Physics 1240
Hall Chapter 4 Notes
1
Focus questions and learning goals
Chapter 4 focus questions:
1. How does sound travel?
2. How does the physics of sound propagation affect our perception of music?
Chapter 4 learning goals. After studying this chapter, you should be able to:
1. Predict how a given sound will bounce from a wall (and decide what properties of the wave
you need).
2. Describe beats, explain how it would be used musically, predict the beat frequency (given
details of the two pitches involved).
3. Predict which wavelength will bend more when coming out of a hole or around a corner.
4. Explain in simple terms the advantage (or disadvantage) to using a small versus a large
opening for an instrument,
5. Explain in simple terms the effects of using a small (or large) roughness on the wall.
6. Explain what physical features of a band shell help improve sound quality for the audience.
7. Predict which of two band shell shapes (or position of musicians) would make the sound
louder for audience members in various particular spots.
8. Predict the (change in) pitch of sound arising from a moving source, or heard by a moving
listener.
This chapter contains some stuff that we won’t focus on (like refraction), and the most essential
points are a little hidden. So I recommend reading it in a funny order, starting with 4.1 and 4.5,
and then going back to 4.2 (which we’ll cover only very loosely), and then 4.3 which is kind of a
cool application! Section 4.4 is fun too, but really not all that important of a point for sound and
music, we’ll just talk about it briefly without worrying about the formulas.
2
Reflections
To understand the first section of chapter 4, remember that sound is a wave, a pressure wave in a
medium (usually air). Sound arises from the steady propagation of a disturbance. The disturbance
travels with a well-defined speed. It travels outwards from sources (in an expanding sphere of
influence) Sound will reflect or echo from hard, flat surfaces. If the surface is bumpy, sound waves
will reflect every which way, in many different directions; this is called diffuse reflection. If the
surface is smooth and hard, the sound will reflect much like you would expect billiard balls to reflect
off a wall (see Fig 4.1a in your text).
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Chapter 4 notes
We need to discuss this figure some more, though. What the heck are those arrows in Fig 4.1?
To explain, first go way back to Fig 1.6a. If you focus on the curved (but mostly “up and down”)
lines in 1.6a, those are the ones we’re used to thinking about to represent sound waves—they look
like the lines of constant high pressure in the PhET simulation you used for a homework. The
arrows (pointing to the right-ish in that picture) are “rays of sound”, they represent the direction
of motion of sound. This is a rather abstract idea: it’s not something physical! It just shows
you how the sound is moving. Start at 1.6a, and try to mentally visually sound “flowing with the
arrows”, with high pressure fronts following low pressure fronts. (Does that picture make sense?)
So now go back to Fig 4.1a. Those incoming arrows represent a sound wave which is heading in
towards the wall. We’re not showing the pressure at all any more, just the “flow” of sound. The
arrows show the direction the sound is moving. The sound bounces (echoes) and now it’s heading
away from the wall; that’s what the outgoing arrows represent.
2.1
Diffuse versus specular reflections
When a sound hits a wall and reflects, what happens depends on the size of the bumps on the wall.
But, how bumpy is bumpy? The answer is subtle, but basically, it’s the wavelength of the sound
that determines if a surface is “smooth” or not. If the bumps are smaller than the wavelength of
the sound, then the bumps are so small they don’t matter, the wall seems smooth. If the bumps
are bigger than the wavelength of the sound, they’re big enough that Fig 4.1b is the more realistic
picture. (Why is this? I’m going to leave it as a puzzle for you to mull over for the moment, it’s
really not so important for where we’re heading.)
Remember the key formula from chapter 2: frequency times wavelength = speed, or f λ = v =
344 m/s (at room temperature). So think about how higher frequency sound always has a smaller
wavelength. For example, if f = 344 Hz, then λ = 1 meter. That’s a middle note, the F right
above middle C on a piano.
Calculation: Find out the frequency of the lowest note on a piano, of middle C, and of the
highest note on a piano. What wavelength corresponds to each of these frequencies? What
are physical objects that are comparable in size to these wavelengths?
If you go up to f = 3,444 Hz = 3.4 kHz, then λ = 0.1 meters (convince yourself of this). That’s a
note from the very high end of the piano keyboard, producing waves 10 times smaller in wavelength,
about the size of your fist. On the other end of the keyboard, down at about f = 34 Hz, λ = 10
meters—this wavelength is larger than the piano itself!
What this all means is that a wall might be effectively smooth for the low frequency sound
hitting it, but at the very same time, in the very same spot, act bumpy for the higher frequency
sound hitting it! Think for a minute about how strange this is: different sounds (of different
frequencies) can behave differently when striking the same surface. If the wall has bends, angles,
or bumps that are a few feet in size, then the notes on the high end of the piano keyboard will
“notice” them (because the bumps are bigger than their wavelength), and bounce at crazy angles.
But the notes from the low end of the piano will still see a smooth wall (because the bumps are
smaller than their wavelength) and reflect in a simpler way. When you design rooms for playing
music, you have to think about this:—the way different frequencies reflect from different surfaces
is a key element of concert hall acoustics.
Note that there is also absorption at the boundary. Some sound energy will always get
absorbed, it doesn’t all echo perfectly. One way to say this is that the amplitude of the reflected
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Chapter 4 notes
wave going out will be smaller than the amplitude you had coming in. Bigger amplitude means
louder sounds means more energy flowing.
2.2
Multiple reflections
Figure 4.2 of the text shows multiple reflections. When sound is emitted from a source, it heads
out in all directions. The sound waves hit walls and floors and ceilings: some energy gets absorbed,
but if the walls are fairly smooth and hard, quite a bit of sound will bounce (echo, reflect). So there
are lots of possible paths on which sound waves can travel. The sound waves then bounce and
bounce and reflect around. Think of pinballs in a pinball machine, bouncing all around. (Please be
very careful with this analogy. It’s almost painful for me to compare sound to pinballs—I’m asking
you to think of sound as THINGS, like pinballs?! Isn’t that a bad idea? Sound is a wave, not a
pinball! But it’s OK, sometimes it’s not the worst mental image in the world to think about. But
remember the difference.)
Multiple reflection is one of two main reasons why you can hear sounds from other rooms, even
if there’s no direct “line of sight” from you to the source. Even if light cannot go straight from
the source to you, the sound can bounce around a little and reach you through a slightly more
complicated path. (Look at fig 4.2 again!) Sound travels so fast (344 meters in one second) that
the slight delay from having to bounce a few times and take a longer path is not noticeable in
ordinary sized areas or ordinary circumstances. Note that there is another way that sound can get
to you (by bending around corners!) that is quite different. We’ll return to this soon.
The text discusses refraction next, which yet another kind of “bending” of sound paths. Feel
free to look it over, but it’s just not all that important a physical effect for sound or music in most
circumstances, and I’m not going to talk about it.
3
Interference and superposition
Next I want to skip right to the heart of this chapter, which is section 4.5. This talks about
interference of sound (or, in general, any kind of wave). This is such a huge idea, that I’m
surprised the text doesn’t focus a little more on it. So let’s talk about it here a bit.
Let’s first think about a wave pulse traveling along a slinky or string, from left to right. Picture
it—a little “upwards blip”, traveling to the right steadily. A pulse like this is the world’s simplest
wave—see my drawing in panel A of the figure below. (I’ve drawn a square pulse because it’s easier
to draw; you can think of it as a rounded one if you like.) The arrow shows that the pulse is heading
to the right. If you looked at it a moment later, the blip would have moved further right.
Now imagine someone on the far side of the slinky has sent a pulse heading left, coming towards
my pulse—see panel B. I’ve make the two pulses with different amplitudes so you can tell them
apart. These two waves are coming towards one another on the same slinky. (They travel at the
same speed!) What happens when the two waves (blips) pass each other on the slinky? If waves
were particles, they would bounce, and maybe squish. But waves are not particles! They basically
pass through one another, and they interact in a simple way. The slinky shape will basically always
be exactly the sum of the two waves. It’s as though each wave is continuing on its own, unchanged,
but what you see is the numerical sum of the displacements of the two blips. Look at the figure,
where I try to trace this through as the pulses move. Each lower drawing—panels C, D, E, and
F—represents a later moment in time.
I have drawn the original pulses as light dots (for the big, right-moving blip) and light dashes
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Chapter 4 notes
(for the smaller, left-moving blip), but the sum of these two (the solid line) is what the slinky
shape really looks like at that instant. In the case of these two blips, when the leading edges come
together (panel C), you’ll see a little “spike” growing in the middle. The spike is just the sum of
the left wave and the right wave.
As the two waves continue to move,
they fully overlap, so it’s no longer just
the leading edges that interact. At the
moment they fully overlap, you just see
the simple sum (panel D). At this one
instant in time, the sum looks like a single, taller pulse.
But the two waves continue along:
they don’t stay perfectly overlapped for
long! Now only the trailing edges will
still overlap, and you’ll see something
like what I’ve drawn in panel E of the
figure. You really need to think carefully about this—draw it for yourself,
neatly, perhaps on a piece of graph paper. Just let each wave continue along,
and at any moment in time, you simply add the two of them to figure out
what the total wave looks like. You have
to make sense of these pictures, because
it’s the essence of how waves interact.
What happens later? Each blip continues along cheerfully on it its way. After they pass by one another, there is no evidence they ever
interacted! So later, you’ll see a picture like in panel F of the figure, where each pulse is continuing
on its way.
What I’ve just described is called the superposition of waves. It’s a description of how waves
interact: they superpose (add), but are individually unaffected. Almost all waves act like this.1
Slinkies really do this, sound does this, water waves do this, light does this. Superpositon is a
simple, lovely idea that is a description of a lot of phenomena in the world!
Simulation: Pulse superposition. You can make pulses and watch them interfere, just like in the
figure above. Go back to the wave on a string PhET simulation, linked in this footnote2 , or
see the link to this simulation from the course web page (click on “Wave on a string” under
“Resources”). I recommend that you go to “pulse” mode, and turn the damping down to zero
(so the pulses don’t die away). Then you can create pulses at different times, watch them
travel down the string, watch them reflect from the ends, and watch how they interfere. It
can be useful to pause the simulation at different points, and then click the “step” button to
move the simulation forward slowly, one time step at a time. See if you can understand how
the superposition is formed from the individual pulses.
1
There is a caveat: for superposition to work, the waves do have to be regular, “linear” waves—but this is the
usual case, and that’s what we’ll talk about in this course.
2
http://phet.colorado.edu/simulations/sims.php?sim=Wave on a String
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3.1
Chapter 4 notes
Constructive and destructive interference
When two positive-displacement waves pass each other (as in the above example), the superposition
(the new wave, the total wave, the resulting wave), is “doubly positive”: the waves have added
where they overlap. We call this constructive interference, because the two waves are interfering
with each other, building something larger. It only lasts as long as the waves overlap! If a positivedisplacement wave meets a negative-displacement wave (imagine the slinky is flicked downwards,
or think of a pressure rarefaction rather than a compression) then when you add plus to minus,
they cancel! This is called destructive interference.
An example: suppose a loud sound is coming at you from the left. Another different loud sound
comes at you from the right. For most observers (in most spots), the two waves pass by at different
times, one before or after the other: each sounds normal. If you’re the one lucky person standing at
the right place and right time where they overlap, and if the pressure of these two sounds happen
to both be positive (overpressure), then you’ll hear an extra loud sound. But if one of them was a
positive pulse, and the other negative, then this lucky spot now hears nothing! Everyone else hears
the normal sound, but the two waves destructively interfere at the particular time and place they
pass through one another.
This is the basis for those cool noise-canceling headphones. They are not blocking sound at all.
In fact, they are adding sound: they are producing sound! What they do is monitor the noise from
the outside world, and then add some more sound waves heading towards you which are exactly
out of phase (exactly upside down, that is, they are low pressure where the original sound was
high, and vice versa). The result is that right at your eardrums those two sounds cancel, and you
hear...nothing, silence! You can’t cancel out sounds at all places and times. But you can cancel
out sounds at one particular place, that’s destructive interference!
We’ve been talking about “pulses”, but lots of waves (like sound!) are more continuous. An
ideal wave would be a sinusoidal or sine wave. Adding those is mathematically harder, although
you can make sense of it in the same way as the square pulses above. Figure 4.15 of the text is
the author’s attempt to draw superposition for sinusoidal waves. Look at 4.15a first. There are
two low sine waves that are almost overlapping. The sum is drawn (alas, in the same color, though
maybe slightly thicker) and you can see that it’s a similar looking wave but twice as tall: the two
have added. This is pure constructive interference.
In Figure 4.15b, Hall has drawn two sine waves that are exactly out of phase (one goes up
where the other goes down). Can you convince yourself that those two waves cancel everywhere?
In Fig 4.15c, there are again three curves. Two of them (slightly lighter) are the smaller ones. The
third one (slightly darker) is supposed to be the sum, the superposition of the other two. Trace it
out, and think about how the two lighter curves add up to give the third one.
In Fig 4.15d, the two curves which we’re superposing are both light, one of them is quite tall,
the other is not as tall, and is out of phase (upside down) from the first. The sum (superposition,
total) curve is darker, it’s not as tall as the tall one because the tall one is being partly cancelled
out by the second one. Think about these, draw them for yourself, and try to make sense of it!
3.2
Standing waves
Let’s think more about what happens if you have a lovely traveling sine wave heading to the right
on a slinky, and a lovely traveling sine wave heading to the left on the same slinky, and they pass
through each other. The simulations really help in visualizing this. If you try to do this on paper,
it’s hard—you’d want to draw a sine wave on two pieces of paper and slide them past each other,
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Chapter 4 notes
sketching the sum at various moments in time. You’ll need to convince yourself of what happens—
it’s a bit of a surprise. Look below at panel A in the figure: this is a picture at one moment in
time: the dotted curve is moving left, the solid curve is moving right. At this moment in time,
they are out of synch, and cancel. The sum (superposition) is just a flat line. The wave appears
gone at this moment. But it’s really still there, the slinky is moving, it just happens to be passing
through zero at this moment.
Now, let’s wait a moment (in fact, one fourth of a period). At this moment, as shown in panel
B of the figure, each wave has slid over by one fourth of a wavelength in its respective direction.
You have to first convince yourself that I really did slide the two curves the right amount (one
left, the other right). And then you have to convince yourself that adding them gives the dot-dash
curve—a doubly large sine wave.
And now you have to imagine going smoothly
from one of these pictures to the next. What you get
in this case is a standing wave. Panel C of the figure shows just the total (superposed) wave at three
different moments in time. The dark black curve is
one moment, the horizontal line is one fourth of a
period later, the faint grey is another one fourth of
a period later still. It no longer looks like something
moving left, or moving right, it looks like a wave
wiggling in place. It looks just like a violin string—
or the demonstration we did in class of waves on
a string! There are places where the waves always
cancel, they never move. These are called nodes.
There are other places where the wave oscillates up
and down a lot, called antinodes.
Simulation: Standing waves on a string. Go back
to the wave on a string PhET simulation, linked in this footnote3 , or see the link to this
simulation from the course web page (click on “Wave on a string” under “Resources”). I
recommend that you go to continuous mode, and turn the damping down to zero (so the
wave doesn’t die away along the length of the string). Then you can start the wave, see how
it reflects from the ends, and see how the interference creates a standing wave. Again, it’s
useful to pause the simulation at different points, and then click the “step” button to move
the simulation forward slowly, one time step at a time. Can you see the string changing in
time as shown in panel C of the figure? Can you find the nodes, and see how the string at
the nodal points never moves up and down?
Try to visualize this (the simulations are helpful here). Picture a string which is smoothly
flipping up and down at the antinodes. This standing wave is what happens when two idenitcal,
continuous waves pass through each other. In a violin, if you pluck it, you send waves left and right.
The waves reflect off the ends, and pass through each other (over and over). So it’s no accident
that the violin string wiggles like this—it really is a standing wave.
3
http://phet.colorado.edu/simulations/sims.php?sim=Wave on a String
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3.3
Chapter 4 notes
Beats
We can now understand a phenomenon we’ve seen (heard) in class. If you produce two sounds
which are nearly but not quite the same frequency, you hear beats. The sound seems to get louder
and softer with time. This happens when you tune a guitar, and two strings are nearly the same
frequency, but not quite: you can hear the beating. The explanation is drawn in Figure 4.16, but
it is a little hard to follow. Let’s walk through it. In the top part of the figure, there are two
different sine waves, which represent the two different sound waves. One of them is slightly higher
frequency than the other—it wiggles up and down just a little faster. So if you start drawing the
two sine curves, although you might start drawing them on top of each other, one of them slowly
but steadily gets out of synch (or out of phase). After a while, they are totally out of phase, one
is up when the other is down. But as you continue on (in time, which is the horizontal axis), they
drift back into phase again, because remember, one of them wiggles a little faster than the other.
So stare again at the top part: it’s two sine curves, each one of which is perfect and ideal with one
constant amplitude and frequency. But the two frequencies are off, so one of them wiggles faster,
and so they get in and out of synch. Can you see it?
Now look at the curve in the lower part of Figure 4.16. It’s the sum of the two waves above. At
every time, the two sounds superpose, they simply add. When they’re in synch, they “double up”,
making a nice tall sine wave. When they’re out of synch, they cancel out. And they go steadily
back and forth from one to the other, building up, then canceling, smoothly. That’s the solid curve
drawn below, that looks like a sine wave that gets bigger, then smaller, then bigger, then smaller.
Now, the author has added some dashed lines that kind of show the “envelope” or amplitude of
this sum. You can see that the amplitude of the sum gets bigger and smaller, itself in a sine wavey
kind of way! So what you hear is a sound of fixed frequency—pretty much the average of the two
frequencies, which were nearly the same anyway—but getting louder and softer—the amplitude is
getting bigger and smaller, remember, that’s what loud and soft corresponds to. That’s beating!
The dashed curve is itself a sine wave, it has a frequency which is the difference in frequencies of
the two original sounds. That’s called the beat frequency. It tells you how rapidly the sound gets
louder and softer. The closer the two original sounds are to one another, the smaller the difference,
that means the beating is happening less frequently. So when you tune a guitar, you listen to the
beats (you listen for the loud-soft-loud-soft pattern). If that pattern is happening slowly, i.e., it
takes a long time to beat, to go from loud to soft, then you know you’ve gotten the two frequencies
really close to one another. If they match exactly, the beat frequency goes to zero, there is no
more beating, they just superpose to make a louder, steady sound at the common frequency. Then
they’re in tune!
So, beating is a temporal thing. It happens when two sound sources are at the same place
(or nearly so), and they’re producing sound which is superposing first constructively (loud), then
destructively (soft), over and over. As long as the beat frequency is low (say, from half a Hz to
maybe 5 Hz), you can detect it. If the beat frequency is higher than maybe 10 Hz (i.e., if the two
sources are getting pretty far apart in frequency), then the beating happens too rapidly for you to
perceive it directly, you just hear a kind of mishmash sound. (The text says 30 or 40 Hz is the
limit, I think it might depend on you, and also on the particular value of the average frequency
too.)
If the beats are slower than 0.5 Hz, you hardly notice it because it’s such a gradual drift in the
loudness it’s hard to spot. So, when you tune a guitar, if the two strings agree to less than about
0.5 Hz, you’ll probably call it in tune.
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3.4
Chapter 4 notes
Spatial interference
There is another kind of interference that can happen—if two speakers are producing sound that
is exactly the same frequency, and exactly the same amplitude. So, at first, you might think that
you’re just going to double up the sound: they’re both the same, doesn’t adding a second speaker
just always make the sound louder? But if the speakers are spatially separated, you can get some
interesting effects from interference!
Look at the panel A of the figure: it’s a picture of a
sound wave produced by the left speaker alone. I’m graphing air over-pressure as a function of position in the room
at one particular instant in time. So this is a snapshot
of air pressure. Where the curve is high means the pressure is high. Where the curve is low, pressure is low. I’ve
drawn you (the happy face) at some point away from the
speaker. At this instant in time, you feel overpressure.
Panel B shows the wave at a later time. Imagine the
movie: this wave would appear to be moving steadily to
the right, like our wave-on-a-string simulation. You are
now located at a node, a spot where pressure is normal
(not high or low). This is exactly one quarter of a period
after the previous snapshot—the wave has changed one
quarter of the way from one high to the next.
Panel C shows the sound wave produced by the right
speaker alone. At this time, you feel overpressure, you’re
located where the wave is peaking. Now consider listening
when you are right smack in between the two speakers,
as shown in panel D. Since the distance to each speaker
is identical, then when one speaker is producing a high
pressure at your location, so is the other one. They’re in synch, they add up, the speakers are
doubling up the sound. If you wait half a cycle later in time, each speaker is now producing a
low pressure at your location. But both are doing that, and two lows superpose (add) to make the
pressure doubly low, as shown in panel E.
If you add (superpose) the two graphs, the fluctuations are twice what they would be from one
of the speakers alone. So indeed, it just sounds louder: two speakers are louder than one, it all
seems logical! (The sum of these two traveling waves is actually a standing wave, exactly what we
described a couple pages ago. It’s two traveling waves passing through each other. The person is
sitting at an antinode, where the fluctuation is high—it’s loud there.)
But now imagine stepping a little to one side. You’re still in between the speakers, but you’re a
little closer to speaker number 1. If you go just one fourth of a wavelength over, you move from an
antinode to a node. At that spot, it is silent. There is no variation in the pressure at any time: at
a node, the pressure stays normal all the time. At that place in space, a microphone or ear would
detect nothing. So there are loud spots and quiet spots in the room, in between the speakers. If
you placed a chair at a node, you could crank the speakers as loud as you want, and still hear
nothing (in an ideal world, of course, with no other reflected sounds coming at you, and with your
head “localized” at the node).
This last example was one dimensional. What about in the real world where you are located
somewhere else in the room, not in the line between the speakers? The same effect occurs—there
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Chapter 4 notes
will be “hot spots” where the waves constructively interfere, and nodes where they destructively
interfere. The pictures are a little more complicated to draw. Hall tries to show this in two pictures,
figure 4.13 (which shows only the “high pressure lines” in a top down view) and figure 4.14, which
shows some individual sound waves (kind of a sideways view). You need to just stare and think
about those figures to make sense of them.
Simulation: Spatial interference. Go back to the webpage in this footnote4 , or see the link on
the course web page (click on “Sound waves” under “Resources”). Click on the “Two Source
Interference” tab and play with it for a while. I recommend that you turn on the audio as
heard by the listener (check the “Listener” button), and turn the frequency down to around
200 Hz, so the wavelength of the sound is fairly long. Move the listener around and see what
happens. Can you see the places where the picture of the sound pressure has a high contrast,
where there are alternating light and dark blobs? Place the listener in that area, and you
should hear the sound get louder. Then can you see the places in the picture where the
contrast is low, where it just looks gray? Position the listener in one of those areas, and the
sound should get soft. Can you find a spot to place the listener where there is pure destructive
interference, where the sound disappears completely?
Here’s the bottom line. If you are located at some point in the room, draw a line to each speaker.
If the distance to each speaker is exactly the same, then the waves that come from them will be in
synch and add up—the highs are higher, the lows are lower, and you get a double amplitude sound.
This would also be true if the distance to one speaker was exactly one full wavelength longer than
the other. (Because after traveling one more full wavelength, a high is again a high, and a low is
again a low.) If, on the other hand, the distance to one speaker is half a wavelength different, then
the wave that has traveled half a wavelength more is now out of synch—half a wavelength takes
you from a high to a low—and they cancel.
So it’s a geometry puzzle now. Find all points in the room which are exactly half a wavelength
farther from speaker #1 than they are from #2. (If the distance is one and a half, or two and a half
wavelengths, that also works.) Those will be nodal points, and it will be quiet there (assuming
the two speakers are producing the same exact sound, in phase). Notice that the location depends
on the wavelength, so although one pitch (one wavelength) might have a node at some point, a
different pitch (different wavelength) might be at an antinode at that same point! So this is not the
easiest effect to notice in real life. It also gets complicated by multiple reflections! Still, if you’re
setting up speakers in a fancy room designed for good sound reproduction, you have to think about
these effects, and take care that you don’t put people in spots where there might be noticeable
nodes!
This section was filled with fairly complicated ideas, involving wave properties, geometry, and
visualization of the time dependence of waves. Take a look again at the textbook (section 4.5).
It might make more sense on the second pass through. Try to make sense of it! This idea of
waves interfering is quite important when you try to understand sound, and music. Indeed, the
standing wave is a very central idea behind understanding what is going on in the strings of string
instruments, or the air inside a wind instrument , so that idea also will come back many times in
this course. Ask, if you’re still puzzled. It’s not easy!
4
http://phet.colorado.edu/simulations/sims.php?sim=Sound
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4
Chapter 4 notes
Diffraction
Moving on (back) to section 4.2, the main point of this part of the chapter is to point out that
sound waves can do funny things! We’ve already looked at interference effects (that was the whole
last part of this chapter), but there’s another oddball behaviour of waves when they go through
small holes, or reach sharp corners. Waves can bend. Just look at water waves in a pond: they
really do curve around corners! Look at Fig 4.5 and 4.6 of the text— you can start with pretty
“flat”, ordinary waves coming in, traveling basically in a straight direction (from the right in 4.5,
and then bending down around the sharp corner), or from the top in Fig 4.6 (notice they are coming
in totally straight from the top in that figure, imagine waves at a beach, or sound waves that come
from far away and are heading straight down the page). There’s a small hole, and then the waves
“spread out” again from the hole. All of these bending effects are called diffraction. The small
hole acts like it was a point source of sound, and of course sound spreads out spherically from a
point source (like a speaker: a point source means that the sound is generated at one point).
There is some fancy math associated with diffraction which we won’t go into. The big idea
which you should be aware of is this: the amount of bending depends on the wavelength! If the
wavelength is small (high frequency), then the waves don’t bend much. If the wavelength is large
(low frequency), then the waves bend a lot. This may or may not be intuitive. (Remember that
it was the small wavelengths that were more likely to notice if the wall was rough, and scatter
diffusely from it.) Small wavelength (high frequency) sound tends to go through the hole the way
you might expect a beam of particles to behave: the hole makes a sharp shadow. In a sense, the
hole looks big to the small wavelength sound, and a big hole doesn’t do anything weird. The sound
doesn’t bend, it behaves more like a particle (like a tennis ball: if the hole is larger than the ball,
the tennis ball just flies through the hole with no change to its motion). But large wavelength (low
frequency) sound acts more “wavey”, bending and coming out like the hole was a point source.
(The small hole really does act like a point when compared to large wavelength sound.)
Take a look at Fig 4.7. It’s similar to 4.6, except the wavelength is smaller (the high pressure
lines, moving down the page from the top, are more closely spaced). And they don’t bend as
much—there’s a “shadow region” behind the hole (which is exaggerated in size in Fig 4.7). All
waves diffract like this, even light! It’s hard to notice, though, because the wavelength of light is
really small. So, light behaves mostly like it was a beam of particles, and you get sharp shadows
behind holes. But if you make the hole small enough, you can get awesome diffraction effects, where
the light bends around corners. Laser light from a laser point (which sure seems to travel in a very
straight line!) will bend if you shine it through a small hole, and head out in all sorts of different
directions on the other side!
Diffraction explains a lot of maybe surprising things about sound: first, sound can go through
open doors or windows, and it goes in all different directions on the other side of the opening. You
don’t have to be in the line of sight of the speaker. The sound diffracts through the doorway.
Higher frequencies diffract less: they cast sharper shadows. So people speaking in high voices
are harder to hear through the doorway if they’re not in a straight line—they are quite muffled
and faint. (Remember that multiple reflection might still let their voices reach you. That’s not
diffraction, it’s just sound bouncing off walls!) On the other hand, a low tone can be heard coming
through that same door at all sorts of different angles, the sound paths bend and spread out.
The long-wavelength, low-frequency sound (remember, wavelength is already 1 meter, much bigger
than the door, even for ordinary sounds around 344 Hz) acts like the door was effectively a point,
spreading out every which way from it.
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Home experiment: Play around with sound diffraction yourself. To do this, you need a stereo
system in a room with a door: play some music and then listen both inside the room and
at different points outside the door. For this to work well, make sure the music you are
listening to has a range of frequencies, both treble and bass. It also helps if you can avoid
multiple reflections, for example by making sure the room outside the door is quite large (or
is outside). How does the music sound different when you are in the room with the stereo,
versus outside the door? Can you hear the effect of the bass notes “bending” (diffracting)
at the door and heading out in all directions? How does it change if the door is partially
closed—can you hear a difference in which frequencies reach your ear?
If you turn and face away from me (even outside where there are no walls to make multiple
reflections), I can still hear you, because the sound diffracts out of your mouth and goes in all
directions (even backwards!) Your mouth is quite small, so even high frequencies will still diffract
coming out of it. Remember, you diffract if the hole is smaller than the wavelength! So smaller holes
cause more bending! Bass sounds coming out of big speakers, likewise, will go every direction. But
very high pitched sounds will go more preferentially in a straight line, they won’t ”bend” around
the speaker as much. So, you use smaller speaker openings for the treble sounds, so they’ll go out
in more directions!
Diffraction also affects sound perception and how well we can hear the directions that sounds
come from. Low pitch sounds will diffract around your head, so a low sound coming from your right
side will sound almost equally loud in both ears. But high pitched sounds will bend less around
your head: you cast a bit of a sound shadow. So there will be a stronger difference in loudness
between your two ears for a sound coming from one side. This is one way your brain can help you
localize where a sound is coming from—and it will be different for different pitches!
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Outdoor music
The next topic in this chapter I want to cover (Section 4.3) is some applications of all these ideas—
to outdoor music. Although the pictures are a little complicated, I’m hoping that this introduction
should make that section a little more straightforward. The author is going to come back to
acoustics of music halls later; this is just some initial observations.
The key points of this section are the following:
1. Sounds are weaker outside. There are fewer multiple reflections! So, you need loud music and
lots of amplification to make it sound good.
2. Sounds are “deader” outside. This has to do with the fact that indoors, the sound comes to
you from various paths (because of multiple reflections), and therefore gets a little spread out
in time. But outside, each sound comes and goes, there’s much less reverberation or richness
to the sound. (But this richness can be electronically mimicked.)
3. It’s harder to make everyone have the same listening experience outside. Indoors, the multiple
reflections tend to eliminate soft spots and loud spots, but outdoors, if there’s a bad spot,
there’s no obvious way to fix it.
The author then shows some geometrical tricks that help make outdoor sound a little better. A
reflecting shell behind the musicians can send “backwards going” music (sound moving away from
the audience) forwards (toward the audience). This increases the energy of sound available to the
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Chapter 4 notes
listeners, and adds some multiple reflections to fix some of the problems above. Of course, amplifiers
and speakers will also change this story totally. This section is talking about live, acoustic music—
go out to the band shell in downtown Boulder and look at how it’s designed. You’ll recognize the
story of Figure 4.9 and 4.10 there!
The shape of a band shell can matter—you can make it a kind of “lens” to make sound that
starts at one point either “focus” at some other point (like in Fig 4.10a) or come out more evenly
(4.10b). Clearly, the latter would be better for a big, spread-out audience. For this to work, the
musicians better sit in the designated location!
In addition, you can try to capture the “upwards moving” sound and bounce it back to the
audience too—hence, the roof above the musicians matters also. Figure 4.11 shows several designs,
and points out that musicians need to hear each other too! So, the shape may have to be a
compromise between getting the sound out into the audience, and helping the musicians hear each
other.
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Doppler shift
The chapter ends with the Doppler shift. This is fun—read it if you’re interested, and I’ll
demonstrate it in class. The main idea is that sound waves travel, but if the source of sound is also
traveling, you get some interesting effects. Fig 4.12 shows where the high pressure fronts are if the
source is moving. In the direction it moves, the high pressure fronts are more closely spaced—that
means smaller wavelength, higher pitch! The person who the source is approaching hears a higher
pitch! This works in the opposite way on the back side. If the source runs away from you, you
hear a lower pitch, longer wavelength, more distance between peaks. Hence, the classic effect of an
ambulance siren (or police siren) sounding higher as they approach, and lower after they pass you.
For years, I thought they were really changing their pitch, and never noticed the odd coincidence
that they always changed pitch right when they passed me!
The figure (4.12) is pretty helpful, but you do need to try to think about a time-lapse movie
of it to really fully make sense of it. The point is that if the source sits still, the wave fronts move
out at some speed (the speed of the wave), and form growing circles around the source. A listener
perceives these fronts traveling past, at some frequency (the same frequency as they are produced).
If the source makes 440 wiggles each second, the listener hears 440 wiggles each second. This is
true no matter how far away they are. You need to think about that and convince yourself. The
wave fronts all move outwards at the same (constant) speed, so if they wiggle 440 times per second
at ONE place, they will wiggle 440 times per second farther out too.
If the source is moving towards you, every time it makes the next high pressure front, it’s closer
than it would have been if it sat still. So each consecutive wave front has less distance to travel
towards you, and so it gets to you earlier than it would have. If wave fronts reach you sooner, that
means you perceive them as coming more often: that means higher frequency! Your eardrum is
wiggling more times each second. That’s really all there is to the Doppler shift—it’s just the fact
that when there’s relative motion between the source (or listener) and the medium, the waves will
come more often (or less often, if it’s moving away from you), and that means the frequency is
different.
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7
Chapter 4 notes
Summary
Let me wrap up by reminding you of the bottom line that you should take away from Chapter 4,
and remind you of some questions you should be able to answer now.
Sound waves reflect (bounce) off surfaces. Those reflections can be diffuse or specular: it depends
on whether the surface is bumpy or smooth. If the size of the bumps is small compared to the
wavelength of sound, the surface looks smooth to the sound wave. If the size of the bumps is large
compared to the wavelength of sound, the surface looks rough to the sound wave. This means that
the same surface can diffusely reflect some wavelengths, and specularly reflect other wavelengths!
Sound waves also undergo multiple reflections—they can bounce off walls/ceilings/floors of rooms
many times.
When two different sound waves move through the same place, they interfere. To find out what
happens, you can just add up (superpose) the displacements or overpressures of the two waves.
Interference can be constructive (the two displacements add, leading to a bigger total displacement)
or destructive (the two displacements have opposite signs and they cancel, leading to a smaller total
displacement). Temporal interference helps create standing waves; it also produces the phenomenon
of beats. Beats occur when two slightly different frequency sound waves interfere, and what you
hear is a slow variation in the sound at the difference in the frequency of the two waves. Waves
can interfere spatially, particularly when you play the same pure tone (one frequency, in synch,
same amplitude) from two spatially separated speakers. This means that there will be loud spots
at some places in the room (where the two waves constructively interfere) and soft spots at other
places in the room (where the two waves destructively interfere).
Diffraction is the way that sound bends around corners and after traveling through holes (like
doors or windows). Long wavelengths bend around corners more than short wavelengths: the short
wavelengths experience more of a “shadow” due to obstacles. When sound travels through openings,
the important thing is the size of the hole compared to the wavelength of the sound: when the
opening is large compared to the wavelength, the sound wave doesn’t really “see” the edges of the
hole and it travels through mostly in a straight line. But if the opening is small compared to the
wavelength of the sound, the sound comes out of the other side of the opening as if it were being
emitted from a point source: the sound spreads out in all directions.
Sound reflection, interference, and diffraction all affect the way outdoor music sounds. Band
shells are designed to help improve the sound of outdoor music by reflecting sound toward the
audience.
When sound is emitted from moving sources (or any time the source and listener are moving
relative to each other) the sound experiences a Doppler shift: the perceived frequency of the sound
is different from the emitted frequency.
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