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Unformatted text preview: 50 Harmonics 501 Musical tones Pythagoras is said to have discovered the fact that two similar strings under
the same tension and diﬂering only in length, when sounded together give an
effect that is pleasant to the car 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 in
terval 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’ dis
covery 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 mathe
matical 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 oﬁ" 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 presentday 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 moreorless 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 50—1 50—1 Musical tones 50—2 The Fourier series 50—3 Quality and consonance
50—4 The Fourier coefﬁcients
505 The energy theorem 50—6 Nonlinear responses PRESSURE (a) A NOISE PRESSU R E TIME
I‘ T —‘I
(b) A MUSICAL TONE Fig. 50—1. Pressure as a function of
time for (a) a noise, and (b) a musical
tone. 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. 50l(b). Musicians will usually speak of a musical tone in terms of three character
istics: 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 re
quired 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—2 The 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 coo, 2w0,
3mg, . . . The most general motion of a plucked string, therefore, is composed of
the sum of a sinusoidal oscillation at the fundamental frequency (.00, another at the
second harmonic frequency Zwo, another at the third harmonic 3020, etc. Now the
fundamental mode repeats itself every period T1 = 21r/w0. The second harmonic
mode repeats itself every T2 = 27r/2w0. It also repeats itself every T1 = 2T2,
after two of its periods. Similarly, the third harmonic mode repeats itself after a
time T1 which is 3 of its periods. We see again why a plucked string repeats its
whole pattern with a periodicity of T1. 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 cot—for each of the various
harmonic frequencies. If the period of the vibration is T, the fundamental angular
frequency will be w = 27r/T, and the harmonics will be 2w, 3w, 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 (wt + ¢). It is, however, simpler to use instead
both the sine and cosine functions for each frequency. We recall that cos (wt + ¢) = (cos ¢ cos wt — sin :1» sin at) (50.1)
50—2 and since 45 is a constant, any sinusoidal oscillation at the frequency to can be
written as the sum of a term with cos wt and another term with sin wt. We conclude, then, that any function f(t) that is periodic with the period T
can be written mathematically as N) = ao
+ a1 cos wt + b1 sinwt
+ a2 cos 2wt + b2 sinZwt
+ a3 cos 3wt + b3 sin 3wt +... +. where w = 21r/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 “zerofrequency” term do 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, an and b", 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. (50.2) 50—3 Quality 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 pro
portion of harmonics than does an oboe. We can “manufacture” various musical tones if we connect several “oscilla
tors” to a loudspeaker. (An oscillator usually produces a nearly pure simple
harmonic function.) We should choose the frequencies of the oscillators to be 0),
2w, 3w, 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 oscilla
tor 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 ampli
tudes 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. 503 ml
7 t
r + ‘1'.va +°"er(
+ etc. + etc. Fig. 50—2. Any periodic function f“)
is equal to a sum of simple harmonic
functions. 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 funda
mental—that is, the “quality”—changcs as we change pitch. Apparently the mech
anism by which we recognize speech is not based on speciﬁc harmonic relation
ships. 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 fre
quencies 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 proper
ties 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: c —2 G — 3
0—4 G’— 6
C”—8 G”—12 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 in
struments 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)“ 12. A ﬁfth no longer
has the frequency ratio 3/2, but 27’ 12 = 1.499, which is apparently close enough
for most ears. 504 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 har
monics—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 arith
metic when we decide that we like a sound. 50—4 The 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 a0 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
righthand side of Eq. (50.2) is zero, except for a0. (Recall that we must choose
at = 27r/ T.) Now the average of a sum is the sum of the averages. So the average of f(t) is
just the average of a0. But a0 is a constant, so its average is just the same as its
value. Recalling the deﬁnition of an average, we have 1 T
a0 = T/O f(t) dt. (50.3) 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 cos 7wt. We have then f(t)  cos 7wt = a0  cos 7wt
+ a1 cos wt  cos 7w! + b1 sinwt  cos 7wt
+ a2 cos Zwt  cos 7wt + b2 sin 2wt  cos 7wt +... +...
+ a7 cos 7wt‘cos 7wt + b7 sin 7wtcos 7wt
+"' + (50.4) Now let us average both sides. The average of a0 cos 7wt over the time T is pro
portional 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
al term. We know, in general, that cos A cos B = % cos (A + B) + %cos (A — B). (50.5) The a1 term becomes
%a1(cos 8w! + cos 6wt). (50.6) 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 al term is therefore zero. For the a 2 term, we would ﬁnd a 2 cos 9w! and a 2 cos 5wt, each of which also
averages to zero. For the ag term, we would ﬁnd cos l6wt and cos (—2wt). But
cos (—2wt) is the same as cos Zwt, so both of these have zero averages. It is clear 50—5 that all of the a terms will have a zero average except one. And that one is the
a7 term. For this one we have §a7(cos l4wt + cos 0). (50.7) 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 $07.
The b terms are even easier. When we multiply by any cosine term like cos mot,
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
cos 7wt and average, all terms drop out except a7, and we ﬁnd that Average [f(t) ~ cos 7wt] = a7/2, (50.8)
or T
(17 = % f0 f(t)cos7wtdt. (50.9) We shall leave it for the reader to show that the coefﬁcient b7 can be obtained
by multiplying Eq. (50.2) by sin 7wt and averaging both sides. The result is T
b7 = %/o f(t)  sin 7wtdt. (50.10) 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 0: = 21r/T, then T
I. / sin nwt cos mwt dt = O. (50.11)
0
T
11. / cos mot cos mwtdt = 
0 O 1fn 75 m. (50.12)
T T/Z ifn = m.
111. / sin mot sin mwt dt =
0
1v. f(t) = a0 + 2 an cos mot + E b. sin nwt. (50.13)
n=1 n=1
1 T
V. 00 = T] f(t)  dt. (50.14)
0
2 T
an = T/ f(t)cos nwt dt. (50.15)
0
2 T
bu = T / f(t)  sin nwtdt. (50.16)
0 In earlier chapters it was convenient to use the exponential notation for repre
senting simple harmonic motion. Instead of cos wt we used Re ei‘“, 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 f(t) = Re i anew”, (50.17) n=0 where 21,. is the complex number an — ibn (with b0 = 0). If we wish to use the
same notation throughout, we can write also 2 T
an = 7/0 ﬁneWart (n 2 1). (50.18)
50—6 We now know how to “analyze” a periodic wave into its harmonic compon
ents. 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 j umps 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 S t < to, and f(t) = lfor to g t S T, the Fourier
sum will give the right value everywhere except at to, where it will have the value %
instead of 1. It is rather unphysical anyway to insist that a function should be
zero up to to, but 1 right at to. 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 Tin the usual
way. The integrals are easy, however, if we separate them into two parts: the
integral from zero to T /2 (over which ﬁt) = l) and the integral from T /2 to T
(over which f(t) = — l). The result should be f(t) = % (sin cot + §sin 3m + g sin 5w)? + .), (50.19) where w = 27r/ 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 I.
Let us choose t = T/4, or col = 7r/2. We have 4 .7r 1.31rl . 571'
f(t) — ;(Sln§+§SIn—2—5—Sln7+ ) 4 1 1 1
_7r(1_§+§_7+..> (50.21) The series* has the value 7r/4, and we ﬁnd that f(t) = 1. 505 The 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 f: f 2(t) dt.
We can also relate this energy to the Fourier coefﬁcients. We write T T 00 no 2
f 1%) d; = / [a0 + 2 an cos nwt + 2b,, sin mot] dt. (50.22)
0 0 n=1 n=l When we expand the square of the bracketed term we will get all possible cross
terms, such as a5 cos 5001‘ b7 cos 7wt. We have shown above, however, [Eqs.
(50.11) and (50.12)] that the integrals of all such terms over one period is zero. 1* The series can be evaluated in the following way. First we remark that 0 [dx/(l + x2)] = tan—1x. Second, we expand the integrand inaseries 1/0 + x2) =
l — x2 + x4 — x6 + . . . We integrate the series term by term (from zero to x) to
obtain tan”1 x = l — x3/3 + x5/5 — x7/7 + .. . Settingx = l,we have the stated
result, since tan‘1 1 = 1r/4. 507 fm Fig. 50—3. ﬁt)
f(f) [I [I function. Squarewove
+1 forO <1 < i/2,
—1forT/2 < t < T. hut
xin
lo) LINEAR m NONLINEAR
xwﬁ KxhI XM leln+ Elf")
Fig. 50—4. Linear and nonlinear re
sponses.
1am NONLINEAR Fig. 50—5. The response of a non
linear device to the input cos wt. A
linear response is shown for comparison. We have left only the square terms like a§ cos2 Scot. The integral of any cosine
squared or sine squared over one period is equal to T/2, so we get T
f0f2(t)dt=Ta3+§(ai+a§++b¥+b§+) 2 T w 2 2
Ta0 + E n; (a. + bn). (50.23)
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 [)‘(t)]2 = l we get T_ I.(ﬂ>2(1+L+L+L...)
_ 2 1r 32 52 72 ’
so we learn that the sum of the squares of the reciprocals of the odd integers is
1r2/ 8. In a similar way, by ﬁrst obtaining the Fourier series for the function and using the energy theorem, we can prove that l + 1/24 + 1/34 + ~  ' is 7r4/90,
a result we needed in Chapter 45. 50—6 Nonlinear 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 x0“, at the time t, is determined
by the input xin at the time t. For example, xin might be the force and xout might
be the displacement. Or x1" might be the current and xout the voltage. If the device
is linear, we would have xouto) = KxinO): where K is a constant independent of t and of xi... Suppose, however, that the
device is nearly, but not exactly, linear, so that we can write xouta) = lem(t) + 6x340], where e 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 xm = cos wt. If we plot x0.“ 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
xin = cos cut with Eq. (50.25), we have (50.24) (50.25) x...t = K(coso.>t + e cos2wt). (50.26)
From the equality cos2 0 = %(l —— cos 26), we have
x0... = K(c05wt + g — ECOSZwt) . (50.27) 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 50—8 appeared at the output a constant term K(e/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 harmon
ics, nonlinearities of higher order—those which have terms like x3, and x3, 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 xin = A cos wlt +
Bcos wzt, where now w, and (.02 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 compo
nent in the output given by X0“ = K6 (A cos wlt + B cos (.0202 (50.28)
= Ke(A2 cos2 colt + B2 cos2 wzt + 2AB cos wlt cos w2t). (50.29) 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 colt cos wt in two ways. First,
if the two frequencies are widely diﬂ‘erent ‘(for example, if (.01 is much greater than
(.02) 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: AB cos wlt cos wgt = C(t) cos colt, (50.30)
with
C(t) = AB cos wzt. (50.31) We say that the amplitude of cos (.01 is modulated with the frequency (.02.
Alternatively, we can write the cross term in another way: AB cos wltcos (921? = A73 [cos (col + w2)t + cos (wl — w2)t]. (50.32) We would now say that two new components have been produced, one at the sum
frequency (wl + (.02), another at the diﬂerence frequency (001 — (.02). We have two different, but equivalent, ways of looking at the same result.
In the special case that an >> 602, we can relate these two different views by re
marking that since (wl + (.02) and (col — wz) 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 ml by onehalf the difference
frequency 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 6, but also to the product of two amplitudes—either
A2, 32, 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 soundreproducing 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 objection
able. It is for this reason that “HiFi” equipment is designed to be as linear as 50—9 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 trans
mitter 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 diﬂerence 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! 5010 ...
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 Spring '09
 LimKong
 Physics, Fourier Series, Periodic function

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