Unit Three: Waves and Sound
Chapter Six, Seven and Eight
6.1 Vibrations
Waves transfer energy over a distance in the form of a
disturbance. This disturbance can be caused by the
vibration of an object. A vibrating object has a ‘rest’
position and will periodically shift from this state to
another. Think of the type of vibrational motion exhibited
in the following examples.
A weight hanging from a spring.
A weight hanging from a rope and the rope is twisted.
A pendulum.
A piece of wood floating in the ocean.
Periodic motion occurs when an object repeats a pattern of
motion. The vibration, or oscillation is repeated over and over with
the same time interval.
Types of Vibrations
There are 3 basic types of vibrations: transverse vibrations,
longitudinal vibrations, and torsional vibrations. A transverse
vibration occurs when an object vibrates perpendicular to its
axis at the normal rest position (such a pendulum).A
longitudinal vibration occurs when an object travels parallel
to its axis at the rest position.
axis
movement
axis
movement
A torsional vibration occurs when an object twists around
its axis at the rest position, for example a human twisting
to the left or the right periodically.
Observe the twist to the left, then the rest position, then
the twist to the right.
One complete oscillation of a vibrating object is called a
cycle. A cycle would be complete relative to a point in
the object’s vibration. This is when the object travels
past that point in the same direction.
Possible cycles:
1,2,3,2,1
3,2,1,2,3
or
2,3,2,1,2
or
Not 2,3,2 !!!
The number of cycles which occur per second is called
the frequency of vibration (f). Frequency is measured in
hertz (Hz) or s-1. The period of vibration (T) is the time
required to complete one cycle. Period is measured in
seconds.
# cycles
f 
t
1cycle
f 
T
t
T
# cycles
1
T
f
The distance in either direction from the equilibrium, or
rest position to maximum displacement is called the
amplitude (A) of vibration.
Objects that are ‘in phase’ vibrate with the same period
and pass through the rest position at the same time and
in the same direction.
In phase
In phase
p. 198 1-8 p.202 1-11
Out of phase
Inertial Balance Activity
Inertial Balance Activity
Inertial balances are useful for measuring the mass of items when the force of
gravity is not present or strong enough to do so.
Inertial balances vibrate back in force after being set in motion. If the amplitude
is not large then the period of vibration is constant even while the amplitude
diminishes.
Obtain 8 sets of data using the inertial balance. Record the mass of plates,
washers, nut and bolt. Record the number of oscillations (at least 10) and the
time of oscillations.
Prepare a observations results table with the following headings: mass, #
oscillations, time, period, period2, frequency (include units).
Make a graph of mass (y) vs. period2 (s). Draw in a line of best fit and calculate
the equation of the line.
m
k 2
T  2
or m  2 T
k
4
6.2 Wave Motion
Waves transfer energy over a distance in the form of a
disturbance. In transverse waves, particles in the medium
move at right angles to the direction in which the wave
travels. The high section of the wave is called the crest and
the low section is a trough. The rest position is known as
equilibrium. A wave that consists of a single disturbance is a
pulse (may be positive or negative). Periodic waves originate
from periodic vibrations where the motions are continuous
and repeated in the same time intervals.
Crest
Wavelength (λ)
Amplitude
Trough
Wavelength (λ)
As the wave travels through any medium, its amplitude
will decrease due to energy loss from friction. If no
energy were required to overcome friction, an ‘ideal
wave’ would be present.
Longitudinal waves
In longitudinal waves, particles vibrate parallel to the
direction of motion of the wave. The most common
longitudinal waves are sound waves. In a longitudinal
wave, regions where the particles are closer together
than normal are known as compressions and the regions
where they are farther apart than normal are known as
rarefactions.
The amplitude will be the change in a physical quantity
from the rest position to a maximum compression or
minimum rarefaction. For sound this quantity would be
air pressure.
Wavelength (λ)
Compression
Rarefaction
Compression
Rarefaction
v
Wavelength (λ)
Homework: p.208 3,4
Demonstrate pulses, waves, reflections, reflection transmission, speed of medium then
change, superposition. Phet.
6.3 The Universal Wave Equation
A wave travels one wavelength in one
period of time.
d
v
t
v

T
v  f
f = frequency
1
but f 
T
= wavelength
The speed of a wave is a property of the
medium it travels in.
The distance between successive crests in a
series of water waves is 4.0m. The crests travel
9.0m in 4.5s. What is the frequency of the crests?
Δd = 9.0m
Δt = 4.5s
λ = 4.0m
f=?
d
 f
t
9
 f ( 4)
4 .5
f  0.5 Hz
Homework: p.211 1,2 p, 211 1-8 (hand in)
6.4 Transmission and Reflections of Waves
Waves behave in various ways A change in the medium in
which a wave is traveling often results in a change of the
speed of the wave. The frequency of the wave or disturbance
will never change however the wavelength of the wave must
change if the Universal Wave Equation is to be upheld.
Changes to the speed of the wave will ensure proportional
changes in the wavelength of the wave.
When waves undergo fixed end reflection the pulse is
inverted. If the reflection occurs from a free end then there is
no inversion. In both fixed end and free end reflection there
is no change in frequency or wavelength. There is also no
change in the speed of the pulse since the medium is the
same.
Fixed end
Free end
Inverted reflected pulse
Reflected wave is
not inverted
When a wave travels into a new medium partial reflection
occurs. Some of the energy of the wave is transferred
through to the new medium, but some is also reflected back
into the original medium. Of course, since the wave is
traveling into a new medium there will be a change in its
wavelength and speed. The loss of energy also means a
decrease in amplitude for the wave.
Fast to Slow Medium (Less dense to more dense Medium)
The slow medium acts like a rigid obstacle, and the reflected wave
is inverted. The transmitted wave is not inverted, travels with
reduced speed and wavelength and has a diminished amplitude.
Slow to Fast Medium (More dense to less dense Medium)
The fast medium acts like a free end and the reflected wave is not
inverted. The transmitted wave is not inverted, travels with
increased speed and wavelength and has a diminished amplitude.
Incident Pulse
Fast Medium
Reflected pulse
Slow Medium
Transmitted Pulse
Incident Pulse
Fast Medium
Transmitted pulse
6.6 Interference of Waves
Wave interference occurs when two waves act
simultaneously on the particles of a medium. There are two
types of interference: constructive and destructive.
Destructive interference occurs when a crest meets a trough.
Constructive interference occurs when crests meet crests
(supercrests) or troughs meet troughs (supertroughs).
The concept of adding the amplitudes of waves is known as
the superposition principle. It states that at any point the
resulting amplitude of two interfering waves is the algebraic
sum of the displacements of the individual amplitudes.
Homework: Superposition worksheet
p. 222 1-3 (just sketch)
Extra p. 221 1,2 p. 222 4,5
Constructive Interference
Waves
approach
Waves occupy
same space
Waves diverge
Destructive Interference
Waves
approach
Waves occupy
same space
Waves
diverge
Waves approach each other
Overlap
Resulting Wave
Pattern
Remember this pattern only appears for
an instant!
6.7 Mechanical Resonance
Resonance is the response of an object that is free to
vibrate to a periodic force with the same frequency as
the natural frequency of the object. Therefore resonance
is also a transfer of energy from one object to another
having the same natural frequency. If the two objects are
touching, it is known as mechanical resonance.
Every object has a natural frequency at which it will
vibrate. A swing’s natural frequency will depend on the
length of the chains. A window rattles with its natural
frequency. Bridges, propellers, blades, turbines, glasses
and many types of equipment all have a natural
frequency. Read p. 223-224 for examples.
If you push someone on a swing at the
right time they will travel higher and
higher on a swing (with the swing’s
natural frequency). Think what would
happen if a bridge got “pushed” at the
right time over and over . . .
Tacoma narrows, singing rod, swing set, army marches
When an object vibrates in resonance
with another, it is called a sympathetic
vibration.
6.8 Standing Waves – A Special Case of 1
Dimensional Wave Interference
The amplitude and the wavelength of
interfering waves are often different.
However if the conditions are such that two
waves have the same amplitude and
wavelength and travel in opposite
directions, then a special interference
pattern known as a standing wave occurs.
Try the standing wave worksheet.
The resulting wave pattern is known as the standing
wave interference pattern.
Node (N): point that remains at rest
Anti-node: point midway between nodes where
maximum constructive interference occurs
N
N
N
λ
2
N
The distance between two successive
nodes in a vibrating string is 10cm. The
frequency of the source is 30 Hz. What is
the speed of the waves?
f = 30 Hz Distance between successive
nodes is ½ λ
λ=?
½ λ = 10 cm
v=?
λ = 20 cm
Hmwk. p.229 1-4
extra p.230 1-4
v=fλ
v = (30 Hz)(0.20m)
v = 6.0 m/s
7.1 SOUND
Sounds are a form of energy produced by
rapidly vibrating objects. Sound needs a
material medium for its transmission. Sound
cannot travel through a vacuum. The
vibrating object causes compressions and
rarefactions in the medium. A receiver
senses the sound by sensing the
compressions and rarefactions.
The amplitude of a sound is its loudness. The amplitude
of sound in air depends on the size of the pressure
changes in the air. The frequency of sound is often
referred to as pitch (however this is subjective).
Young people can hear a wide range of sound, from 20
Hz to 20 000 Hz. Sounds with frequencies less than 20
Hz are infrasonic while sounds above 20 000 Hz are
ultrasonic.
p. 238-241 in text
p. 241 1,2,3
extra 4
p. 242 1,4,5
extra 2,3,6,7
7.2 THE SPEED OF SOUND
m
m/ s
V  332  (0.59 o C )T
s
Air pressure and elevation do not
significantly affect the speed of sound in air.
p. 243-246
p. 243 1-4, p.246 3-5
extra p.246 1,2,6-8
Notice any patterns?
THE INTENSITY OF SOUND
p. 247-248
p. 248 1-4, p. 249 2-4
Sound intensity is the power of sound per
unit area (W/m2). Sounds can be emitted
with an extremely large variance in intensity
and likewise humans can sense extremely
soft sounds as well as loud sounds.
The quietest whisper is about 10-12 W/m2
while a sound with an intensity of 104 W/m2
will instantly perforate an eardrum.
The decibel scale is utilized for sound
intensity and gives an easy scale to judge
relative intensities. The least intense sound
we can hear is given the intensity of 0 dB.
For every 10 dB increase in intensity the
sound increases its true intensity by 10X.
The scale is logarithmic so if the intensity
increases by 30 dB then the true intensity
has increased by a factor of 1000X.
The intensity of sound we hear depends on
the power of the source and the distance
2
between us and the source. I  1 / r
read 7.5 The Human Ear p. 249-253
-will not test on parts of the human ear
read 7.6 The Reflection of Sound Waves p. 254-257
-understand echoes and echo problems
(remember to double distance)
-p. 257 1-4, p. 258 1-3
-know echolocation and who uses it
-know ultrasound applications
Not responsible for 7.7 Diffraction and Refraction of Sound
Waves p. 258-260.
Not responsible for section on 7.8 The Interference of Sound
Waves p. 260-263.
Responsible for 7.9 Beat Frequency p.264-266
-p. 266 1,2 p. 266 1-7
The Doppler Effect (p. 267-272)
The apparent changing frequency of sound in relation to
an object’s motion is called the Doppler effect, named
after Christian Doppler (1803-53). If a sound emitter is
moving towards a listener (or vice versa) then the
listener hears a higher frequency than is actually
emitted. If a sound emitter is moving away from a
listener (or vice versa) then the listener hears a lower
frequency than is actually emitted.
The Doppler effect (Doppler shift) has been used to
estimate the speed of distant stars and galaxies (using
light waves) relative to our solar system. The Doppler
shift is also used in police radar for speeding.
v
f 2  f1 (
)
v  vs
Vs is speed of source or
listener
V is speed of sound
The Mach Number is the ratio of an object’s velocity to the
speed of sound.
When flying at Mach 1, an object is flying as fast as the
sound it gives off. When the object emits another sound the
crest will alongside the original crest so these crests pile up,
producing an area of very dense air. This intense
compression of air is called the sound barrier. Extra thrust is
needed to break through this barrier. Objects must be
designed to cut through this dense air leading to sleek and
pointy shapes. At hypersonic speeds, the crests are left
behind the object which constructively interfere with other
crests to create a double cone. This intense acoustic
pressure is called the sonic boom.
p.269 1,2 p.270 3-5 p.272 1-6
Good websites for , interference, Doppler Shift and Breaking the Sound Barrier.
http://www.phy.ntnu.edu.tw/oldjava/waveSuperposition/waveSuperposition.html
http://www.kettering.edu/~drussell/Demos.html
http://www.answers.com/topic/sonic-boom
http://library.thinkquest.org/19537/java/Doppler.html
8.1 Music and Musical Notes
It’s important to realize the difference between what is
music and noise. Music is sound that originates from a
vibrating source with one or more frequencies (usually
harmonious and pleasant). Noise on the other hand is
sound that originates from a source with constantly
changing frequencies and is usually not ‘pleasant’ to the
ear. On an oscilloscope, noise would not have a
constant wave form or pattern.
There are three main characteristics of musical sounds:
pitch, loudness and quality. Each of these characteristics
depends not only on the source of the musical sound,
but also on the listener. Thus, they are called subjective
characteristics.
Which of the following are musical and which
are noise?
Pitch is the perception of the highness or lowness of a sound; it
depends primarily on the frequency of the sound.
Loudness is the perception of the intensity of sound.
Sound Quality is a property that depends on the number and
relative intensity of harmonics that make up the sound.
In music, a pure tone is a sound where only one frequency is
heard. Musical sounds are not normally pure tones; they usually
consist of more than one frequency.
In general, two or more sounds have consonance if their
frequencies are in a simple ratio (simpler ratio produces more
consonance). Harmonious pairs of sounds have high consonance;
unpleasant pairs of sounds have high dissonance, or low
consonance.
Unison is a set of sounds of the same frequency. An octave has
sounds with double the frequency of the sounds in another
frequency. For example, a 200-Hz sound is one octave above a
100-Hz sound.
The two common musical scales are the scientific musical scale,
based on 256 Hz, and the musicians’ scale, based on 440 Hz.
p. 278
2
p. 280
3,4
p. 281 1-4
8.2 Vibrating Strings
Vibrating strings (examples?) are often used to produce musical
sounds. The frequency of a vibrating string is determined by four
factors: length, tension, diameter, and density. All of these factors
are taken into consideration when designing stringed musical
instruments, such as the piano, guitar, cello, harp, lute, mandolin,
banjo and violin.
Increase length ->
decrease frequency
Increase tension ->
increase frequency
Increase diameter ->
decrease frequency
p. 283 1-5
Answer
qualitatively!
Increase density ->
decrease frequency
8.3 Modes of Vibration – Qualities of Sound
When a string, stretched between two fixed points, is plucked a
standing wave pattern is produced. Nodes occur at both ends.
Different frequencies of varying amplitudes may result depending
on how many nodes and antinodes are produced. The resulting
note is the sum of all of these different vibrations of the string.
In its simplest, or fundamental mode of vibration, the string
vibrates in one segment. This produces its lowest frequency,
called the fundamental frequency ( fo).
If the string vibrates in more than one segment, the resulting
modes of vibration are called overtones. Since the string can only
vibrate in certain patterns (always with nodes at each end) the
frequencies of the overtones are simple to determine.
1st overtone (f1)
f1 = 2fo
These vibrations are also referred to as harmonics.
Fundamental freq.
First overtone
Second overtone
Third overtone
fo
f1 (2fo)
f2 (3fo)
f3 (4fo)
First harmonic
Second harmonic
Third harmonic
Fourth harmonic
Stringed instruments vibrate in a complex mixture of overtones
superimposed on the fundamental frequency. Very few vibrating
sources can produce a note free of overtones. An exception is
the tuning fork, but even it has overtones when first struck.
However, because the overtones disappear quickly, the tuning
fork is valuable in studying sound and tuning musical
instruments.
The quality of a musical note depends on the number and
relative intensity of the overtones it produces along with the
fundamental frequency. The quality enables us to distinguish
between notes of the same frequency and intensity coming from
different sources; for example, we can easily distinguish between
middle C on the piano, on the violin, and in the human voice.
8.4 Resonance in Air Columns
Closed Air Columns
When a sound wave is sent down an air column (closed at one
end) the end of the tube reflects the sound waves back. Certain
frequencies produce standing wave patterns (through
interference) that amplify the original sound. The closed end is
fixed so a node is located there. The open end of the column is
free to vibrate so an anti-node is located there.
Resonance first occurs when the column is (1/4) λ in length. The
next possible lengths are 3/4 λ, 5/4 λ, etc. check wooden box with tuning fork
2nd Resonant Length
1st Resonant length
Sample Problem:
The first resonant length of a closed air
column occurs when the length is 16 cm.
(a) What is the wavelength of the sound?
(b) If the frequency of the source is 512 Hz,
what is the speed of sound?
(a) first resonant length
=¼λ
¼λ
= 16 cm
λ = 64cm
(b)
v
=fλ
= 512 Hz (64cm)
= 32 768 cm/s (327.7 m/s)
Open Air Columns
Resonance may also be produced in an
open air column(open at both ends).
Antinodes occur at free ends. This means
the first length at which resonance occurs
is 1/2 λ. Resonance will next occur at
lengths of λ, 3/2 λ, 2 λ, etc.
test air tubes
1st Resonant Length
2nd Resonant Length
Sample Problem:
The third resonant length of an open air
column occurs when the length is 50cm.
(a) What is the wavelength of the sound?
(b) If speed of the wave is 300 m/s, what is
the source frequency?
(a) third resonant length
= 3/2 λ
3/2 λ = 50 cm
λ = 0.33 m
(b)
f = v/ λ
= 300m/s / (0.33m)
= 9.0 x 102 Hz
p. 290 1-7, p. 292 1-7, 9