Physics 1240
Hall Chapter 1 Notes
1
Focus questions and learning goals
Chapter 1 focus questions:
1. What is the connection between music and acoustics?
2. What are the physical characteristics of sound?
Chapter 1 learning goals. After studying this chapter, you should be able to:
1. Describe waves.
(a) Represent waves in various ways, especially using graphs.
(b) Describe how small particles in the medium are moving for various kinds of waves, such
as sound or ripples on the surface of a pond.
2. Predict how the speed of sound will change if you change various aspects of the sound itself
or the room. For example, how does the speed of sound change if you:
(a) Raise the pitch?
(b) Make the sound louder?
(c) Heat the room?
3. Predict how the pitch and loudness of a sound you hear will depend on the frequency and
amplitude of a pure tone from a sound synthesizer.
4. Calculate the distance from a source (for example, a lightning strike) if you hear a sound at
a different time than you see the even that caused it.
2
Acoustics and Music
This course will start with some big questions about sound and music. What is sound? How do
you define music? How do acoustics fit in? This is partly definitions (that is, human conventions)
which I’m not so concerned about: it’s the ideas that matter!
Sound can be defined in different ways. It can refer to a physical disturbance (a pressure
wave) that travels from a source. It can also refer to your sensation when that pressure wave
wiggles hair cells in your inner ear. (The difference between these two definitions is the essence of
the old puzzle that asks “If a tree falls in a forest, and there is no one there to hear it, does it make
a sound?”)
2.1
The three ingredients
We’ll need to study three things to get a handle on sound:
1. The source, or creation, of sound waves.
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Chapter 1 notes
2. The nature of these waves, and how they are transmitted.
3. The detection mechanism (usually this is your ear).
This is it—that’s what this course is all about. We’ll talk about all three ingredients, first at an
overview level for the first month or so of the term. We’ll then circle back with more details and
specifics, maybe even several “passes” at deeper levels, as we go through the course (see the figure
in the preface of the book).
2.2
Acoustics
Acoustics is the science of sound. It’s a branch of physics—like studying light, or electricity,
or liquid crystals. There are people who study acoustics for a living (about 7000 members of the
Acoustics Society of America). Acoustics sometimes is related to psychology (what sounds good),
sometimes engineering (designing better sonograms for medicine), and sometimes it’s pure physics
(understanding the wave phenomena from a source). Topics of study in acoustics might involve
health (speech, hearing), perception (neural processing, auditory illusions), architecture (sound
mitigation), sound reproduction, and digital synthesis, just to name a few.
There are different types of sound. Music is a special type of sound which is intentional,
ordered, and (in some respect) aesthetic. Speech can be musical or not, but its purpose is usually
communication. Noise is random and usually unwanted (but not always). These definitions are
subjective and variable! We’ll talk about all of them more this term.
2.3
The three ingredients overview
Fitting with our list above, our study (in this course) of musical acoustics is about (1) production
(2) propagation (3) perception of sound. As I said, we’ll spiral around these three ingredients all
term. Here is a super-quick first-pass overview of all three.
1. Production of sound comes from rapidly vibrating things. Feel your throat as you talk: your
vocal cords vibrate (buzz). All sound comes from some kind of vibration. Watch a guitar
string vibrate when you strum it. Can you see the vibration that causes the sound? Stand
near the speakers at a concert and you can feel the bass vibrations in your body.
2. Propagation of sound depends on the medium, which is the material the sound moves
through. Mostly, we’ll think of sound in air. We’ll talk about how sound can travel through
air—we speak of the sound wave, which travels. (In air, the speed of sound is about 344
meters/second.) What’s waving? How does it move, and why at that speed? We’ll get there
soon! It’s all very real, and well understood, but it’s a little subtle. This is one of the first
topics we’ll cover.
3. Perception of sound happens at our ears. There are physical interactions with features inside
your ear, which result in electrical signals to your brain, which are interpreted, as “sound”.
Some aspects of this are very well understood, but the “higher level” (neural/cognitive)
aspects are still very much under study, which means scientists don’t fully understand them.
We might add electronic processing as a fourth ingredient which is different from the three main
topics above. We’ll talk about this as we go along, and we’ll use electronics to help us image and
understand all three of the key ingredients above.
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Chapter 1 notes
Sound, vibrations and frequency
Sound arises when something vibrates. We’ll talk lots more about this, but it’s a simple observational fact to start with. Vibration means something going back and forth at some rate, at some
frequency. The vibration is called periodic if it keeps moving back and forth in the same cycle,
over and over, passing the starting position at some frequency. Suppose we observe an object which
vibrates 1 time per second (also written as 1 time/sec, 1 cycle/sec, 1 vibration/sec). The fancy
name for this unit is Hertz: 1 Hz = 1 cycle/sec. So 10 Hz = 10 vibrations each second.
Home experiment: How fast can you rhythmically tap on a table? 1 Hz? 5 Hz? 10 Hz?
My voice box can vibrate from between a couple 100 Hz to a couple 1000 Hz. How do I know this?
We’ll measure it, soon.
Normal hearing for humans ranges from about 20 Hz to about 20,000 Hz. How do I know this?
Thought experiment: Design an experiment you could to to determine the hearing range for
humans.
Things can vibrate slower that humans can hear; these sounds are called infrasonic; things can also
vibrate faster than humans can hear, and these sounds are called ultrasonic. The vibrations still
occur—you just can’t hear the sound that results. (What experiment could be done to demonstrate
this?) You may have seen medical images called ultrasonograms or ultrasounds, which use highfrequency sound waves to make images of soft internal organs. I’ve seen an ultrasound image of
my kidney.
If something wiggles at 10 Hz (that is, 10 times a second), how long do you have to wait for one
wiggle? Think about this carefully, because we’ll use this idea a lot in the course. 10 wiggles take
one second. So 1 wiggle takes 1/10 sec (one tenth of a secon). We say the period of the vibration
is 0.1 sec. The period is the time it takes for one full wiggle to occur.
Calculation: If something vibrates at 2 Hz, what is the period of oscillation?
Calculation: Can you convince yourself that period = 1/frequency, always?
The formula P = 1/f is just a convenient way to think of this connection. In this formula P
represents period and f represents frequency.
3.1
Frequency and pitch
High frequency means fast wiggles. It also sounds high to your ear. We call that sensation pitch.
Pitch is what you feel that associates with the physical quantity of frequency. High pitch implies
high frequency, and high frequency implies high pitch. They’re not the same: one is what you sense,
the other is physical and measurable. As a scientist, it’s easier for me to think about frequency—
precisely because it’s so easy to measure!—but what you care about, in some cases, is pitch. We’ll
talk about these kinds of connections often in this class.
Since P = 1/f , high pitches (high frequency) mean small (or short) periods. A useful way to
remember this is that 1 divided by a really big number is a really small number: 1/large is small. A
small period means it’s quick, with very little time for each cycle of vibration. Similarly, 1 divided
by a really small number is a really large number: 1/small is large. So a low frequency means a
long period.
Calculation: Test the rule of thumb that 1/large is small (1 divided by a large number is a
small number). What is 1 divided by 1000? 1 divided by 30,000? 1 divided by 5 million?
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3.2
Chapter 1 notes
Sound waves
I’ve been a little sloppy so far in this discussion: I’m mixing up vibration of the source and vibration
of the resulting sound wave. One produces the other, and their frequencies are the same number,
but there’s a difference. A vibrating object, wiggling at 400 Hz, is making a sound. What’s the
sound? It’s a pressure variation in the air, called a sound wave or pressure wave.
Here’s the main idea: as something wiggles (this might be a vocal chord, a guitar string, a bell
surface, or something else) it pushes on air molecules. First it pushes one way, “squeezing” the air,
then it moves back and “rarefies” the air. The vibrating object makes first a compression, then a
rarefaction, of air. If the vibration is at 400 Hz, the compression and rarefaction are also coming
400 times a second, they’re also at 400 Hz! Sound waves are simply these small compressions and
expansions of air, repeating and traveling.
There’s an important consequence of our picture of sound waves as small compressions and
expansions. Sound waves can only propagate when you have compressible substances. Consider our
example of a sound started by a vibrating object. The air molecules that get pushed bump into
their neighbors (compressing the air a little to one side), which then bump into their neighbors,
and their neighbors. So we get a propagating disturbance, or a traveling wave.
Thought experiment: Imagine a sound wave that starts on earth and travels up though the
atmosphere and out into space, where there is no air. What happens to the sound wave as
the atmosphere grows thinner and eventually disappears?
Note that actually convincing you that sound is a pressure disturbance is not so trivial. Not
all concepts can be simply shown with a single, easy demo. You can’t see air, so it’s hard to do a
demo like with the slinky! What’s worse is that these pressure fluctuations are about one millionth
of the normal background air pressure. It’s a subtle story! So let’s leave it for a bit, but we’ll come
back to it soon.
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Interlude: how to learn science
In these notes, I’ve told you a bunch of facts. You can memorize them, but that’s not learning
science. Let’s consider some questions about the last couple pages of notes. Why do you believe
all this? Who says sound is a pressure wave in air? Is it a definition, or can we demonstrate it in
an experiment?
Rather than telling you the answer, I’d rather pose more questions (because that’s really how
scientists learn):
1. Is air needed for sound to propagate?
2. How do I know what the frequency of human voices is?
3. How do I know that higher frequency means higher pitch?
4. How do I know that the pressure of the air is changing?
5. What is pressure, anyway?
If you haven’t been asking these kinds of questions, the time to start is now. Asking questions is
your goal in this class!1 Good questions lead to new ideas and new ways of understanding. If I don’t
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Maybe in life! Question everything. Question authority.
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Chapter 1 notes
answer all your questions, keep thinking about them yourself. You’re perfectly capable of figuring
out the answers, in almost every case. The scientists who came up with this stuff hundreds of years
ago were no different than you—they just got curious enough to mess around, ask questions, and
make “mental models” (hypotheses and theories) to help them understand the behavior of sounds
and instruments. And then they did more experiments, and made more measurements to check,
refine, and verify those theories.
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Waves
Since you now know we’re going to try to conclude that sound is a wave (of some kind), let’s talk
about vibrations and waves by themselves. I’m trying to visualize, characterize, and describe them.
Waves are everywhere, never mind sound! Think water waves, waves in a flag, “the wave” in a
stadium.
5.1
Transverse waves
Think first about a simple wave: picture a slinky stretched out on a table. You take one end and
start to wiggle it back and forth. A wave will begin to “slide” across the table. There’s a picture
in the book (Fig 1.3) that shows a snapshot of what I’m talking about. The slinky metal actually
moves up and down, but the wave (the disturbance) travels to the right. This shows an important
and confusing property of some types of waves: The motion of the wave and the motion of the
medium (the material, the stuff that’s waving) are in different directions. We call this a transverse
wave. Waves on the surface of a pond are like that too—drop a pebble in the pond. Waves move
outwards in circles (parallel to the surface), but the actual water droplets are moving up and down.
Water is not flowing “outward” with the wave: you don’t end up with more water at the outside
and less in the inside. A duck in the pond bobs up and down as the wave passes underneath it (Fig
1.5 in the book).
Home experiment: Try dropping pebbles (or marbles, or coins) into water in a pond (or bathtub,
or pot of water). Can you see the waves moving out, away from the source of the disturbance?
Can you see (or convince yourself) that individual bits of water move only up and down, not
out away from the source?
5.2
Amplitude and wavelength
If you look at a picture like Fig 1.3 of the text, you can see that there are two distances that
seem important. First, there’s the “up and down” motion of the slinky material itself. The largest
distance of travel is called the amplitude. A wild wave (where you jerk your hard really far
up and down over and over) will have a big amplitude. There’s another totally different distance
though—that’s the horizontal distance from one crest of the waveto the next one. That’s called the
wavelength, the length of one wave. It’s the horizontal length, the distance you must go sideways
to get from one crest to the next. If you’re looking down at the pond, you’ll see the ripples, and
if you focus on the “high spots”, the horizontal distance from one to the next is the wavelength.
(See Fig 1.6a in the text, think of that as looking down at a piece of these ripples) We usually give
it the Greek letter lambda (λ) as a symbol. (See Fig 1.6b for another picture.)
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5.3
Chapter 1 notes
Longitudinal waves
There’s another kind of wave, where the motion of the medium is in the same direction as the
motion of the disturbance. It’s called a longitudinal wave. The slinky is again the best way
to see one: instead of whipping the end up and down, jerk it forward and backwards in the same
direction as the slinky itself stretches. You’ll see a little compression pulse travel along. You can
get this kind of wave in materials that are easy to compress and expand. Air will do this too—in
fact, I claim that’s what sound is! A little compression pulse of high density (or high pressure) air,
which spreads out sort of like ripples in a pond.2
5.4
Representations of waves
The textbook has a bunch of different ways to “represent” waves—see Fig 1.2 through 1.6. Look
at them, think about them, try to picture what the “real life” wave looks like, and how this
representation tries to show you the characteristics of the wave. We’ll be thinking and talking about
waves all term, and you need to be comfortable thinking about them in these slightly different ways.
It’s worth taking a few minutes to stare at the figures and try to understand as many aspects of
the diagram as you can.
There’s one slightly confusing thing in the book, especially in Fig 1.2. Stare at it—it’s a snapshot
of a sound wave—represented by dark and light “bands” representing high and low pressure. You
have to imagine carefully what’s going on there. First of all, as time goes by, picture it as a movie:
the bands all move outwards (to the right, in that figure). The wavelength is the distance from a
dark band all the way out to the next dark band. (A to B in that figure is only half a wavelength,
from a “high” to a “low”. You need to keep going to the next “high” for a full wavelength.) Now,
the actual motion of air molecules is not shown in that figure, and it’s a little subtle. The air
molecules move right and left, they jiggle. This is just like how the pieces of a slinky jiggle back
and forth as a pulse passes along. The slinky material is not “flowing” or moving from left to right,
and the slinky metal doesn’t get thinner at the start. It’s the wave that moves, not the medium!
This point is important, and I found it confusing for a long time myself. Think about it, talk about
it, make sure you get what I’m saying. Air molecules do not go from point A to point B. They
jiggle back and forth—a molecule that starts at A will wiggle around point A all the time. The
wave passes from left to right, but the air itself is not moving in bulk, anywhere. When I talk, the
wave travels across the room to your ear. But individual air molecules do not!
Simulation: Transverse waves. There’s a nice web resource to supplement these static pictures
in the book. Go to the webpage in this footnote3 , or see the link to this simulation from
the course web page (click on “Wave on a string” under “Resources”). If you look at the
simulation of a wave on a string, you can watch waves, slow them down and speed them up.
This is a transverse wave. You’ll see what wavelength means (you can measure it on the
screen) and how it’s different from amplitude. Watch a green dot—that’s a piece of string.
As a pulse goes by, which way does the pulse move? Which way does the string move?
Simulation: Soud waves. Now look at the webpage in this footnote4 , or see the link to this
simulation from the course web page (click on “Sound waves” under “Resources”). This is
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But remember that sound and ripples in a pond are different, because sound is longitudinal, not transverse.
http://phet.colorado.edu/simulations/sims.php?sim=Wave on a String
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http://phet.colorado.edu/simulations/sims.php?sim=Sound
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Chapter 1 notes
like a movie of Fig 1.2 from the text. You can mess around with it in various ways. Give it
a try, you’ll learn a lot with just 5 minutes!
6
Sound speed
Let’s leave waves behind for a second. So never mind what sound is (for a bit), let’s think about how
it behaves. We’ve argued that it’s produced when something vibrates in air. When you “smack”
the slinky, you get a single pulse that travels along. How fast? It depends on the material. But,
how about sound? Does it travel instantly, or does it have a speed? How could you tell? (Think
about it!) The speed of sound in air is represented by the symbol v:
vsound in air = 344 meters/sec = 770 mi/hr = 0.21 mi/sec.
This is the speed of sound in air in different units. Notice that the speed is the same, but the
number depends on the units used. The equals signs in this equation is telling you that 770 miles
per hour is the same speed as 0.21 miles per second.
The speed of sound is fast! A speed of 0.2 mi/sec means sound travels 1 mile in 5 sec.
Calculation: Use the speed of sound to determine a “rule of thumb” for figuring out how far away
a lightning strike is. In other words, suppose you see a lightning strike and then hear the
thunder t seconds later. How far away is the strike?
Equivalently 300 m/s is about 0.3 km/sec, which means sound travels 1 km in 3 sec.
Calculation: Suppose that you’re at a concert, where you sit 34 meters (just over 100 feet) back.
How long does it take for music produced on stage to reach you?
To do this, remember that speed is distance/time. In symbols, v = d/t. If v = 344 meters/sec,
what is the time to go 34 meters?
Answer: t = d/v = 34 meters/(344 meters/sec) = 0.1 seconds. This is a big enough time for
you to notice! You might even be able to notice something a little bit shorter than that, in the
right circumstances. You’re used to it, but a problem occurs if they set up a speaker near you (at
the Pepsi Center, for example). Then you hear the speaker sound first, and the “real” sound 0.1
sec later. (The electronic signal that feeds into the speaker travels much master than the speed
of sound!) This leads to a weird and uncomfortable effect. This speed plays a real role in the
architecture of theatres, concert halls, stadiums, etc.
Remember that when a wave travels, the wave (disturbance) moves, but the medium is not
traveling in the same way. (What examples that we’ve discussed demonstrate this?) Is the speed
of sound the same as the speed of the air molecules?
6.1
Variation in the speed of sound
What does speed of sound depend on? Think carefully about what it could depend on, based on
your experience.
It turns out that the sound speed is independent of the frequency (pitch) of sound.
Thought experiment: How can you tell that the sound speed is independent of frequency? What
would be different if this weren’t true? Why is it good that the sound speed doesn’t depend
on frequency?
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Chapter 1 notes
The sound speed is also independent of loudness of sound. The sound speed does depend (weakly!)
on temperature. I’ve assumed the room temperature is 20 degrees C, or 68 degrees F, at which
the speed of sound is what I’ve been saying: 344 m/s. Every degree Fahrenheit above room
temperature, sound is 1.1 ft/sec faster; or every degree Celsius above room temperature the speed
goes up by 0.6 m/s. (If the temperature is below room temperature, the speed goes down by those
numbers.) What are the consequences of this for musicians? Well, if the temperature increases 10
degrees C above room temperature, sound moves about 6 m/s faster. That’s about a 2% increase.
Most people can detect the resulting change in sound. We’ll see how this affects the sound of some
instruments: wind instruments must be warmed up, especially on cold days.
6.2
Echoes
If sound hits a solid, smooth wall, it bounces! This is because hard solid surfaces don’t absorb
much energy, like bouncing a ball off a cement wall (hard) versus a padded one (soft). We can use
echoes to learn some interesting physics!
Calculation: You’re playing trumpet in the Taj Mahal, and want to make a cool sound effect by
playing notes fast enough to be exactly in sync with the return echo from the ceiling. How
long is the echo delay? How fast should you play your notes? The Taj Mahal’s ceiling is
about 34 meters high.
Answer: d = 2 × 34 meters (up and back!) = 64 meters. So t = d/v = 64 meters/(344 m/s)=
0.2 seconds. This is the echo delay. Therefore if you play 5 notes/sec (tum-tum-tum-tum-tum) the
echoes will sit exactly on top of the next note.
Thought experiment: Could you use echoes to estimate the speed of sound? How would you
design the experiment?
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Pressure and force
Now that we’ve talked about waves, and sound, let’s talk a little more about pressure, so we’re
better equipped to make sense of my claims that sound is a pressure wave.
Air is compressible. If you squeeze it, it shrinks (the pressure goes up, the volume goes down),
then expands (the pressure goes back down). If you let it expand, air will, to fill up the available
space.
Pressure is a measure of force over area. Pressure = force/area, or in symbols, P = F/A. A
large force on a small area makes lots of pressure. (Remember the rule of thumb: 1/small is large.
Think of high heels sinking into mud.) A large force spread out over a large area is less pressure.
Units of force: we measure force in Newtons. 1 Newton is a light touch. Holding up one
pound requires 5 Newtons. (Note that pounds is also a unit of force, but in this course let’s stick
with metric! If you’re totally unfamiliar with metric units—let me know, maybe we’ll have a little
mini-lecture on it. In the meantime, check out Appendix B of the book.)
Units of pressure: consider one Newton spread out over a square that is 1 meter on a side.
We will also write this area as 1 meter × 1 meter = 1 square meter = 1 m2 . This pressure—one
Newton spread out over one square meter—is 1 N/m2 = 1 Pascal (1 Pa). This is a very light, gentle
pressure.
We sit under a giant ocean of air that is 50 km deep. All of this air is heavy, and weighs down
on us. In fact, the pressure of air is 100,000 Pa. (This pressure is also called 1 atmosphere or 1
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Chapter 1 notes
atm = 100,000 Pa.) So 100,000 Newtons over 1 m2 , that’s about 22,000 pounds (or a very large
truck) supported on a square about 3 feet by 3 feet.
I weigh about 650 Newtons. The area under my feet might be about 0.04 m2 , so the pressure
on my soles due to my weight is about 500 Newtons/.04 m2 = 12,500 Pa. That’s way less than one
atmosphere.
Calculation: What would be the pressure on my soles due to my weight if I wear high-heeled,
stiletto shoes? Imagine for this calculation that I balance on my heels so all the weight is on
the stiletto points.
7.1
Pressure in sound waves
What does pressure have to do with sound waves? Let’s put it all back together. Vibrating surfaces
make high and then low pressure regions in air, which travel at 344 m/s outwards. This traveling
wave of compression and rarefaction is a pressure wave—first the air is slightly over normal
pressure, then slightly under normal pressure. (To truly measure this pressure wave is a hard subtle
experiment that we won’t do. But think about how you might!)
How big is this overpressure of sound waves? It depends on how loud the sound is. The louder
the sound, the more the overpressure. Loud but normal sounds might be 1 N/m2 above (and then
below) normal pressure (which, recall, is 100,000 N/m2 ). So it’s a tiny variation: between 100,001
to 99,999 N/m2 . This is 1 part in 100,000, or .001%. And that’s a loud sound!5
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Summary
Let me wrap up by reminding you of the bottom line that you should take away from Chapter 1,
and remind you of some questions you should be able to answer now.
Sound requires the three ingredients. Ingredient 1 is production, which is due to vibrations.
How do you describe vibrations? Ingredient 2 is propagation. Air compresses and expands, in
regions of high and low pressure. How do you describe a wave? The speed of sound is a constant in
air, except for a slight dependence on temperature. Ingredient 3 perception in our hearing system.
What do higher frequency and higher pressure mean for perception?
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The fact that the overpressure is so tiny is why the experiment to measure it is a little hard. In this case, you’ll
have to take my word for it. But that’s exactly what the oscilloscope in our demos is doing—the microphone has a
little membrane that flexes due to a shift in pressure, and you’re looking at the result of the flex directly.
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