have
harmonics. We can find what the harmonics are. Using $x_{\text{in}} =
\cos\omega t$ with Eq. (50.25), we have
\begin{equation}
\label{Eq:I:50:26}
x_{\text{out}}(t) = K(\cos\omega t + \epsilon\cos^2\omega t).
\end{equation}
From the equality $\cos^2\theta = \tfrac{1}{2}(1 + \cos2\theta)$, we
have
\begin{equation}
\label{Eq:I:50:27}
x_{\text{out}}(t) = K\Bigl(\cos\omega t + \frac{\epsilon}{2} +
\frac{\epsilon}{2}\cos2\omega t\Bigr).
\end{equation}
The output has not only a component at the fundamental frequency, that
was present at the input, but also has some of its second
harmonic. There has also appeared at the output a constant
term $K(\epsilon/2)$, which corresponds to the shift of the average value,
shown in Fig. 50–5. The process of producing a shift of
the average value is called *rectification*.

A nonlinear response will rectify and will produce harmonics of the frequencies at its input. Although the nonlinearity we assumed produced only second harmonics, nonlinearities of higher order—those which have terms like $x_{\text{in}}^3$ and $x_{\text{in}}^4$, for example—will produce harmonics higher than the second.

Another effect which results from a nonlinear response is
*modulation*. If our input function contains two (or more) pure
tones, the output will have not only their harmonics, but still other
frequency components. Let $x_{\text{in}} = A\cos\omega_1t +
B\cos\omega_2t$, where now $\omega_1$ and $\omega_2$ are *not*
intended to be in a harmonic relation. In addition to the linear term
(which is $K$ times the input) we shall have a component in the output
given by
\begin{align}
\label{Eq:I:50:28}
x_{\text{out}} &= K\epsilon(A\cos\omega_1t + B\cos\omega_2t)^2\\[.5ex]
\label{Eq:I:50:29}
&= K\epsilon(A^2\cos^2\omega_1t + B^2\cos^2\omega_2t +
2AB\cos\omega_1t\cos\omega_2t).
\end{align}
\begin{align}
\label{Eq:I:50:28}
x_{\text{out}} &= K\epsilon(A\cos\omega_1t + B\,\cos\omega_2t)^2&\\[1ex]
&= K\epsilon(A^2\cos^2\omega_1t + B^2\cos^2\omega_2t \notag\\[.5ex]
\label{Eq:I:50:29}
&\phantom{=K\epsilon}+\,2AB\cos\omega_1t\cos\omega_2t).
\end{align}
The first two terms in the parentheses of Eq. (50.29) are
just those which gave the constant terms and second harmonic terms we
found above. The last term is new.

We can look at this new “cross term” $AB\cos\omega_1t\cos\omega_2t$
in two ways. First, if the two frequencies are widely different (for
example, if $\omega_1$ is much greater than $\omega_2$) we can
consider that the cross term represents a cosine oscillation of
varying amplitude. That is, we can think of the factors in this way:
\begin{equation}
\label{Eq:I:50:30}
AB\cos\omega_1t\cos\omega_2t = C(t)\cos\omega_1t,
\end{equation}
with
\begin{equation}
\label{Eq:I:50:31}
C(t)=AB\cos\omega_2t.
\end{equation}
We say that the amplitude of $\cos\omega_1t$ is *modulated* with the
frequency $\omega_2$.

Alternatively, we can write the cross term in another way:
\begin{equation}
\label{Eq:I:50:32}
AB\cos\omega_1t\cos\omega_2t =
\frac{AB}{2}\,[\cos\,(\omega_1 + \omega_2)t +
\cos\,(\omega_1 - \omega_2)t].
\end{equation}
\begin{gather}
\label{Eq:I:50:32}
AB\cos\omega_1t\cos\omega_2t =\\
\frac{AB}{2}\,[\cos\,(\omega_1 + \omega_2)t +
\cos\,(\omega_1 - \omega_2)t].\notag
\end{gather}
We would now say that two *new* components have been produced,
one at the *sum* frequency $(\omega_1 + \omega_2)$, another at
the *difference* frequency $(\omega_1 - \omega_2)$.

We have two different, but equivalent, ways of looking at the same
result. In the special case that $\omega_1 \gg \omega_2$, we can
relate these two different views by remarking that since $(\omega_1 +
\omega_2)$ and $(\omega_1 - \omega_2)$ are near to each other we would
expect to observe beats between them. But these beats have just the
effect of *modulating* the amplitude of the *average*
frequency $\omega_1$ by one-half the difference
frequency $2\omega_2$. We see, then, why the two descriptions are equivalent.

In summary, we have found that a nonlinear response produces several effects: rectification, generation of harmonics, and modulation, or the generation of components with sum and difference frequencies.

We should notice that all these effects (Eq. 50.29) are
proportional not only to the nonlinearity coefficient $\epsilon$, but
also to the product of two amplitudes—either $A^2$, $B^2$,
or $AB$. We expect these effects to be much more important for
*strong* signals than for weak ones.

The effects we have been describing have many practical
applications. First, with regard to sound, it is believed that the ear
is nonlinear. This is believed to account for the fact that with loud
sounds we have the sensation that we *hear* harmonics and also
sum and difference frequencies even if the sound waves contain only
pure tones.

The components which are used in sound-reproducing
equipment—amplifiers, loudspeakers, etc.—always have some
nonlinearity. They produce distortions in the sound—they generate
harmonics, etc.—which were not present in the original sound. These
new components are heard by the ear and are apparently
objectionable. It is for this reason that “Hi-Fi” equipment is
designed to be as linear as possible. (Why the nonlinearities of the
*ear* are *not* “objectionable” in the same way, or how we
even know that the nonlinearity is in the *loudspeaker* rather
than in the *ear* is not clear!)

Nonlinearities are quite *necessary*, and are, in fact,
intentionally made large in certain parts of radio transmitting and
receiving equipment. In an am transmitter the “voice”
signal (with frequencies of some kilocycles per second) is combined
with the “carrier” signal (with a frequency of some megacycles per
second) in a nonlinear circuit called a *modulator*, to produce
the modulated oscillation that is transmitted. In the receiver, the
components of the received signal are fed to a nonlinear circuit which
combines the sum and difference frequencies of the modulated carrier
to generate again the voice signal.

When we discussed the transmission of light, we assumed that the induced oscillations of charges were proportional to the electric field of the light—that the response was linear. That is indeed a very good approximation. It is only within the last few years that light sources have been devised (lasers) which produce an intensity of light strong enough so that nonlinear effects can be observed. It is now possible to generate harmonics of light frequencies. When a strong red light passes through a piece of glass, a little bit of blue light—second harmonic—comes out!

- The series can be evaluated in the following way. First we remark that $\int_0^x[dx/(1 + x^2)] = \tan^{-1} x$. Second, we expand the integrand in a series $1/(1 + x^2) = 1 - x^2 + x^4 - x^6 \pm\dotsb$ We integrate the series term by term (from zero to $x$) to obtain $\tan^{-1} x = x - x^3/3 + x^5/5 - x^7/7 \pm\dotsb$ Setting $x = 1$, we have the stated result, since $\tan^{-1}1 = \pi/4$. ↩