Physics 1240
Hall Chapter 8 Notes
1
Focus questions and learning goals
Chapter 8 focus questions:
1. Why do different instruments sound different even when they are playing the same note?
2. How is a sound spectrum physically & mathematically described?
Chapter 8 learning goals. After studying this chapter, you should be able to:
1. Understand the connection between harmonic number, pitch, and octave, so you can describe
what notes you are hearing when a string makes many tones at the same time.
2. Relate a sound waveform and the spectrum of frequencies that make it up. If you are given
a spectrum, describe how the resulting sound is perceived, and what the impact of various
changes like adding or subtracting various harmonics would have on the waveform. Which
alterations to a spectrum change the period? the pitch? the loudness?
3. Explain how you might build other waveforms (like square waves) up from different harmonics,
and describe qualitatively why and how a square wave sounds different from a sine wave with
the same period.
4. Define sine wave, and relate back to chapter 1 and 2. Define and distinguish sine, square,
pulse, and triangular waves.
2
Timbre and harmonics
We just learned that if you wiggle a string, only certain special frequencies, the harmonics, will
resonate. All other frequencies tend to die away almost instantly. So if you pluck a string, you will
hear the fundamental frequency f1 , and the higher harmonics (2f1 , 3f1 , 4f1 , 5f1 , and so on) all at
the same time, all superposed. If you look back to Chapter 2 (section 2.2, on waveforms), there
were pictures of complex waves that had some basic (fundamental) frequency, but didn’t look like
a simple sine wave.
Here’s one of the most remarkable things about waves (discovered by a mathematician named
Fourier): If you superpose a bunch of waves (all harmonics of a given fundamental frequency f1 )
you can generate any possible waveform that repeats itself at frequency f1 .
In other words, all those complex waveforms shown back in Chapter 2.2. of the text can be
thought of as arising from some particular simple sum of harmonics of the basic frequency f1 . A
given string will produce a particular characteristic sum of harmonics (e.g., the higher harmonics
might die away quickly in amplitude), and that will yield a “characteristic sound”, or timbre
associated with the string. You perceive the fundamental frequency f1 as the pitch, and all the
higher harmonics add in just to change the shape of that tone from the whiny, annoying pure sine
wave to a richer, fuller tone. The point is, if a string plays A440, you hear a pitch of 440 Hz, but
the character of the sound depends on how many (and which, and how strong) harmonics you add
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Physics 1240
Chapter 8 notes
in. If you don’t add in any, you hear a very “electronic” sound. If you add in higher harmonics in
just the right way, you can produce an A that sounds like a violin, or a saxophone, or....just about
anything! The difference is not in the frequency, but in the mix of higher harmonics!
By the way, if you add in random frequencies, that is, frequencies which are not an exact multiple
of the fundamental (something between f1 and 2f1 , for example) then you get anharmonic sound.
It can be interesting, some drums and bells have this kind of sound, it generally doesn’t have a very
clear pitch associated with it. It can also sound pretty bad, like noise. If you play the piano and
play two notes which are right next to each other, they clash, they are dissonant. In part, that’s
because the frequencies of one are not related in a simple, nice way, like harmonics are.
Here’s a note on one other cool thing about our hearing. If I play 300 Hz and 600 Hz together,
they sound nice together. The 600 Hz is a higher harmonic (n = 2) of the first. The interval is an
octave. If I play, say, 300 Hz and 900 Hz together, it sounds nice too. 900 Hz is not the same note
(it’s not an octave, or two octaves, but someplace in between), but because it’s a higher harmonic
(n = 3), they still sound very nice together. If I play 300 Hz and 450 Hz together, we again get
something nice. But in this case, 450 is not a harmonic at all. But both of these tones are harmonics
of a common, lower, fundamental (150 Hz, in this case: 300 is twice 150, and 450 is three times
150, this other fundamental). Now things are getting really interesting! You play 300 and 450 Hz,
and your brain sort of vaguely senses 150, which isn’t there but you “feel” its presence.
Why does your brain do this? Part of the reason is that 150 Hz signals repeat every 1/150 of
a second. So do 300 Hz signals! They repeat every 1/300 of a second, which means you’re also
back to where you started after 1/150 sec. Similarly, 450 Hz signals repeat every 1/450 sec, which
means you’re back to where you started (for the third time) after 1/150 sec. For some reason, we
seem to like hearing sounds that repeat themselves! If you play tones at 300 and 320 Hz together,
it doesn’t sound so nice. It’s fairly dissonant. (This is about like playing two notes right next to
each other on a piano. Kind of jarring). You could argue that these are still both harmonics of
a common frequency, but that common frequency is going to be 20 Hz, very far away from either
one.
Section 8.1 of the text starts off by showing you various wave forms in Figure 8.1, all of which
(except the last) have the exact same frequency. They all repeat themselves a little more than
three times in the time shown in the graph. The details are totally different, one is a sine wave,
but you can also see a square wave, a pulse wave, a triangular wave, and a sawtooth wave.
But they will all sound like the exact same frequency! They just have a different timbre. They are
each built up by superposing a bunch of different harmonics of that same (common) fundamental
frequency. The mathematics here are a little fierce, but the details don’t matter so much. We’ll
focus on the pictures and the concepts.
Simulation: Go to the web page in this footnote1 , or from the course web page under “Resources”
choose the PhET simulation “Fourier: making waves”. Using this simulation, you can build
up complicated waves by adding in different strengths of harmonics. I strongly suggest you
go play with this yourself!
None of the waves shown in Figure 8.1 of the text sound especially nice. They all sound artificial or
computer-like. A little harsh to my ears, anyway. Waves with rapid changes (“sharp edges” in the
waveform, like in Fig 8.1c) have lots of higher frequencies, lots of high harmonics. (Remember
1
http://phet.colorado.edu/simulations/sims.php?sim=Fourier Making Waves
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Chapter 8 notes
that these are also called overtones.) It’s a funny thing—you are adding up smooth waves, and
yet you can build up these sharp edges. The PhET Fourier sim will help you see how that works!
3
Sound spectra
When you add a bunch of harmonics, you need to be careful to note two things about every
frequency that you add in. First, the amplitude. That’s just the strength. So, e.g., if you start
with a pure sine wave of frequency f1 , and amplitude 1 (in some units, maybe you’re talking about
pressure waves so this is 1 N/m2 overpressure). Then you could add in a sine wave of frequency
2f1 , and you can choose any amplitude you want. If you have an amplitude much bigger than 1,
it will tend to dominate. If you add in this higher harmonic with amplitude much smaller than
1, it will subtly alter the timbre of the sound. In most natural instruments, this is how it goes
automatically—higher harmonics tend to have smaller and smaller amplitudes.
There is another thing you can alter, the phase of the wave. Remember that just means shifting
the sine wave left or right. So if two waves are in synch, in phase, they both start at zero at a
common time. But if you shift one of them, if you change its phase, then one wave might, for
example, be zero when the other is not. Phase changes makes for subtle shifts, which we’re not
going to pay too much more attention to. But for real sound analysis, you do have to be aware of
this too.
So if you want to describe a periodic wave, you should tell me the fundamental frequency, and
then tell me the amplitude of each of the harmonics you add in. We could represent this in a simple
graph, which is shown in a box at the bottom of Figure 8.4. You need to stare at it and make
sense of it! Each of the vertical bars in that graph, which is called the spectrum, represents the
strength of a harmonic. Going to the right on that graph is going up in frequency. The lines occur
only at discrete values of frequency: f1 , 2f1 , 3f1 , 4f1 , etc, the allowed harmonics. The height of
the line is just how much of that frequency you’re adding.
When you see a complex waveform, you can always think of it as a (unique!) spectrum. There
is one way (and only one way) to add all the harmonics to get to this wave. So the spectrum gives
you a quick and simple way of seeing how much of each harmonic is present. (What it doesn’t
show you is the phase, that makes life more complicated, so in this course we’re not bothering with
phase.)
Look carefully at figure 8.4. Look first at the upper left. That shows (from the top down) the
first 6 harmonics. It draws each one (above each other, just to separate them). So I see that the
fundamental (low frequency, long wavelength) is not very strong, it has a small amplitude. Now
go down below to the spectrum (it’s in the box, with an x-axis labeled f and a y-axis labeled dB
for strength). Notice that the very first vertical line is not so big, that represents a not-so-big
amplitude. Now look back at the original wiggles. The second one down has twice the frequency
of the fundamental, it wiggles 4 times for every two of the wave above it. It must be n = 2. And
it has a nice big amplitude. That’s shown, below in the box again, as the second line being quite
tall. And so on. Do you see how each wiggly graph represents the next harmonic, and the strength
of that wiggle is represented by the height of the line in the spectrum? If you can understand this
figure, you’ve gone a long way to making sense of interpreting the spectrum. The PhET Fourier
simulation will also help—the bar graph at the top is the spectrum! The spectrum is kind of a
fingerprint; it uniquely identifies and defines the timbre of a sound.
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3.1
Chapter 8 notes
Synthesis versus spectral analysis
When you “build up” a sound by adding up a bunch of harmonics, we might call this synthesis
or Fourier synthesis. Nowadays it’s quite simple to do this electronically, and so you can, in
principle, artificially generate any possible timbre at a given frequency. This is the power of
computer synthesized music. What’s remarkable is also how often it doesn’t sound all that nice,
unless you reproduce the pattern (the spectrum) of a real instrument!
You can go the other way: if you play music into a Fourier analyzer or spectrum analyzer,
you will get an output that looks very much like the graph at the bottom of Fig 8.4. You can find
software for computers that will do this for you. I found a free program called Visual Analyser for
PC (I’ll show it in class). Another free program called AudiXplorer is available for Mac.
Home spectral analysis: Download a spectrum analyzer program to your computer; Visual
Analyser (PC) or AudioXplorer (Mac) are two free possibilities. Try some spectrum analysis
for yourself. Play different sounds into your computer and look at their spectra. Do the
spectra make sense? Are there peaks where you think they should be? Are there any features
of the spectra that are surprising to you? I strongly suggest you go play with this yourself!
In Figure 8.6, you see the spectrum of each of the waves we looked at in Fig 8.1. The first one (a)
should make total sense—it’s a pure sine wave, so there is only one frequency. The others are more
complicated. I want to focus for a minute on the square wave
3.2
Square wave
The square wave is demonstrated in the PhET simulation, and also
in figure 8.6(b). You need lots of high harmonics, dying off slowly in
amplitude as you go up in frequency, to generate this shape. Also, you
only want every other harmonic. This can be understood in terms of
symmetry.
Here’s the basic idea of how to build a square wave: Imagine taking
a sine wave, and adding in a sine wave with exactly twice the frequency.
I sketch that at right: first I just draw n = 1 (the fundamental) as a
solid, and n = 2 (the first harmonic) as dashes. Now if you add these
two curves together, what do you get? Over on the left half, they’re
both positive, and add up. But over on the right half, one is plus and
the other is minus, they tend to cancel. So you get a weird wave which
has lost the lovely symmetry of the fundamental. (I tried to draw it
here, the second panel down at right—my drawing is not perfect, but
you get the idea). The fundamental is the same to the left and right
of the peak, but the sum is not. So if we’re trying to “square up” the
sine wave, we still want it to be symmetrical around the middle, and
so we don’t want any n = 2 added in!
But if you go to the next wave, n = 3, that’s cool. See the drawing,
the third panel down on the right. It is once again symmetrical about
the middle (the midpoint). So when we add some n = 1 to n = 3, we
change the shape, but do not lose the lovely symmetry we want. (I didn’t try to draw the sum
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Chapter 8 notes
this time, but think about what it will look like. On the left third, you are adding some dash to
the solid, bringing the solid up a little. In the middle third, you are canceling some of the solid,
bringing the top of the peak down. In the right, you are again lifting it up. So, in the end, what
happens is that you create a new curve, still symmetric, that is “flatter”. You’ve lifted up the low
parts of the fundamental, and lowered down the highest parts.
Simulation: Go to the web page in this footnote2 , or from the course web page under “Resources”
choose the PhET simulation “Fourier: making waves”. Play with the PhET simulation, try
to make something square-wave-like and see for yourself how it looks. Try adding some n = 1
and n = 3. Vary the relative amplitude of n = 3. What happens if the amplitude of n = 3 is
too big or too small? What do you need to do to make it look more like a square wave?
That’s the reason why the square wave in 8.1(b) contains only every other frequency. (There are
woodwind instruments that naturally do this, unlike strings, you can build a wind instrument
that tends to avoid every other harmonic. And the sound of, say, an oboe does sound a little bit
square-wave-like.
3.3
Real spectra
If you look at real-life instruments, you will find that there are, indeed, anharmonic (not-harmonic)
frequencies, so the spectrum is not a bunch of spikes at given frequencies, but instead itself becomes
a smooth curve. Figure 8.7 shows some realistic spectra of real instruments. The general features
of this graph will be: you get strong “bumps” or peaks at the harmonic frequencies. The size of
the spike tends to get smaller as you go up in frequency.
Can you make sense of those two features?
If you go back to Chapter 10, and look at Figure 10.4(c), you’ll see a spectrum for a plucked
violin string. This spectrum is ideal rather than real, so it’s not smooth, you get perfect spikes
only at the harmonics. The first 2 harmonics are strong, then they die off pretty quickly as we go
up in frequency. There are a couple of missing harmonics, because of where exactly we plucked.
When you pluck a string somewhere, you know that no harmonic which has a node, or zero, at
that particular point, will contribute. It’s very much like the symmetry argument we just made.
There’s no way to get any of that particular frequency.
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Modulation and radio
All these ideas about spectra and adding up waves are used in generating/detecting radio waves.
We won’t have time to discuss this in depth in class, but here’s the idea—read it if you’re curious.
When radio stations send music through the air, they are not sending pressure waves. They
are sending electromagnetic waves. You cannot hear or detect electromagnetic waves with any of
your senses, the frequency is in the Megaherz (MHz) range, and no organs in our bodies detect
this form of electromagnetic wave. (You can detect much higher frequencies, that’s what your
eyeballs do! Visible light has frequencies up in the 1015 Hz range, though!) Your ears do not
detect elelectromagnetic waves at all, they can only detect air pressure variations. The radio waves
jiggle electrons, they make electrical currents, so your radio (or TV, if you still have rabbit ear
2
http://phet.colorado.edu/simulations/sims.php?sim=Fourier Making Waves
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Physics 1240
Chapter 8 notes
antennae) picks up the electromagnetic signals, which makes electrical currents that wiggle back
and forth. If you amplify this signal and send it through wires to speakers, you can then easily
build a mechanical device which will physically wiggle in response to the electric currents. This
physical wiggle can couple to, say, a cardboard speaker cone, which wiggles the air, which makes a
sound. The problem is that if you did this directly, you’d be converting MHz electrical vibrations
into MHz physical vibrations, and that is way above anything your ear can detect, even after the
electromagnetic wave is “converted” into air motion. So instead there is a trick: you modulate
the high frequency (MHz) signal with a much lower frequency signal. It’s this lower one that is
the music you want to hear, in the 20-20,000 Hz range.
There are many ways to modulate, but the two simplest are AM and FM. AM stands for
amplitude modulation, and FM stands for frequency modulation.
4.1
AM: Amplitude modulation
The dashed line is a low frequency signal. The solid
line is the high frequency carrier wave. (Note that
this drawing is not to scale! The difference between
the low and high frequencies is much higher than I
can draw.) So the idea is that the high frequency
signal (in the MHz range) is relatively slowly varying
in amplitude. (Hence the name, amplitude modulation.) What you see in the picture is a sinusoidal
variation in the amplitude. If you build an electronic device which filters away the high frequency
signal, then you are left with just the envelope, the lower frequency signal, which might be directly
converted into speaker motion and thus sound. Here, you would hear a pure sine tone, corresponding to the frequency of that dashed gray envelope. But any sound is just some variation of this,
some repeating waveform, and so if that waveform is the envelope, it’s what you’ll end up hearing.
4.2
FM: Frequency modulation
In an FM signal, it’s now the amplitude of the high
frequency (carrier wave) signal that is constant. The
frequency is nearly constant at the carrier frequency,
but with some variation that encodes the signal.
(Again, my drawing is not to scale!) So you’re
slightly varying the frequency of the signal, a little high, a little low, a little high, a little low. This
variation (the modulation of the frequency) becomes
your desired signal! The circuitry required here is a little fancier, more tricks are required to convert
frequency variation into physical motion of the cone. One advantage of this technique is that you
don’t care about the strength of the signal as much—the amplitude is irrelevant, only the frequency
matters. So if weather, or obstructions, or shadows, accidentally absorb some signal strength, that
won’t change what you hear. So FM radio tends to sound “cleaner”, with less static and noise than
AM radio signals.
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