Download 08 Waves Chapters 8_-_waves_combined

TableofContents
Harmonic Motion ....................................................................................................................... 211 Harmonic Motion Terms ......................................................................................................... 212 Frequency and Period ............................................................................................................. 216 Energy and Amplitude ............................................................................................................. 216 Waves ......................................................................................................................................... 221 Transvers and Longitudinal Waves ......................................................................................... 223 Reflection ................................................................................................................................ 227 Wave Speed ............................................................................................................................. 228 Principle of Superposition ....................................................................................................... 229 Standing Waves ....................................................................................................................... 230 Sound .......................................................................................................................................... 234 The Doppler Effect ................................................................................................................... 236 Resonance ............................................................................................................................... 243 Waves Wrap-up .........................................................................................................................254
AP Physics - Harmonic Motion
We have been dealing with straight line motion or motion that is circular. There are other types of
motion that must be dealt with. Specifically, a motion that repeats itself over and over. Such a
thing has its very own fancy physics name. “The name?” the desperately curious, interested
advanced physics student asks? It is called, dear student, periodic motion.
Periodic Motion  motion in which a body moves back and forth with a fixed path over a
definite interval of time.
A special case of periodic motion is known as harmonic motion.
Harmonic Motion  Periodic motion with no friction that is produced by a restoring
force that is directly proportional to the displacement and oppositely directed.
Don’t you hate definitions like that? The Physics Kahuna is so sorry, but what can you do?
Let’s see how it works, which is really the best way to learn about it.
Okay, see, the uh, restoring force pushes and or pulls the object undergoing the harmonic motion
and is the thing that makes it happen.
Let’s look at a simple example of harmonic motion. A sphere is attached by a spring to a solid
block. The sphere (and spring) is free to slide back and forth on a smooth, frictionless surface.
F
F=0
F
The drawing above gives you an idea of the sequential motion of the system. Initially the spring is
compressed. This is shown in the first drawing. The ball is released and the spring pushes it
outward. In the second drawing, the spring has reached its normal displacement and is no longer
exerting a force on the ball. Ah, but the ball’s inertia and the good old first law ensure that the ball
keeps moving. As it does this, it stretches the spring. Of course this causes the spring to exert a
force in the opposite direction which slows the ball. This is the restoring force. Anyway, eventually
the spring force has had enough time to stop the ball. This is depicted in the third drawing. Once
the ball stops, the spring will pull it back in the opposite direction and the process will repeat itself.
211
Here are some critical terms that have to do with such motion.
Equilibrium position  this is the point in the motion where the object would be if it were
not subject to any forces.
Amplitude (A)  The maximum displacement from the equilibrium position.
Cycle  One complete iteration of the motion sequence.
Period (T )  The time it takes to do one cycle.
Frequency ( f )  The number of cycles per unit of time.
The drawing below gives us a depiction of the amplitude of the system.
A
A
A simple way to understand this motion is to look at a graph of displacement vs time. The graph
looks like this:
x
A
t
1 cycle
T
212
The displacement is plotted along the y axis and time is plotted along the x axis. The amplitude is
the maximum displacement value. On the graph, one cycle is the segment of the curve from in
phase point to the next sequential in phase point. In phase points? Okay, okay already. These are
points along the displacement path where the object is doing the same thing.
No better way to understand the thing than to look at another example. Here we have us a weight
attached to a vertically mounted spring. The weight is given an initial small displacement (which
will be the amplitude) and then released. It bobs up and down. Its velocity will vary as it moves
through the cycle. The velocity is zero at the top of the motion and at the bottom. The maximum
velocity occurs at the zero displacement position.
1
2
3
4
5
v= 0
vMax
v= 0
Acceleration and displacement: We can analyze the motion more fully by looking at the forces in
the system, the accelerations that are taking place, and the displacement.
We begin with Hooke’s law, which we studied when we dealt with energy. This is the force that is
exerted by the spring.
Hooke’s Law
Fs  kx
Recall that the minus sign simply means that the force exerted by the spring is always opposite to
the displacement.
The maximum force will occur at the maximum displacement, which is the amplitude, so we can
write an equation for the maximum force.
FMax  kA
Where FMax is the maximum restoring force, k is the spring constant and A is
the amplitude.
213
We can now apply the second law to find the maximum acceleration.
FMax  maMax
aMax 
FMax
m
We can plug FMax   kA into the acceleration equation we just developed:
aMax 
FMax
m
aMax  
  kA 
1
m

kA
m
kA
m
The acceleration is not constant because the force is not constant. We can write an equation for the
acceleration as a function of displacement.
a 
k
x
m
At zero displacement, the acceleration will be zero, since, as you can see, the displacement x is
zero. The acceleration is varying then between its maximum value, which takes place at the
maximum displacement (the good old amplitude) and zero.
The acceleration ranges from 
k
k
k
A to  A . In between it is given by: a   x
m
m
m
a Max
a= 0
- a Max
214
In the drawing above, we can see the positions that have the maximum acceleration. We have
chosen the coordinate system so that an acceleration to the right is positive and an acceleration to
the left is negative.
Another important idea is that the initial displacement given the system – we could call this the
release point – will determine the amplitude. So you pull the spring out and give it a displacement
of 5.0 cm - that means that the amplitude has to be 5.0 cm. Do you see why this is so?

(a)
A spring has a constant of 125 N/m. A 350.0 g block is attached to it and is free to slide
horizontally on a smooth surface. You give the block an initial displacement of 7.00 cm. What
is (a) the maximum force and (b) the maximum acceleration acting on the block?
To find the force we use Hooke’s law:
N

FMax  kA   125   0.0700 m  
m

8.75 N
Note that the Physics Kahuna cleverly gave the thing a negative displacement so it
would all come out positive.
(b)
To find the acceleration we use the second law:
F  ma
a
F
m

8.75 N
0.350 kg

8.75
kg  m
s2
0.350 kg

25.0
m
s2
Obnoxious Facts:








Windmills always turn counter-clockwise--except for the windmills in Ireland.
Zero is the only number that cannot be represented by Roman numerals.
Janet Guthrie was the first woman to race in the Indianapolis 500.
Karate actually originated in India, but was developed further in China.
The expression getting someone's goat is based on the custom of keeping a goat in the
stable with a racehorse as the horse's companion. The goat becomes a settling influence
on the thoroughbred. If you owned a competing horse and were not above some dirty
business, you could steal your rival's goat (seriously, it's been done) to upset the other
horse and make it run a poor race. From goats and horses it was linguistically extended
to people: in order to upset someone, get their goat.
The game of squash originated in the United Kingdom. It came about after a few boys,
who were waiting for their turn to play racquets, knocked a ball around in a confined
area adjoining the racquets court.
The hammer throw is illegal as a high school sport in all states except Rhode Island.
A cubic mile of fog is made up of less than a gallon of water.
215
Relation between Frequency and Period:
The frequency of the system is the number of cycles per unit time. We can write an equation for
this:
f 
number of cycles
time
f 
1
T

1 cycle
Period
Therefore:
f is the frequency and T is the period.
T
Also
1
f
The unit for frequency is the Hertz. (It used to be a cps, a cycle per second, i.e.,
cycle
, but they
s
changed it back in the turbulent 60’s.)
1 Hz 

1
s
The period of a system is 0.00210 s. What is its frequency?
f 
1
T

1
0.00210 s

476 Hz
Energy and Amplitude:
The total energy of a system undergoing harmonic motion must,
by law, remain constant. But of course the type of energy can be transformed from one type or
types to another.
Clearly the energy of the system must be all potential energy when the displacement is at its
maximum value. This is because the velocity of the system is zero. This will occur when the
displacement is equal to the amplitude. Once the mass is moving away from maximum
displacement, some of the potential energy is converted to kinetic energy. The kinetic energy
increases and the potential energy decreases until the displacement is zero. At this point all the
energy is kinetic and the object is moving at maximum speed. The kinetic energy then begins to
decrease and the potential energy increases until the amplitude is reached again. And so on. It
looks like this:
216
PE Max
KE Max
PE Max
The potential energy for a spring is given by this equation: (the Physics Kahuna is positive that you
remember the equation with joy)
US
1
2
 kx 2
If we plug in the amplitude for the displacement, we get an equation for the energy of the system.
U
1
2
 kA2
This is the maximum amount of energy that the system can have – the system will get no other
energy.
The Law of Conservation of Energy tells us that the total energy of the system must remain
constant, so this equation gives us the energy that the system has at any point in the cycle.
Also not that the energy is proportional to the square of the amplitude.
We can plot kinetic energy vs time, potential energy vs. time, and displacement vs. time and then
compare them.
When we do this, we get three graphs that look like this:
217
x
Kinetic
Energy
Potential
Energy
t
t
t
You should be prepared to mark up a drawing or a graph of a harmonic motion situation and locate
the places where the kinetic energy is at a maximum or minimum, the potential energy’s maximum
and minimum values, where the velocity is zero or maximum, where the acceleration is minimum or
maximum, and where the forces are minimum or maximum.
Calculating Energy:
From the law of conservation of energy, we know that the energy must
stay constant for the system. This means that:
 K  U  U S i   K  U  U S  f
But since we are ignoring gravitational potential energy, the expression is simplified to:
 K  U S i   K  U S  f
We also know that the total energy of the system must be:
US
1
2
 kA2
The energy of the system for any displacement must be:
1 2 1 2 1 2
kA  mv  kx
2
2
2
kA2  mv 2  kx 2
218
Period of spring system:
The period for a spring system moving with a small displacement
is given by this equation. (You will be provided with this equation on the AP Physics Test).
T  2
m
k
T is the period, m is the mass of the weight (we are ignoring the mass of
the spring, as it is usually insignificant), and k is the spring constant.

A 345 g block is part of a spring system, is oscillating. The spring constant is 125 N/m. What
is the period of the system?
T  2
m
k


1
T  2 0.345 kg 
 125 kg  m  1

m
s2








0.330 s
In the real world, we have to deal with friction and energy losses, so that the actual graph would
look like this:
X (m)
t (s)
The amplitude decreases over time, although the period and frequency do not change. This
decrease in amplitude deal is called dampening or attenuation (takes your pick). We will, in
solving our little problems, ignore this dampening effect.
Period of a Pendulum: Another simple harmonic system is a pendulum.
pendulum is constant for small angle swings (for large angle
swings, the period is not a constant). Galileo is credited with the
discovery that the pendulum period is constant. He supposedly did
this in a cathedral. Apparently not paying as much attention to the
priest as church officials would desire, he grew bored and noticed
that a chandelier was swinging back and forth. Using his pulse as a
timing device, he discovered that the chandelier took the same
amount of time to make each swing regardless of how big the swing
was. Later he used pendulums to time his experiments. Eventually
pendulums were used to regulate the motion of clocks. This led to
the first accurate clocks. (One can argue that this led to the
horrible, frantic modern world with its insane preoccupation with
The period of a
 
219
time, but he Physics Kahuna will not go there.)
The period of a pendulum is given by this formula (which will also be available to you on the AP
Physics Test).
T  2
L
g
T is the period, L is the length of the pendulum, and g is the
acceleration of gravity.
Note that the period of a pendulum depends only on its length.
Pendulums are commonly used to experimentally find a value for g, the acceleration of gravity in
physics experiments. How would you go about doing that?

A tall tower has a cable attached to the ceiling with a heavy weight suspended at its bottom near
the floor. If the period of the pendulum is 15.0 s, how long is the cable?
This is a simple problem – we use the period equation and solve it for the length of the pendulum.
L
T  2
g
T  4
2
m
2

 9.8 2  15.0 s 
s 
L
4 2

2L
L
g
gT 2
4 2
55.9 m
220
AP Physics - Waves
The class, this AP Physics thing, has been terrific so far, hasn’t it? Motion and mechanical energy
were awesome were they not?
It’s time to build on the energy thing and keep the excitement moving along. This will surely
happen with our next topic – WAVES. Waves turn out to be one of the ways that energy can be
transferred from one place to another.
Out at sea, the waves roll on.
Waves  disturbances that travel through space transferring energy from one place to
another.
Sound, light, and the ocean's surf are all examples of waves.
A wave is a disturbance that travels through space, but what do that mean?
Think of how gossip travels through the country. This was an example that Einstein used to explain
waves. A rumor starts in one of the math classes among the students – Mr. Gunderson says. “Hey,
221
I’m thinking about getting a sports car.” One of the students hears the tail end of the statement and
tells a friend out in the hall, “Mr. Gunderson just bought a sports car, I think he got a Ferrari.” The
Physics Kahuna supposes that this is possible. Anyway, two hours later one of the students in the
Kahuna Physics Institute tells the Physics Kahuna, “Hey, Mr. Gunderson just got a Ferrari.” This
student was not in Mr. Gunderson’s class. So how did the gossip get from the math wing to the
most southern part of the building? Very likely it happened through a series of conversations.
Student A told student B who told student C and so on until the Physics Kahuna heard about it. No
one student physically traveled the whole distance from one end of the school to the other with the
information, but the scuttlebutt (the proper Navy term for gossip) did. This is how waves work. The
medium would be the mass of students teeming in the halls of our beloved CCHS. The disturbances
would be the little conversations between people.
Have you ever played the gossip game? You have a whole bunch of people and they are all placed
in a line. The first person is given a bit of gossip and repeats it to the next person who repeats it to
the next person and so on until it finally reaches the last person in the line. The object of the game
is to see if there are any changes to the scuttlebutt as it travels down the line. But the Physics
Kahuna’s object in bringing the thing up is to show you how it behaves like a wave – a bunch of
disturbances that pass through a medium.
Well, back to the waves. Here is where we introduce the facts.
There are two species of waves, mechanical waves and electromagnetic waves. Mechanical waves
require a medium that the wave will then travel through, or rather, the disturbance will travel
through. Electromagnetic waves do not require a medium. The disturbance that travels is a
changing magnetic and electric field (both of which you, lucky student, will get to study in the near
future). We’ll examine these waves later on in the course.
A key concept here is that the only things that moves is the disturbance. The medium itself does not
move.
That waves carry energy should be obvious. Picture the waves on the ocean. Waves are generated
far out at sea mainly by the wind. The wave travels through the water for hundreds or even
thousands of miles. Finally it reaches the shore where the waves pound against the beach. They
have enough energy to break down the coastline and erode away continents.
Traveling Waves: The traveling wave is a sort of bump that travels through a
medium. A good example of a traveling wave would be a pulse sent down a rope. Such a thing is
shown in the drawing below.
Note that the rope itself does not travel, just the pulse, which is the disturbance.
222
You will have seen traveling waves in one of the class demonstrations on a long slinky spring.
Another type of wave is the continuous wave. Sometimes these are called wave trains. A
continuous wave requires a periodic source of energy and a medium for the wave to travel through.
There are two types of mechanical waves, the transverse wave and the longitudinal wave.
Transverse Wave - The disturbance direction is perpendicular to the wave
direction
Longitudinal Wave - The disturbance direction is parallel to the wave direction
Transverse Wave
Wave direction
Disturbance direction
Longitudinal Wave
Wave direction
Disturbance direction
The Physics Kahuna will have shown you a lovely demonstration of these two types of waves.
Here are some examples of these waves:
Transverse waves – water waves, waves on a string or spring, seismic waves, and
electromagnetic waves.
Longitudinal waves – well, the best example would
be sound waves, but you saw them on springs as
well.
crest
trough
223
A transverse wave is made up of a series of positive pulse and negative pulses. The positive pulses
are called crests and negative pulses are called troughs. The height of the crest and the depth of the
trough is of course the amplitude of the wave.
Longitudinal waves are a bit different. Basically the medium gets scrunched or pushed together and
then pulled apart – stretched out. The areas of increased medium density are called compressions.
The compression is surrounded on either side by an area where the medium is stretched out. These
areas of low medium density are called rarefactions.
Rarefaction
Compression
A really important, like key-type concept is that it is only the disturbance that moves, not the
medium. Take you your basic water waves for example. These waves can travel hundreds of miles
across the sea. But only the wave travels. The disturbance in water waves is the changing height of
the water – the water goes up and it goes down. A duck sitting on the water would not be swept
forward with the wave. Instead the duck would bob up as the crest of the wave passed beneath it
and then the duck would go down as the trough went past. The Physics Kahuna has made a
sequential type drawing of the very thing.
Water Wave Passing
Under a Duck
224
Dreadful trivia:













It's possible to lead a cow upstairs...but not downstairs.
The Bible has been translated into Klingon.
Humans are the only primates that don't have pigment in the palms of their hands.
Ten percent of the Russian government's income comes from the sale of vodka.
Ninety percent of New York City cabbies are recently arrived immigrants.
On average, 100 people choke to death on ballpoint pens every year.
In 10 minutes, a hurricane releases more energy than all the world's nuclear weapons
combined.
Reno, Nevada is west of Los Angeles, California.
Average age of top GM executives in 1994: 49.8 years.
Elephants can't jump. Every other mammal can.
The cigarette lighter was invented before the match.
Five Jell-O flavors that flopped: celery, coffee, cola, apple, and chocolate.
According to one study, 24% of lawns have some sort of lawn ornament in their yard.
The frequency of a traveling wave is simply the number of cycles divided by the time they occur
within (or something that sounds more scientific than that, the old Physics Kahuna is stretching – at
the limits of his writing skills).
f 
n
t
Here f is the frequency, n is the number of cycles (and has no unit) and t is
the time.

A speed boat zooms by you as you lie on your floating mattress. You find yourself bobbing up
and own on the waves that the boat made. So, you decide to do a little physics experiment.
You count the waves and time how long it takes for them to go past. Six wave crests go by in
five seconds. So what is the frequency?
f 
n
t

6.0
5.0 s

1.2 Hz
Below is the plot of a transverse wave. The displacement is plotted on the y axis and distance is
plotted on the x axis. The amplitude, A, is shown. This is the maximum displacement, just as it
was for periodic motion. The other thing that is shown on the graph is the wavelength - . The
wavelength is the distance between two in phase points on the wave.
225
Y
A
X

The wave is traveling at some velocity v. We know that velocity is given by this equation:
v
x
t
We also know that the wave travels a distance of  in the period, T.
We can plug these into the equation for velocity:
v
x
t


so
T
v

T
But we also know that the period is given by:
T
1
f
so
We can plug this in for T in the velocity equation we’ve been working on:
v

T
 f 
This gives us a very important equation:
v f 

A middle C note (notes are these musical frequency kind of deals) has a frequency of
approximately 262 Hz. Its wavelength is 1.31 meters. Find the speed of sound.
v f

 262
1
1.31 m  
s
343
m
s
A wave has a frequency of 25.0 Hz. Find the (a) wavelength, (b) period, (c) amplitude, and (d)
velocity of wave. A graph of this wave is shown below.
226
35.0 cm
Y
12.0 cm
X
(a) Amplitude:
We can read the amplitude directly from the graph:
12.0 cm
This can be read directly from the graph as well.
35.0 cm
(b) Wavelength:
(c) Period:
The period is the inverse of the frequency, which we know.
T
1
f

1

1
25.0
s
0.0400 s
(d) Velocity: We use the wave velocity equation.
v f
 25.0
1
 0.350 m  
s
8.75
m
s
Here’s another problem.

The speed of light is 3.00 x 108 m/s. What is the wavelength for an FM radio signal broadcast at
105.3 MHz? (Note, radio waves all travel at the speed of light.)
v f

v
f




1
m
  3.00 x 108 

s  105.3 x 106 1 

s 

 0.0285 x 102 m

2.85 m
Wave Dynamics:
When a wave is busy traveling through a medium, it’s a beautiful thing.
But what happens when a wave travels from one medium to another? What happens when two
waves meet up? These are good questions and the answers are even better.
Reflection: When a wave traveling through a given medium encounters a new medium, two
things happen: some of the energy the wave is carrying keeps going on into the new medium and
some of the wave energy gets reflected back from whence it came. If the difference in the wave
velocity is large, then most of the wave will be reflected. If the difference in velocity is small, most
227
of the wave will be transmitted into the new medium. The junction of the two mediums is called a
boundary.
If there is no relative motion between the two mediums, the frequency will not change on reflection.
Also, and this is a key thing, the frequency does not change when the wave travels from one
medium into another. It stays the same. This means that the wavelength does change.
There are two types of reflection. The type of reflection
depends on how the mediums at the boundary are allowed to
move. The two types are: fixed end reflection, and free
end reflection.
For fixed end reflection think of the medium as being
constrained in its motion. In the picture to the left you see a
string that is securely fixed to the wall. The string (the old
medium) is free to move up and down, but at the boundary
where it meets the new medium (the wall) it is constrained –
the string can’t really move up and down like it could
before. In fixed end reflection, the wave that is reflected
back is out of phase by 180. In the drawing you see an
erect pulse traveling down the string. When it is reflected it
ends up inverted. It will have the same speed going in as
coming out. So in fixed end reflection an erect pulse would
be reflected as an inverted pulse.
In free end reflection, the medium is free to move at the
boundary. The reflected wave will be in phase. In the
drawing on the right, you see an erect pulse traveling into
the boundary being reflected with no phase change. The
pulse went in erect and came out erect. Water waves
reflecting off a solid wall are a good example of free end
reflection.
Wave Speed: For a wave on a string, the speed of
the wave is directly proportional to the tension in the
string. Increase the tension and the wave velocity will
increase.
The speed of sound waves in air is directly proportional
to air temperature and directly proportional to the air
density. In other words, as the temperature of the air increases, the speed of sound increases. As
the density of the air increases, the speed of sound also increases. For a given air temperature, the
speed of sound would be less in Gillette than it is in Orlando because the air is less dense in Gillette
than in Orlando due to the greater altitude of our fair city.
228
Principle of Superposition:
What happens when two or more waves encounter each other
as they travel through the same medium? The waves can travel right through each other. As they
do this, they add up algebraically to form a resultant wave.
Toss a pebble in a pond and it makes a series of waves that spread out
in expanding circles. You can see a drawing of the resultant wave
pattern from such an event the waves travel outward in a series of
expanding wave fronts. In the drawing, each of the dark lines
represents a wave crest. As the wave front expands, the energy of the
wave gets spread out and the wave crest decreases in amplitude.
Eventually the energy is so spread out and diluted that the wave will
cease to exist. This decrease in amplitude from spreading is called
dampening or attenuation. This is why you couldn’t hear your mom
calling you home when you were a wee tyke several blocks away.
What happens when two pebbles hit the water? Both produce waves and where the waves meet
they produce interference patterns.
Here is a drawing showing a set of interference patterns from the two pebbles in the water deal.
These interference patterns occur where the wave crests and troughs meet each other. The
interactions behave according to the law of superposition
Law of superposition  when 2 or more waves meet, the resulting
displacement is the algebraic sum of the individual separate wave
displacements.
These interference patterns will be of great importance later on when we study light.
Basically, the waves add up or cancel each other out. Waves can add up constructively - we get
constructive interference, or they can add up destructively - destructive interference.
Where two crests meet, they add up to make an even larger crest. Where a crest and a trough meet,
they add up destructively - subtract from each other. So if a wave meets another wave that has the
229
same amplitude but is out of phase (crest to trough, so to speak) they will completely cancel each
other out.
+
+
=
=
cancellation
Constructive interference
v
Destructive interference
v
v
v
Meeting waves
out of phase
Meeting waves
in of phase
Standing waves:
If you take a long, slinky spring and fix one end of it to a wall and then
shake the free end you produce a pulse that travels down the spring. The pulse will be reflected
when it reaches the end of the spring. This would be fixed end reflection so it would be out of
phase. If you just wave the end of the spring up and down, you get a very confused, chaotic looking
thing. But, if you wave the end of the spring at just the right frequency, you can produce a standing
230
wave. You produce an incident wave that travels down the rope. If the frequency is the correct
value, the incident wave and the reflected wave will alternately interfere with each other
constructively and destructively. The effect is that parts of the spring will not move at all and other
parts will undergo great motion.
The two waves moving in opposite directions will form a standing wave. The law of superposition
acts and we get constructive and destructive interference, which forms the standing wave.
You will have observed standing waves in class with some of the demonstrations that we did. The
parts of the wave that don't seem to be doing much are called the nodes and the places where the
wave is undergoing maximum movement are called antinodes. The end of a string with such a
wave that is attached to the wall would have to be a node, would it not?
You can produce a variety of standing waves by controlling the frequency of the wave. You will
have seen a delightful demonstration of standing waves in action.
1
2
3
5
6
7
8
4
Time Lapse View of Standing Wave
Musical instruments produce standing waves. Piano strings, the interior of a tuba, a flute, and
violin strings all produce standing waves. Buildings being buffeted about by the wind also have
standing waves. Both transverse waves and longitudinal waves can form standing waves.
231
The standing wave to the left represents one half of the
wavelength of the wave or ½ .
This would be a complete wave cycle or 2/2  or 1 .
This would be 3/2  or 1 ½ wave.
The lowest frequency standing wave for a system is called the fundamental frequency or the first
harmonic.
Fundamental Frequency  Lowest frequency of vibration
Integer multiples of the
fundamental frequency are called
harmonics. The first harmonic is
the fundamental frequency. The
second harmonic is simply two
times the fundamental frequency.
The third harmonic is three times
the fundamental harmonic. And
so on.
fundamental frequency
1st harmonic
2nd harmonic
3rd harmonic
Dear Doctor Science,
My local radio station is broadcast by megahertz. I would like to know what these are and
where they come from.
-- Kerry Quanbeck from Juneau, AK
Dr. Science responds:
Hertz is more than a rental car company; it's a unit of frequency. Frequent means often, and the
megahertz is no exception, giving you one million hertz per second. That's a lot of hertz. You
don't have to be a scientist to know that when you say something hurts, it frequently means it
causes pain. Radio is a source of pain to many people. This is where we get the term, "Silence is
golden." The hertz itself is actually silver in color, and originates from a mysterious object in
space called the Hertz Donut. The Hertz Donut can be artificially replicated in a laboratory, but
it's very dangerous. If a scientist should ever ask you if you want a Hertz Donut, for your own
safety, you'd better say no.
232
Dear Doctor Science,
I have noticed while driving around the country, that the best radio reception always seems to be on
religious radio stations. Does God have anything to do with this?
-- Rick Urban from Urbandale, Iowa
Dr. Science responds:
Oddly enough, no. God tends to favor stations that play early rythmn and blues recordings, and
maintains a limited interest in hip hop, or rap music. By the way, Solomon was the first rapper,
and if read in the original language, the Song of Solomon is very close to something Snoop Doggy
Dog or Puff Daddy might intone into the microphone. Last time I checked, God's favorite song
was "At Last" by the young Etta James. God finds most religious music to be a crashing bore and
given his soulful musical leanings, it's not hard to see why.
Loverly Trivia:
















A neutron star has such a powerful gravitational pull that it can spin on its axis in
1/30th of a second with tearing itself apart.
There are 1,575 steps from the ground floor to the top of the Empire State Building.
At one point, the Circus Maximus in Rome could hold up to 250,000 people.
Buckingham Palace has over six hundred rooms.
Built in 1697, the Frankford Avenue Bridge which crosses Pennypack Creek in
Philadelphia is the oldest U.S. bridge in continuous use.
Close to 35,000 people used to work in the World Trade Center, 110 story, twin
towers in New York.
Due to precipitation (snow and ice), for a few weeks each year K2 (a mountain in
the Himalayas) is taller than Mt. Everest.
Every year, an igloo hotel is built in Sweden that has the capacity to sleep 100
people.
Harvard uses Yale brand locks on their buildings; Yale uses Best brand.
Hawaii's Mount Waialeale is the wettest place in the world - it rains every single
day, about 460 inches per year.
If a statue in the park of a person on a horse has both front legs in the air, person
died in battle; if the horse has one front leg in the air, the person died as a result of
wounds received in battle; if the horse has all four legs on the ground, the person
died of natural causes.
In 1980, a Las Vegas hospital suspended workers for betting on when patients
would die.
If you attempted to count the stars in a galaxy at a rate of one every second it would
take around 3,000 years to count them all.
In Czechoslovakia, there is a church that has a chandelier made out of human bones.
Each person shares a birthday with at least nine other million people in the world.
Stainless steel was discovered by accident in 1913.
233
AP Physics - Sound
Sound is a longitudinal mechanical wave. For most of the time, what we will be talking about is a
wave that travels through air. Sound can travel through other mediums - water, other liquids,
solids, and gases. We can hear sounds that travel through other mediums than air – put your ear to
the wall and hear the sounds on the other side. You hear sounds when you are under water although not as well as in air. This is because our ears are set up for listening to sounds that move
through the atmosphere.
Rarefaction
Compression
v
Wavelength
The disturbance which travels through air is the compression of air molecules – they are squeezed
together and pulled apart. Sound is a series of traveling high pressure and low pressure fronts.
Sound waves are frequently graphed with pressure on the y axis and time on the x axis. This makes
the wave look like a transverse wave - a sine wave shape on the graph. But in this depiction,
changes in pressure are being plotted Vs time and it is not a depiction of the disturbance itself,
which is longitudinal. So please, the Physics Kahuna begs of you, don't be confused about the
thing.
Pressure
Time
Here we see a graph of pressure Vs time. The compressions are regions where the air pressure is
greater than the ambient pressure of the air. The rarefactions are areas of lower pressure. These
high and low pressure ridges travel outward in an expanding sphere from the sound source.
234
The sound source is simply something that vibrates. It can be the clangor on an alarm clock, a
window shade flapping in the wind, or your vocal cords vibrating because air is passing through
them. The vibrating sound source collides with air molecules, causing them to scrunch together and
pull apart. These scrunches travel through the air. But the air molecules do not physically travel
across the room. They are excited by the sound source and gain kinetic energy. They move
outward and have elastic collisions with other air molecules, which then gain energy, and so on.
Sound waves
propagate as
expanding sphere
Sound source
Air molecule gets excited, gains energy,
collides with another molecule and transfers
energy to it and so on.
The damping of a sound wave (decrease in amplitude) as
it travels is called attenuation. Attenuation depends on
the medium and the frequency of the sound. Low
frequency sounds are attenuated less than high frequency
sounds. Whales make very low frequency sounds (in the
neighborhood of 1 - 10 Hz) which can travel hundreds of
miles through the ocean. It has recently been found that
elephants also employ similar low frequency sounds to
communicate. In air, these sounds can travel many miles.
Audible Spectrum:
The human ear is not the
world's best sound receptor, although it does all right (if
you take care of them). A typical person can hear sounds
whose frequency ranges from 20 Hz to about 20 000 Hz.
This is known as the audio spectrum. Sounds with a
higher frequency are called ultrasonic sounds and sounds
of a lower frequency are called infrasonic sounds.
Other animals have different hearing spectrums. Dogs can easily hear sounds up to 45 000 Hz.
Whales and elephants hear very low frequency sounds (below 10 Hz).
235
The Doppler Effect:
Imagine a water bug floating
motionless on the surface of a calm pond on a lovely
summer day. The bug, bored out of its little bug brain, is
tapping the water with a pair of its little segmented legs,
making a series of waves that radiate outward on the surface
Wave Pattern Created By A Stationary
Bug wiggling its Legs in the Water
- like the ripples on a pond (actually, they are ripples on a pond, ain’t they? Curious, what?). The
bug is unwittingly producing a traveling wave. It would look like this:
The waves spread out in all directions. The distance between the wave crests is the wavelength, .
This wavelength is the same in all directions. Now, imagine that the bug starts swimming in one
direction, but it still makes its little periodic vibrations with its legs at a constant frequency. We
would see a different wave pattern.
Notice that the waves in front of the bug are pushed closer together. Behind the bug, the waves are
stretched further apart. The waves in front have a shorter wavelength, the waves to the rear have a
longer wavelength. Since the speed of the wave is a constant and equal to the wavelength
multiplied by the frequency, this means that the frequency of the waves traveling in front of the bug
is higher. The waves behind the bug are lower in frequency. We call this frequency change the
Doppler shift or the Doppler effect.
236
This happens because the bug makes a wave and then swims after it. So that, when he makes the
next wave, it will start out closer to the first wave and so on. As the wave travels to the rear, it is
already further away from the first wave, so the wavelength is longer and the frequency shorter. All
the waves travel at the same speed so they can't make up the difference.
What happens if the bug swims at the same speed as the wave?
The bug is making a wave and then moving right along with it.
So the bug is riding on top of the wave. Then the bug makes
another wave that is on top of the first, and so on. The bug ends
up riding an enormous wave because all the wave crests are in
phase and add up. This would be tough swimming for the old
water bug.
What happens if the bug
swims faster than the
waves?
The bug makes a wave and
swims through it into clear
water, then it makes the next wave, and so on. The bug is
always in front of the waves in nice smooth water. The waves
propagating behind the bug will have their crests in phase
along a line to either side of the bug that trails back from the
bug - they sort of overlap. They will constructively interfere
with each other and form a V shaped bow wave. The bow
wave will have a very large amplitude as it spreads out behind
the bug. Boats and ships do this all the time. Many harbors
have speed limits for ships because if the ship travels too fast it
will generate a large bow wave that can damage property on
either side of the vessel.
Doppler Shift and Sound: Sound, like all waves, undergoes this Doppler shift.
A car
moving towards you pushes its sound waves closer together in front so the sound you actually hear
has a higher frequency. When it moves away from you its frequency is lower. You can hear this
change when you are near traffic. You can listen to a car and tell from the change in pitch when it
stops coming toward you and starts moving away.
In order to experience the Doppler shift, there must be relative motion between the sound source
and the listener.
If there was no motion between the sound source and the listener, then this equation would hold
true:
v f
so
f 
v

and

v
f
237
But there is motion between them. What happens when the listener is moving towards the sound
source?
The frequency heard must be:
f '
v'


v  v0

The velocity is the sum of the velocity of the listener and the speed of sound.
We know that the wavelength is given by:

v
f
We can plug that into the equation we’ve derived for the new frequency:
f '
v  v0


v  v0
v
 f 
 
 v  v0 
 f

 v 
So the new frequency heard by a moving listener closing on a stationary sound source is given by:
 v  v0 
f ' f 

 v 
f’ is the new frequency heard, f is the frequency actually produced, v is the velocity
of sound, and v0 is the velocity of the listener.
If the listener is moving away from the sound source. It is obvious that the new frequency is given
by:
 v  v0 
f ' f 

 v 
But what about if the sound source is moving and the listener is stationary?
If the sound source is moving towards the listener, the wavefronts that arrive at the listener are
closer together because of the motion of the sound source. The wavelength measured by the
listener is shorter than the wavelength that is actually produced by the sound source. Is this clear?
Think about it and do not proceed until the previous statement is clear. The Physics Kahuna will
patiently wait for you.
So ’ (wavelength collected by listener) is shorter than  (wavelength that is produced). Each
vibration or cycle takes a time that is the period of the wave, T. During this time T, the source
moves a distance of;
238
v
x
t
x  vs t
x  vs t
Since the time is the period we get:
 vsT
T
The period and frequency are related by:
1
f
We can put this together with the equation for distance and we get:
1
 vs  
 f 
x  vsT
x
vs
f
The distance is the change in wavelength, so we can write this as:
 
vs
f
This is the change in wavelength that the listener observes.
The observed wavelength, the one the listener measures is the original wavelength minus the
change in wavelength:
 '    
Because

'  
vs
f
v
f
' 
We can write
vs
f
v
v v
  s
f' f
f
We can solve for f’:
v v 
v  s  f '
f 
 f
vf   v  vs  f '
 v 

f  f'

v
v


s


So, cleaning it up a bit, we get:
 v 
f ' f 

 v  vs 
f’ is the frequency heard, f is the original frequency, v is the speed of sound, and vs is
the speed of the sound source.
239
If the sound source is moving away from the stationary listener, the equation becomes:
 v 
f ' f 

 v  vs 
When solving Doppler problems, we will assume that the speed of sound is 345 m/s.

A train is traveling at 125 km/h. It has a 550.0 Hz train whistle. What is frequency heard by a
stationary listener in front of train?
First, convert the train’s speed to meters per second:
125
km  1 h  1 000 m 
m


  34.72
h  3 600 s  1 km 
s
Then plug the data into the equation which you will have derived (as above). Make sure to use the
proper sign. In this case the train is closing on the listener, so the negative sign is selected.
m

345

 v 
s
f ' f 
  550.0 Hz 
m
 v  vs 
 345  34.72 m
s
s



 


612 Hz
Dubious Facts:






According to ancient Chinese astrologers, 70% of omens are bad.
Airports that are at higher altitudes require a longer airstrip due to lower air density.
Blue is the favorite color of 80% of Americans.
California has issued 6 drivers licenses to people named Jesus Christ.
Deaf people have safer driving records on average than hearing people in the U.S.A.
Five percent of the people who use personal ads for dating are already married.
240
Disk vs Disc: A Public Radio Commentary by Bill Hammack
How you spell the word disk affects how you view laws about copying recordings of music and
movies. Lying in the balance is billions of dollars, and perhaps the cultural legacy we'll pass on to
our heirs.
Disk with a "k" came about in the mid-17th century, modeled on words like "whisk." Disc with a
"c" arose a half-century later from the Latin discus. Most people used these two spellings
interchangeably until late in the 19th century when they began using disc with a "c" to refer to
phonograph records. This usage still persists in Compact Disc, spelled with a "c". Then in the
1940s, when engineers needed a term to describe the data storage devices of their computers they
choose to spell disk with a "k." We still see this in the spelling of the "hard disk" of our
computers.
Today, though, the distinction between these two spellings is no longer meaningful. In the past if
you had a disc with a "c", like a phonograph record, there was no way it could become a part of
your hard disk, spelled with a "k." If you had a record you could make a tape of it, but it was
never as good as the original, and any copies of it were even worse. Now, of course, the digital
revolution has erased the difference between the two spellings of disk. A computer can make a
copy identical to the original.
This, of course, has the entertainment industry terrified, especially when combined with the
Internet, which provides unlimited distribution of these digital copies. Right now they're using
software tricks to reduce copying - certain CDs now work only in players that can't make copies,
but they know these software solutions are only temporary. Computer hackers always come up
with ways to evade these measures. To counter this threat the industry is turning to Congress.
They want intellectual property laws that make all copying illegal. We should be alarmed by these
efforts.
We run the risk that every embodiment of thought or imagination may be subjected to some kind
of commercial control. For example, as books become electronic, readers may lose the rights
they've had since Gutenberg's time. The publishers of an electronic book can specify whether you
can read the book all at once, or only in parts. And they can decide whether you read it once or a
hundred times.
So, the risk is this: The literary and intellectual canon of the coming century may be locked into a
digital vault accessible only to a few. As Congress writes laws to regulate digital intellectual
property and copying, I think they should keep in mind an aphorism from T.S. Eliot. "Good poets
borrow," he said, "great poets steal."
Supersonic travel:
Supersonic motion means that the speed is greater than the speed of
sound. (Figure that sound travels at 345 m/s.) In the past when one talked about supersonic
motion, one was talking about flight, this is because airplanes were the main things that went faster
than sound. (Bullets and projectiles also travel faster than sound.) That is no longer true as in the
past few years goofy daredevils have managed to build cars that travel faster than sound. This was
hard to do because the speed of sound is greater at the earth’s surface than it is at high altitudes.
241
Supersonic motion is a lot like the deal with the
bug swimming faster than the waves it makes.
Supersonic airplanes fly faster than the speed of
sound, so the sound the plane makes expands
outward as do all sounds from a sound source,
except that the sound source is always in front.
The effect is to form a “sound wake” where the
compressions of the sound are constructively
reinforced. This creates a “cone” of sound
energy that trails behind the aircraft. This cone
packs a lot of sound energy, so when it goes by a
listener, a really loud, intense sound is heard.
This sound wake thing is called a shock wave or
sonic boom. Sonic booms can break windows, scare babies and animals, and crack mirrors. For
this reason, airplanes are not allowed to go faster than sound over populated areas.
Dear Doctor Science,
Why do telephone cords coil clockwise?
-- Joan Kozik from Cedar Rapids, IA
Dr. Science responds:
I could get away with saying that all telephone cords are manufactured in the northern
hemisphere, but that would ignore the reality of 200 million Brazilian, Argentinean and
Australian telephone cords. So, I'll tell you the real reason which is that voice transmissions
only travel in a counterclockwise spiral. It has to do with the way the larynx attaches to the
throat. Sonic vibrations are sent spinning from ligaments that connect clockwise and any
attempt to re-direct this transmission induces phase cancellation. All you hear is a bored male
voice telling you to hang up the phone and try your call later.








The national sport of Japan is sumo wrestling.
The only bone not broken so far during any ski accident is one located in the inner ear.
The youngest American female to score an ace was Shirley Kunde in August 1943 at age
13.
There are at least two sports in which the team has to move backwards to win-tug of war
and rowing.
Morihei Ueshiba, founder of Aikido, once pinned an opponent using only a single finger.
Fossilized bird droppings are one of the chief exports of Nauru, an island nation in the
Western Pacific.
Honolulu is the only place in the United States that has a royal palace.
If you leave Tokyo by plane at 7:00am, you will arrive in Honolulu at approximately
4:30pm the previous day.
242
Resonance: The word ‘resonance’ means “resound”.
This is an important topic with physics –
let’s develop the thing a bit.
One of the goofy demonstrations the Physics Kahuna performed involved a tiny little speaker. It
was the cheapest and smallest speaker that he could find. When it was hooked up to a tape player,
the sound quality was terrible. It sounded tinny and weak and really lousy -suitable only for rap
music. But then the Physics Kahuna pulled out a big piece of cardboard that had a small hole in it.
He held the speaker up to the hole and suddenly the sound was ever so rich and nice and much
louder too! So how come that happened?
Natural Frequency:
Every object has a natural frequency at which it will vibrate. How
loud this sound is depends on the elasticity of the material, how long it can sustain a vibration, how
well the whole object can vibrate, how big it is, etc. Some materials vibrate better than others. For
example, a piece of metal, if excited (say you hit it with something), will vibrate. The vibrations
will spread throughout the piece and the whole thing will vibrate. Think of a bell. On the other
hand, a piece of Styrofoam, to look at the other extreme, is not nearly so good at vibrating. You can
bang on it all day and get nothing better than the odd dull thud kind of sound. So bells are made of
metal and not polystyrene foam. At any rate when you bang on an object, it will vibrate at its
natural frequency. This principle is used in many musical instruments – xylophones immediately
come to mind.
Forced Vibrations: The tinny little speaker did produce sound – bad as it was.
Speakers
have a small coil that is set to vibrating by the electrical output of an amplifier. Attached to the
coil is a paper cone. The vibrating coil then forces the paper cone to vibrate. Air molecules in
contact with the speaker cone are then set to vibrating which creates sound waves in the air. These
waves travel through the air to your ears. The cone in this cheap speaker has a very small area and
does not do a very good job of transferring the sound energy into the air. So it sounds lousy and
weak.
When the big piece of cardboard was brought out and the speaker was pressed onto it, the speaker
forced the cardboard to vibrate. Since it was mechanically connected to the cardboard, the
vibrations were easily transferred. The cardboard had a very large area in contact with the air, so it
was more efficient at sending the sound into the air. This is why the sound suddenly sounded so
much better when the cardboard made its appearance.
We call this phenomenon forced vibration.
Forced vibration  The vibration of an object that is made to vibrate
by another vibrating object in contact.
A tuning fork makes a very weak sound - you can barely hear the thing. Place a vibrating tuning
fork against a window or desktop, however, and the sound will become much louder.
Another example of this that you experienced was the coat hanger on a string deal. When the
clothes hanger was hung from a piece of string and struck with something, you couldn’t really hear
anything (unless you placed your ear very close to the hanger). But if you pressed the string into
243
your ear, you heard a deep resonant gong/bell type sound when the hanger was tapped. The string
was forced to vibrate and conducted the sound to your ears.
People do not recognize their own voice. Have you ever heard a tape recording of your voice? Did
it sound like you? Probably not. This is because the sounds that you make travel to your ears via
your skull and not through the air. The vibrations that reach your ear through your bones and tissue
sound slightly different than the vibrations that travel through the air.
Forced vibration is very important in music. Many instruments have sounding boards which are
forced to vibrate to make the instrument sound louder - pianos are a good example of this. Other
instruments have bodies that act as sounding boards. Guitars, violins, banjos, mandolins, and
ukuleles fit into this category. The vibrating strings of these instruments produce a very pitiful
weak sound, but place the same string on your average Martin guitar, and you get a whole different
deal. Much of what makes up the quality of sound produced by an instrument depends on how well
it can transfer sound energy into the air. We've all heard of these fabulous old violins from Italy the best known are the Stradivarius violins - which produce really exquisite sound. Modern
violinists claim that no modern violin can even come close to the quality of sound that these violins
produce. So a Stradivarius violin can sell (if anyone is willing to sell theirs) for millions of dollars.
How these violins were made - the secret that gives them their rich sound is not known. All sorts of
people are desperately trying to duplicate the feat, but so far, no one has succeeded. At least
according to the music experts. The Physics Kahuna himself cannot tell the difference.
Resonance Discussed:
Resonance is sometimes called sympathetic vibration. It means to
"re sound" or "sound again". If two objects which have the same natural frequency are placed near
each other, and one is set to vibrate, the other one will begin to vibrate as well.
What happens is that the first instrument forces the air to vibrate at its natural frequency. These
sound waves travel to the other object and causes it to vibrate at that very frequency. But this is
also its natural frequency. So the waves induce a vibration. Each compression arrives in phase
with the vibrations of the object and adds to its energy, and causes it to build up. So the second
object will begin to vibrate and then vibrate stronger and stronger.
A common demonstration of resonance can be done with a book. The book is hung from a bar. A
person applies a puff of air to the book, causing it to swing a little. When it comes all the way back
from the swing, another puff is applied. It swings just a bit farther out. Apply yet another puff, it
swings more. Eventually you get the book to really swing. What you are doing is applying energy
at the resonant frequency of the system. So the motion builds up and becomes greater and greater.
For this to happen, however, the energy must be fed in at the resonant frequency of the object. We
can associate these resonant waves with standing waves in the object. If you blow randomly, this
will not work.
The other interesting thing is that you can do it at a harmonic frequency. Blow every other swing or
every third swing. Do you see how this would work?
244
Resonant Air Columns: Have you ever blown into a pop bottle and gotten the thing to
make a nice, deep, melodious sound? Bottles can do this because they will resonate. When you
blow across the top of the bottle, you create turbulence – burbles of air – which occur at a broad
band of frequencies. This is
called the edge effect. One of
those frequencies is the bottle’s
resonant frequency. A standing
wave forms in the bottle’s
interior. As energy is fed in from
the blowing thing, the standing
wave gains energy until it is loud
enough to hear.
Close Ended Pipes:
The reason that the bottle resonates is that a standing wave forms in it.
The wavelength of the standing wave has to "fit the bottle", so only the one frequency (or its
harmonics) will resonate and be heard. The other frequencies aren't loud enough to be audible. The
closed end of the pipe is a displacement node because the wall does not allow for the longitudinal
displacement of the air molecules. As a result, the reflected sound pulse from the closed end is
180 out of phase with the incident wave. The closed end corresponds to a pressure antinode.
The open end of the pipe is, for all practical purposes,
a displacement antinode and a pressure node. The
reflected wave pulse from an open end of the pipe is
reflected in phase. The open end of a pipe is essentially
the atmosphere, so no pressure variations take place.
The reflection actually takes place a slight distance
outside the pipe, but we will ignore that.
First harmonic
1
4
Third harmonic
3
4
Let's look at a simple pipe that has a standing wave
within it. There has to be a displacement node at the
Fifth harmonic
closed end and a displacement antinode at the open end.
With this in mind, we can draw in the various standing
5
waves that can form within the pipe. The first one is a
4
quarter of a wave. This is the lowest resonant
frequency that can form a standing wave in the tube. Note that the closed end reflects the sound
wave out of phase - like a fix-ended wave is reflected.

Anyway, the pipe length turns out to be about ¼ of the wavelength. The lowest frequency is called
the fundamental frequency. Its wavelength is essentially ¼ of the length of the pipe.
The next possible frequency will have a wavelength that is ¾ of the pipe's length, then 5/4 of the
length, and so on. You can see that only odd harmonics are resonant in the close-ended pipe.
Only the odd harmonics are present in a resonating close-ended pipe.
The equation that relates wavelength, frequency and wave speed is:
245
v f
For the fundamental frequency (the first harmonic), the wavelength is:
  4l
The frequency in the system must be:
v f
 f  4l 
f 
v
4l
If we want the frequency of the third or fifth or whatever harmonic, we would get:
fn  n
v
4L
n  1, 3, 5, ...
Here fn is the harmonic frequency that resonates in the pipe, v is the speed of sound, L is the
length of the pipe, and n is an integer for the harmonic that you want.
The wavelength for any harmonic would be:
n 
4l
n
n  1, 3, 5, ...
Open Ended Pipes: Open-ended pipes can also
resonate. At both ends of the pipe, the wave is
reflected in phase. The fundamental wave and
associated harmonics would look like this:
The wavelength is approximately twice the length of
the tube. Note also that the open ended pipe has all
harmonics present.
First harmonic
1
2
Second harmonic
2
2
Third harmonic
3
2
Using the same method of derivation as we did with
the close-ended pipe, we can develop an equation for the wavelength for the fundamental
frequency.
Here is the equation. See if you can derive it yourself.
fn  n
v
2L
n  1, 2, 3, ...
A critical difference between the open and close-ended pipes is that the open-ended pipe can have
all harmonics present. The close-ended pipe is limited to the odd harmonics.
All harmonics can be present in a resonant open-ended pipe.
246

A pipe is closed at one end and is 1.50 m in length. If the sound speed is 345 m/s, what are the
frequencies of the first three harmonics that would be produced?
Use the close ended pipe formula to find the first harmonic (the fundamental frequency):
fn  n
f1  345
v
4L
f1 

m
1

 
s  4 1.50 m  
v
4L
57.5 Hz
Recall that close ended pipes only have the odd harmonics, so the next two would be the third and
fifth harmonics:
f3  n  f1   3  57.5 Hz  
172 Hz
f5  n  f1   5  57.5 Hz  
288 Hz
Musical instruments play things we call notes. A note is
a specific frequency from a thing called a scale, which is
Tuning Fork
a collection of eight notes called an octave. The notes
are: A, B, C, D, E, F, and G. A C, for example, on some
scales is 262 Hz. The interesting thing is that if you
multiple the frequency of a note, you get the same note,
Flute
but it is a higher harmonic. When you play a 262 Hz tone
and a 524 Hz tone at the same time, they would give you
the same musical sense, but would sound richer and
fuller. The reason that different musical instruments
sound different – think piano and mandolin, even when
Clarinet
they are playing the same note, is that each instrument
has its own set of harmonics. Some instruments only
produce a fundamental frequency – flutes often do this, while other instruments produce a bunch of
harmonics.
In the examples below, you see a pressure vs. time graph for different things and instruments. A
tuning fork, the first example, produces a single frequency, so its graph resembles a sine wave.
The flute and clarinet do not appear to be sine waves. This is because of the presence of harmonics
and the law of superposition.
247
Tuning Fork
Flute
Clarinet
Intensity
1 2 3 4 5 6
Harmonics
1 2 3 4 5 6 7
Harmonics
1 2 3 4 5 6 7 8 9
Harmonics
The last graph (above) shows the intensity of the different harmonics for the same instruments. The
tuning fork only has the first harmonic. The flute has a strong 2nd and 4th harmonic. These are
stronger than the fundamental frequency. The clarinet has a strong 5th and 1st harmonic. This is
why they each sound different to our ears.
Dear Doctor Science,
If you can hear the ocean in a seashell, why is it that you can't hear the forest in a
pinecone?
-- Enrico Uva from Montreal, Quebec
Dr Science responds:
You used to be able to, but now all pinecones are treated with sound-damping toxic waste
to ensure that they don't harbor the pernicious ear-boring fir beetle. This creature can
scuttle out of a pine cone and into your ear faster than you can say "Paul Bunyan." Once
inside, it quickly bores into the center of your brain, where it lays its eggs. When they
hatch, they feed on the choice parts of your cerebellum, leaving you fit only for life as a
community college administrator. Better content yourself with the harmless seashell.
248
Dear Cecil:
How did they mass-produce those old-fashioned cylinder records? A conventional molding press, like
they use for discs, would leave some sort of line where the two halves met, which would show up as a
click or thump when the cylinder was played. How did they make 3-D moldings of such accuracy in
the 1890s? Or did the artistes just make the same recording over and over again? --Winfield S.,
Chicago
Dear Winfield:
They sure did, at least at the beginning. This is why you didn't see a lot of albums selling 18 million copies
in 1887. The need for a cheap and easy method of reproduction was one of the first problems the early
recording industry faced, and the problem you describe was one of the reasons why cylinders lost out to
discs as the principal recording medium.
In the very beginning, of course, a little thing like a seam on the recording surface didn't matter too much.
On Edison's original phonograph, the ends of the tinfoil sheet that recorded the sound were just tucked into
a slot that ran the length of the metal cylinder that the foil was wrapped around. You did get a click this
way, but since you also got an indescribable barrage of burps, wheezes, and rasps, the first recording
devices being a little on the rustic side, it seems probable that you did not object to the clicks so much.
Later, the recording blanks were made of wax, which could be cast in one piece, eliminating the click, if
nothing else.
When records first began to be sold commercially, the only way to make additional copies was to have the
artistes make the same recording over and over. You would hire, say, a brass band, which you would
surround with a phalanx of recording machines loaded with blank discs, and you'd get some guy with a
suitably stentorian voice to go around to each machine, flip it on for a second, and holler the title of the
piece into the speaking horn. Then you'd turn on all the machines at once, and the band would play as much
of any given tune as would conveniently fit onto the cylinders, which was generally about two minutes'
worth. Then you changed cylinders and started over. Apart from being stupefyingly monotonous for the
performers, this method was very slow.
Eventually somebody hit on the idea of recording additional cylinders off a master cylinder by means of a
pantograph, which was an arrangement of levers and wires that transmitted the sound vibrations from the
stylus on the master disc to that on the receiving disc. This was faster and less boring, but the masters
tended to wear out quickly, and then the band had to go at it again.
Finally, around the turn of the century, Edison's phonograph company developed a reliable method for mass
production. They coated the wax master with a thin layer of gold by an electrical process, coated the gold
layer with a copper layer for strength, then melted out the original wax. This left a negative metal mold.
Then they put a wax blank inside and applied heat and pressure. When the wax cooled, it shrank a little. In
addition, the master and blank were tapered slightly--one end was slightly wider than the other. The
combination of shrinkage and taper was enough to let them slip the master off the copy without (a)
damaging it or (b) leaving a seam.
Actually, this method had occurred to Edison and his buddies fairly early on, but the first recording styli
gouged out such deep grooves that the shrinkage wasn't enough to enable them to clear. The development
of the sapphire-tip stylus, which made shallower indentations, cleared up this problem. Unfortunately,
by the time they got all this worked out, cylinders were beginning to decline in favor of discs,
which were longer playing, among other things. So it was all for nothing, as is often the case in the
record business.
--CECIL ADAMS
249
Dear Cecil:
I have observed that traveling high speeds causes strange things to occur (e.g., the
Charger's hubcaps in Bullitt regenerated twice during the high speed chase scene). If I
traveled down Austin Avenue at a speed exceeding Mach 1 (the speed of sound) would
the noises made by the people on the streets be distorted or totally inaudible to me?
While cruising at this speed, if I blasted my car radio, would the people on the streets
hear it as I would, or would I be traveling faster than radio waves, causing my radio to
die? What phenomenon would occur if I shifted into second around North Avenue and
my Buick exceeded the speed of light? This information will be helpful when I rush my
wife to the hospital sometime in January to deliver our baby.
--Life in the Fast Lane, Chicago
Dear Life:
Nice to hear from a guy who takes a practical view of things. Dealing with the easy parts of
your question first: At the speed of sound you won't hear sounds happening behind you,
because the sound waves won't be able to catch up with you. You'll hear sounds occurring in
front of you, but they'll be increased in pitch about an octave due to the Doppler effect--i.e., as
you move toward a sound source, you crowd up on the oncoming sound waves, which
increases their frequency relative to you. By the same token, people on the street won't hear
you coming, but they'll hear you (and a sonic boom) after you've passed--with the pitch
decreased about an octave.
What was more interesting to the Straight Dope Science Advisory Board was what you would
hear if you turned on your car radio at the speed of sound. If you keep the windows closed
there's no problem, because you, the radio, and the air will all be stationary relative to one
another. With the windows open, however, you wouldn't hear the rear deck speakers at all-the air carrying the sound waves would be blown backward too quickly. However, you'd hear
the dashboard speaker normally (or as normally as you'd hear anything at 700 MPH)--the twin
Doppler effects of source to air and air to observer would cancel out.
Finally, regarding your last question, don't you know the velocity of light is the speed limit of
the universe? Any attempt to exceed it would defy one of the fundamental principles of
creation. Besides, it's hell on the tires.
250
Dear Cecil:
Can operatic sopranos really break glasses with their high notes? What note does the trick?
How come they don't break windows and eyeglasses and whatnot at the same time? Can
women do this better than men? Can I learn how? Or have I been the victim of an elaborate
hoax? --Vox Clamantis, Chicago
Dear Vox:
I dunno--you ever buy whole-life insurance? Now _there_ was a hoax. Shattering glasses, on the
other hand, is entirely legit. Enrico Caruso and Italian opera singer Beniamino Gigli are said to
have managed it, and I seem to remember Ella Fitzgerald doing it once in a Memorex commercial.
The technique is simple. First you find somebody with perfect pitch and leather lungs. Then get a
crystal glass and tap it with a spoon to determine its natural frequency of vibration (this varies with
the glass). Next have the singer let loose with precisely the same note. When he or she is dead-on
pitchwise, the glass will commence to resonate, i.e., vibrate. Then turn up the V. Bingo, instant
ground glass.
What we have here is a graphic demonstration of forced oscillation resonance. If something has a
natural rate of vibration, pump in more energy of the same rate and with luck the thing will vibrate
so bad it'll self-destruct. It's like giving somebody on a swing a good shove at the top of every arc-soon they'll reach escape velocity and soon after that they'll be picking vertebrae out of their
sinuses.
Breaking glasses, however, is strictly light entertainment. For real forced oscillation action you
want a suspension bridge. In 1831 troops crossing a suspension bridge near Manchester, England,
supposedly marched in time to the bridge's sway. Boy, did they get a surprise. Ever since soldiers
have been told to break step when crossing bridges. The same fate befell the Tacoma Narrows
suspension bridge in Washington State on November 7, 1940, only it wasn't soldiers that caused it
to collapse, it was the wind.
But back to the home front. Crystal is more vulnerable than ordinary glass because it has more
internal structure, which allows waves to propagate. (Take my word for it.) But you can annihilate
damn near anything given enough volume. One physicist, obviously one of your classic
Roommates From Hell, claims he inadvertently shattered a glass lamp shade while playing the
clarinet.
Think of the possibilities. Most of us don't have the pipes to break glasses by sheer voice power,
but we all have clarinets, don't we? Unfortunately, none of the standard physics cookbooks gives a
detailed glass-bustin' recipe. Too bad. A fascinating classroom demonstration like this would
surely convince many young people to give up MTV and devote their lives to science.
251
Dear Doctor Science,
Is there a gene missing in men's ear canals that gives them selective hearing?
-- Sue Neff from Missoula, MT
Yes, you could say that. That particular gene resides on the Ear, Nose and Throat
Chromosome and also suppresses emotive speech. When a man suspects that a woman
wants to talk about their relationship, this gene literally takes over the autonomic
nervous system, causing rapid, shallow breathing, a drop in blood pressure, and an
intense desire to watch sports on television. Biogenetic engineering hopes to create a
new "feminist" male, whose genetic makeup will more closely mirror what women
want men to be. So far the only working prototype is celebrity John Davidson and,
based on this data, the FDA is considering banning experimentation on living
organisms.
SHATTERING MYTHS
Dear Cecil:
In the matter of glass-shattering vocalism, Cecil seems to have been led astray by Gunter
Grass's fictional tin drummer, Oskar. In fact, there is no authentic record of glass being broken
by the unamplified human voice. Dorothy Caruso categorically denied rumors that her late
husband had accomplished the feat; a fortiori it was beyond Gigli's comparatively feeble
instrument. Practically speaking, there are reasons to believe the thing impossible, and
without going into technical detail, the following are among them:
(1) Glass is simply much too strong. Try shattering a wine glass in your (gloved)
fingers. Not easy. Now imagine doing the same with the puny little bands of your
vocal cords.
(2) (2) Coupling acoustic energy from larynx-to-air-to-glass is highly inefficient due
to large impedance mismatches; by contrast, marching troops couple very
efficiently to bridge platforms.
(3) (3) In glass shattering attempts, resonance or no resonance, the glass structure
finds other ways to dissipate energy short of fracturing.
Remember the playground swing in which successive small but well-timed swings sent your
sister sailing higher and higher? And the tales of going "over the top" when the process went
critical? Alas! it never happened, because other dynamic processes supervened ("Gee, Mom,
we were just playing") before the longed-for loop could occur.
--Timon, Dallas
252
BRIDGE CRASH NEWS FLASH!
In his recent treatise on whether singers can break glasses with their voices, Cecil mentioned
"forced oscillation resonance," in which an external force amplifies the natural vibration of an
object, sometimes with destructive results. As an example he cited the 1940 collapse of the
Tacoma Narrows bridge. The usual explanation for this disaster is that the wind gusted (to be
precise, "generated a train of vortices") in perfect synch with the bridge's natural rate of bounce,
causing its demise.
Reader Wilbur Pan has alerted us to a recent report in Science News heaping abuse on this
widely held view. Mathematicians Joseph McKenna and Alan Lazer doubt that a storm could
produce the perfectly timed winds required. They're working on a "non-linear" model of bridge
behavior they hope will provide a better explanation. The main problem apparently is that when
the roadway of a lightly constructed suspension bridge flexes, the cables supporting it go slack,
introducing an element of unpredictability in which little causes (i.e., the wind) produce big
results (i.e., a collapsing bridge). They hope to have the mathematical model describing this
effect finished in five years--not the most aggressive schedule in the world, but apparently this is
government work. You'll read about it here first.
--CECIL ADAMS
253
AP Physics – Waves Wrapup
There are four equations that you get to play with. You must recognize them, right? Here they are:
v f
T
This baby relates wavelength and frequency to the velocity of the
wave. Very important equation, not only now, but later on as well.
1
f
This one relates the period of a wave to the frequency – you can flip it
(do a bit of cross multiplying) as well and get frequency equal to the
inverse of the period.
Ts  2
m
k
This third one finds the period for an oscillating mass on a spring. The
period is directly proportional to the mass and inversely proportional
to the spring constant. Increase the mass, increase the period.
Increase the spring constant value, decrease the period.
TP  2
l
g
This fourth one is the period for a pendulum. The period is a function
of the length of the thing. Increase the length and you increase the
period. So the period is directly proportional to the length of the
pendulum.
Here are the things that you are required to be able to do – turns out that there is a lot of stuff in this
unit.
Oscillations
1.
You should understand the kinematics of simple harmonic motion so you can:
a. Sketch or identify a graph of displacement as a function of time, and determine from such a
graph the amplitude, period, and frequency of the motion.
This is not too difficult. We’ve done several of these type things in the previous units.
b. Identify points in the motion where the velocity is zero or achieves its maximum positive or
negative value.
This is also a fairly simple task. Just examine the graph or drawing and pick off the required
values.
254
c. State qualitatively the relation between acceleration and displacement.
The relationship is simple. The maximum acceleration occurs at the points of maximum
displacement. The acceleration is least when the displacement is zero. The maximum
acceleration coincides with the maximum restoring force.
d. Identify points in the motion where the acceleration is zero or achieves its greatest positive or
negative value.
This is pretty much the same stuff as the thing above. Again, the maximum acceleration
occurs at the points of maximum displacement. The acceleration is least when the
displacement is zero.
e. State and apply the relation between frequency and period.
The relationship between frequency and period is given by:
T
1
f
The nice thing is
that the equation will be provided you. You can see that as the frequency gets larger, the
period gets smaller and vice versa.
f. State how the total energy of an oscillating system depends on the amplitude of the motion,
sketch or identify a graph of kinetic or potential energy as a function of time, and identify
points in the motion where this energy is all potential or all kinetic.
The total energy of the system is set by the amplitude. This determines the potential energy the
system has. When the mass is released, the potential energy is converted to kinetic energy. At
zero displacement all the energy is kinetic. When the mass is at the maximum displacement (the
amplitude) all the energy is kinetic. Between the amplitude and zero displacement the energy will
be a combination of kinetic and potential energy.
A graph of kinetic energy, potential energy, and displacement Vs time looks like this:
Picking off the points where the energies are zero should be pie for a superior student such as
yourself.
g. Calculate the kinetic and potential energies of an oscillating system as functions of time,
sketch or identify graphs of these functions, and prove that the sum of kinetic energy and
potential energy is constant.
You can use the equations for the potential energy of a spring and kinetic energy to work
things out.
The potential energy of the spring is given by: U S
1
2
 kx 2
255
When the displacement is equal to the amplitude, we get the maximum potential energy,
which will be the total energy of the system.
U
1
2
 kA2
You can bring in the time element by using the equation for the period of the system.
The reason that the sum of the potential energy and kinetic energy are constants – come on,
does the Physics Kahuna have to give you this one? No? Yes? Right, its because of the law
of conservation of energy.
2. You should be able to apply their knowledge of simple harmonic motion to the case of a
mass on a spring, so you can:
a. Apply the expression for the period of the oscillation of a mass on a spring.
Use the equation for the thing please.
3. You should be able to apply their knowledge of simple harmonic motion to the case of a
mass of a pendulum, so you can:
a. Apply the expression for the period of a simple pendulum.
Use the equation.
b. State what approximation must be made in deriving the period.
The approximation thing? Oh yeah, it’s that the equation is only good for small swings.
1. You should understand the description of waves so you can:
a. Sketch or identify graphs that represent traveling waves and determine the amplitude,
wavelength, and frequency of a wave from such a graph.
p A
A
t
-A

This is pie. You look at the graph and figure out what they want to know. Most graphs will
have time plotted on the x axis and the displacement on the y axis. In the example above
pressure is plotted on the y axis. The amplitude A is the maximum positive or negative
256
displacement from the zero displacement centerline value. This is where the pressure is
zero on the graph drawn above. You just read off the value. To fine the frequency you
have to measure the period T of the wave off the graph. The period is the time for one
cycle. To find the wavelength you need to know the velocity of the wave. You can then use
the v   f equation to find frequency.
Sometimes a graph is made that plots the displacement of the disturbance on the y axis and
distance on the x axis. This might be useful for water waves, things like that. Anyway the
graph would look like this.
y A
A
x
-A

You can read the amplitude and the wavelength directly off the graph. You still need to
know the velocity of the wave to find the frequency.
b. State and apply the relation among wavelength, frequency, and velocity for a wave.
You just be using the equation v   f .
c. Sketch or identify graphs that describe reflection of a wave from the fixed or free end of a
string.
This is pie. Just remember, fixed end reflection – out of phase. Free end reflection – in
phase. Lovely drawings are available in the handout on waves.
d. Know quantitatively what factors determine the speed of waves on a string and the speed
of sound.
Factors for speed of wave on string – tension. Increase tension, increase wave speed.
Factors for speed of wave in air – temperature (and, to a lessor extent, pressure and
density). Increase temperature increase speed of sound.
2. You should understand the physics of standing waves so you can:
a. Sketch possible standing wave modes for a stretched string that is fixed at both ends, and
determine the amplitude, wavelength, and frequency of such standing waves.
257
This is where you draw half a wave, a wave and a half, and so forth. This is pie. Excellent,
clear drawings for what they look like are available in the useful handout.
b. Describe possible standing sound waves in a pipe that has either open or closed ends, and
determine the wavelength and frequency of such standing waves.
Drawing the waves in a pipe is pie. You can easily do that. The equations for wavelength
and frequency are not provided you, however, so you have to be able to develop them
yourself. The Physics Kahuna was kind of enough to show you how to do this.
The main thing here is that the wavelength is four times the length of the pipe for a closeended pipe and two times the length of an open-ended pipe.
Open-ended pipes can resonate at all harmonics. Close-ended pipes can only resonate at
the odd harmonics.
3. You should understand the Doppler effect for sound so you can:
a. Explain the mechanism that gives rise to a frequency shift in both the moving-source and
moving-observer case, and derive an expression for the frequency heard by the observer.
It’s fairly simple to explain how the Doppler shift occurs (see the previous section on
sound). The nasty part is that they expect you to derive the Doppler shift equations
yourself. The Physics Kahuna has kindly shown you how to do it. Seriously now, the
Physics Kahuna has never seen a question on the old AP tests that actually required you to
derive the Doppler shift equations. The Physics Kahuna suggests that you have a good
conceptual understanding of the Doppler shift and how it relates to frequency and
wavelength.
b. Write and apply the equations that describe the moving-source and moving-observer
Doppler effect, and sketch or identify graphs that describe the effect.
This is where you use the equations that you painfully derived. Using them is actually
pretty simple. The Physics Kahuna, out of the goodness of his heart, provided you
numerous example problems to do so you could have lots of practice. “Thanks” – not
necessary. Part of the service.
4. You should understand the principle of superposition so they can apply it to traveling waves
moving in opposite directions, and describe how a standing wave may be formed by
superposition.
This is pretty simple, the idea is that the waves at a particular point add algebraically to
produce a resultant wave. We went over several examples of this in a previous unit.
258
AP Question Time:
From 1998:
 To demonstrate standing waves,
one end of a string is attached to a
tuning fork with frequency 120 Hz.
The other end of the string passes
over a pulley and is connected to a
suspended mass M as shown in the
figure below. The value of M is
such that the standing wave pattern
has four loops. The length of the string from the tuning fork to the point where the string touches
the top of the pulley is 1.20 m. The linear density of the string is 1.0x10-4 kg/m, and remains
constant throughout the experiment.
a. Determine the length of the standing wave.
There are 2 waves in the 1.20 m.
b.
1.20 m

2
0.60 m
Determine the speed of transverse waves along the string.
1

v  120   0.6m  
s

v f
c.

72
m
s
The speed of waves along the string increases with increasing tension in the string. Indicate
whether the value of M should be increased or decreased in order to double the number of
loops in the standing wave pattern. Justify you answer.
To double the number of loops the wavelength must be decreased by a factor of 2.
If wavelength is halved then speed is halved.
(v  f  )
So tension must be decreased to decrease speed.
Thus the mass must be decreased. Mass is proportional to velocity as stated in the
question.
d.
If a point on the string at an antinode moves a total vertical distance of 4 cm during one
complete cycle, what is the amplitude of the standing wave?
Amplitude is the maximum displacement from equilibrium, one way. So it is one quarter
the total vertical motion, during one up down cycle.
The string goes up one centimeter
from its centered position. It then goes down a centimeter to the center position, so it has
traveled two centimeters. Then it goes down one centimeter to the lowest position (a
trough thing), so it’s moved three centimeters. Then it moves back to the center position
– that’s one more centimeter for a total displacement of 4 cm. So the amplitude is 1 cm.
1
  4 cm 
4
1.0 cm
259
From 1995:

As shown, a 0.20-kilogram mass is sliding on a horizontal, frictionless air track with a speed of
3.0 meters per second when it instantaneously hits and sticks to a 1.3-kilogram mass initially at
rest on the track. The 1.3-kilogram mass is connected to one end of a massless spring, which
has a spring constant of 100 Newtons per meter. The other end of the spring is fixed.
a. Determine the following for the 0.20-kilogram mass immediately before the impact.
i.
Its linear momentum.
m

p  mv   0.20 kg   3.0  
s

ii.
kg  m
s
Its kinetic energy.
1
K  mv 2
2
b.
0.60
2
1
m

  0.20 kg   3.0  
2
s

0.90 J
Determine the following for the combined masses immediately after the impact.
i.
The linear momentum
p  p'
ii.
p' 
so
0.60
kg  m
s
The kinetic energy
p '   m1  m2  v '
1
K   m1  m2  v 2
2
m
p'
m
s
v' 

 0.40
s
 m1  m2   0.20 kg  1.3 kg 
0.60 kg
2
1
m

  0.20 kg  1.3 kg   0.4  
2
s

0.12 J
After the collision, the two masses undergo simple harmonic motion about their position at
impact.
260
c.
Determine the amplitude of the harmonic motion.
1
U s  K f  kx 2 x 
2
d.
T  2
2K f

k
2  0.12 Nm 
N 

100 m 



0.050 m
Determine the period of the harmonic motion.
m
k
 2
 0.20 kg  1.3 kg 
kg  m 

100


s2  m 


0.77 s
From 1995:

A hollow tube of length open at both ends as shown above, is held in midair. A tuning fork
with a frequency f o vibrates at one end of the tube and causes the air in the tube to vibrate at its
fundamental frequency. Express your answers in terms of  (length) and fo.
a.
Determine the wavelength of the sound.
In an open tube multiples of half wavelengths cause resonance. Resonance occurs for
the first time with a half wavelength in the tube, so
b.
Determine the speed of sound in the air inside the tube.
vf
c.
  2l
v  2l f o
Determine the next higher frequency at which this air column would resonate.
At the next higher frequency 2 half wavelengths fit so new  l
vf
f new 
v
new
f new 
2l f o
l

2 fo
The tube is submerged in a large, graduated cylinder filled with water. The tube is slowly
raised out of the water and the same tuning fork, vibrating with frequency fO, is held a fixed
distance from the top of the tube.
d.
Determine the height h of the tube above the water when the air column resonates for the
first time. Express your answer in terms of (length).
261
In closed tubes multiples of 1/4 wavelengths cause resonance. So if the tube vibrated at fo with a
wavelength of 2 l when opened ended, then it will vibrate at the same frequency with a
wavelength of 4 l when close ended.
  4h for the tube, and v   f
 4hf 0
The speed of sound is the same for the open ended tube, so:
v  2 l f0
Now we can set the two velocities equal to each
other.
4hf 0  2 l f 0
h
2l
4
h
l
2
262