SPH3UW
Waves and Sound
An Introduction to Waves and
Wave Properties
Mechanical Wave
A mechanical wave is a disturbance which propagates through a
medium with little or no net displacement of the particles of the
medium. A Pulse is a single disturbance which carries energy
through a medium or space
Water Waves
Wave “Pulse”
People Wave
Parts of a Wave
: wavelength
3
equilibrium
2
-3
y(m)
crest
A: amplitude
4
trough
6
x(m)
In the drawing, one cycle is shaded in color.
The amplitude , A , is the maximum excursion of a particle of the medium from
the particles undisturbed position.
The wavelength is the horizontal length of one cycle of the wave.
The period is the time required for one complete cycle.
The frequency is related to the period and has units of Hz, or cycles/second.
1
f 
T
Velocity of a Wave
The velocity of a wave is the distance traveled by a given point on the wave
(such as a crest) in a given interval of time.
v

T
 f
Example 1 The Wavelengths of Radio Waves
AM and FM radio waves are transverse waves consisting of electric and
magnetic field disturbances traveling at a speed of 3.00x108m/s. A station
broadcasts AM radio waves whose frequency is 1230x103Hz and an FM
radio wave whose frequency is 91.9x106Hz. Find the distance between
adjacent crests in each wave.
v

T
 f
v

f
AM
FM
v 3.00 108 m s
 
 244 m
3
f 1230 10 Hz
v 3.00 108 m s
 
 3.26 m
6
f
91.9 10 Hz
• Problem: Sound travels at approximately 340 m/s, and
light travels at 3.0 x 108 m/s. How far away is a lightning
strike if the sound of the thunder arrives at a location
2.0 seconds after the lightning is seen?
The speed of light can almost seem
instantaneous at these distances, so we
need only concern ourselves with the
sound component of lightening.
d  vs t
m

  340   2 s 
s

 680m
Types of Waves
Refraction and Reflection
Wave Types
A transverse wave is a wave in which particles
of the medium move in a direction perpendicular
to the direction which the wave moves.
Example: Waves on a String
A longitudinal wave is a wave in which particles
of the medium move in a direction parallel to the
direction which the wave moves. These are also
called compression waves.
Example: sound
http://einstein.byu.edu/~masong/HTMstuff/WaveTrans.html
Wave types: transverse
Energy transport
A transverse wave is a moving wave that consists of oscillations occurring
perpendicular (or right angled) to the direction of energy transfer. If a transverse
wave is moving in the positive x-direction, its oscillations are in up and down
directions
Wave types: longitudinal
Longitudinal waves, also known as "l-waves", are waves whose
direction of vibration is the same as their direction of travel, meaning that
the movement of the medium is in the same direction
Other Wave Types
• Earthquakes: combination (p and s waves)
• Ocean waves: surface
• Light: electromagnetic
Reflection of waves
• Occurs when a wave strikes a medium
boundary and “bounces back” into original
medium.
• Completely reflected waves have the same
energy and speed as original wave.
Reflection Types
• Fixed-end reflection: The
wave reflects with inverted
phase.
• Open-end reflection: The
wave reflects with the
same phase
Refraction of waves
• Transmission of wave
from one medium to
another.
• Refracted waves may
change speed and
wavelength.
• Refraction is almost
always accompanied by
some reflection.
• Refracted waves do
not change
frequency.
Sound is a longitudinal wave
• Sound travels through the air at approximately 340
•
•
m/s.
It travels through other media as well, often much
faster than that!
Sound waves are started by vibration of some other
material, which starts the air moving.
Hearing Sounds
• We hear a sound as “high” or “low” depending on its
frequency or wavelength. Sounds with short wavelengths
and high frequencies sound high-pitched to our ears, and
sounds with long wavelengths and low frequencies sound
low-pitched. The range of human hearing is from about
20 Hz to about 20,000 Hz.
• The amplitude of a sound’s vibration is interpreted as its
loudness. We measure the loudness (also called sound
intensity) on the decibel scale, which is logarithmic.
Doppler Effect
The Doppler Effect is the
raising or lowering of the
perceived pitch of a sound
based on the relative motion
of observer and source of
the sound. When a
ambulance siren is sounding
when it races toward you,
the sound of its siren
appears higher in pitch,
since the wavelength has
been effectively shortened
by the motion of the
ambulance relative to you.
The opposite happens when
the ambulance moves away.
The Doppler Effect
When a moving object emits a sound, the wave crests
appear bunched up in front of the object and appear to be
more spread out behind the object. This change in wave
crest spacing is heard as a change in frequency.
The results will be similar when the observer is in motion
and the sound source is stationary and also when both the
sound source and observer are in motion.
Doppler Effect
Stationary source
Moving source
Supersonic source
http://www.lon-capa.org/~mmp/applist/doppler/d.htm
The Doppler Effect formula
fo is the observed frequency.
 v  v0 
fo  
 fs
 v  vs 
fs is the frequency emitted by the source.
vo is the observer’s velocity.
vs is the source’s velocity.
v is the speed of sound.
Note: take vs and vo to be positive when they move in the
direction of wave propagation and negative when they are
opposite to the direction of wave propagation.
Example: A source of sound waves of frequency 1.0 kHz is
stationary. An observer is traveling at 0.5 times the speed of
sound.
(a) What is the observed frequency if the observer
moves toward the source?
(b) Repeat, but with the observer moving in the other
direction.
Example Solution: A source of sound waves of frequency 1.0
kHz is stationary. An observer is traveling at 0.5 times the
speed of sound.
(a) What is the observed frequency if the observer
moves toward the source?
fo is unknown; fs= 1.0 kHz; vo = 0.5v; vs = 0; and v
is the speed of sound.
 v   0.5v  
 v  v0 
fo  fo  
 f s  1.5 f  1.5 kHz
 fss  
 v  vs 
 v0 
Example Solution: A source of sound waves of frequency 1.0
kHz is stationary. An observer is traveling at 0.5 times the
speed of sound.
(b) Repeat, but with the observer moving in the other
direction.
fo is unknown; fs = 1.0 kHz; vo = +0.5v; vs = 0; and v is
the speed of sound.
 v   0.5v  
 v  v0 
fo  fo  
 f s  0.5 f  0.5 kHz
 fs  
 v  vs 
 v0 
Pure Sounds
• Sounds are longitudinal waves, but if we graph
•
•
•
them right, we can make them look like
transverse waves.
When we graph the air motion involved in a
pure sound tone versus position, we get what
looks like a sine or cosine function.
A tuning fork produces a relatively pure tone. So
does a human whistle.
Later in the period, we will sample various pure
sounds and see what they “look” like.
Graphing a Sound Wave
Sensitivity of the Human Ear
• We can hear sounds with frequencies ranging from 20 Hz to 20,000 Hz
– an impressive range of three decades (logarithmically)
– about 10 octaves (factors of two)
– compare this to vision, with less than one octave!
An Octave is a series of eight
notes occupying the interval
between (and including) two
notes, one having twice or half
the frequency of vibration of the
other..
Complex Sounds
• Because of the phenomena of “superposition”
•
•
•
and “interference” real world waveforms may
not appear to be pure sine or cosine functions.
That is because most real world sounds are
composed of multiple frequencies.
The human voice and most musical instruments
produce complex sounds.
Later in the period, we will sample complex
sounds.
Speakers: Inverse Eardrums
• Speakers vibrate and push on the air
– pushing out creates compression
– pulling back creates rarefaction
• Speaker must execute complex motion according to desired
waveform
• Speaker is driven via “solenoid” idea:
– electrical signal (AC) is sent into coil that surrounds a permanent
magnet attached to speaker cone
– depending on direction of current, the induced magnetic field
either lines up with magnet or is opposite
– results in pushing or pulling (attracting/repelling) magnet in coil,
and thus pushing/pulling on center of cone
Speaker Geometry
Superposition of Waves
Principle of Superposition
• When two or more waves pass a particular
point in a medium simultaneously, the
resulting displacement at that point in the
medium is the sum of the displacements
due to each individual wave.
• The waves interfere with each other.
Types of interference.
• If the waves are “in phase”, that is crests
and troughs are aligned, the amplitude is
increased. This is called constructive
interference.
• If the waves are “out of phase”, that is
crests and troughs are completely
misaligned, the amplitude is decreased
and can even be zero. This is called
destructive interference.
Constructive Interference
crests aligned with crest
waves are
“in phase”
Constructive Interference
Destructive Interference
crests aligned with troughs
waves are
“out of
phase”
Destructive Interference
Resonance
• Certain devices create sound waves at a
natural frequency. If another object,
having the same natural frequency is
impacted by these sound waves, it may
begin to vibrate at this frequency,
producing more sound waves. This
phenomenon is know as resonance.
• i.e. opera singer shattering glass with
voice
Standing Waves
Standing Wave
• A standing wave is a wave which is
reflected back and forth between fixed
ends (of a string or pipe, for example).
• Reflection may be fixed or open-ended.
• Superposition of the wave upon itself
results in a pattern of constructive and
destructive interference and an enhanced
wave.
• Let’s see a simulation.
Open and Closed Tubes
• Many musical instruments depend on the musician moving air through
the instrument.
• Musical instruments like this can be divided into two categories, open
ended or closed ended.
• A Tube or Pipe can be a musical instrument, it is often bent into
different shapes or has holes cut into it.
• An open ended instrument has both ends open to the air. An example
would be an instrument like a trumpet. You blow in through one end and
the sound comes out the other end of the pipe.
• A closed ended instrument has one end closed off, and the other end
open. An example would be an instrument like some organ pipes, or a
flute. Although you blow in through the mouth piece of a flute, the
opening you’re blowing into isn’t at the end of the pipe, it’s along the
side of the flute. The end of the pipe is closed off near the mouth piece.
Closed Ended Pipes
• Remember that it is actually air that is doing the vibrating as a wave
inside the Pipe.
• The air at the closed end of the pipe must be a node (not moving),
since the air is not free to move past the sealed end it must be
reflected back.
• There must also be an antinode (maximum movement) where the
opening is, since that is where there is maximum movement of the air.
• The simplest, smallest wave that I can possibly fit in a closed
end pipe is shown Below. L
First harmonic
 = 4L
• This is ¼ of a wavelength.
• Since this is the smallest stable piece of a wave we can fit in this pipe,
this is the Fundamental, or 1st Harmonic this is the lowest possible
frequency that any instrument can play.
Closed Ended Pipes
Since the length of the tube is the same as the length of the ¼
wavelength We know that the length of this tube is ¼ of a wavelength so
this leads to our first formula:
1
L 
4
“L” is the length of the tube in metres. On it’s own this formula really
doesn’t help us much. Instead, we have to solve this formula for λ and
then combine it with the formula v=fλ to get a more useful formula:
1
L 
4
  4L
v f
v  f 4L
v
f 
4L
L
When the wave reaches the closed
end it’s going to be reflected as an
inverted wave . This does not
change the length of the wave in
our formula, since we are only
seeing the reflection of the wave
that already exists in the pipe.
Closed Ended Pipes
So what does the Next Harmonic look like?
I know this name might seem a little confusing
(I’m the first to agree with you!) but because of
the actual notes produced and the way the
waves fit in, musicians refer to the next step up
in a closed ended pipe instrument as
the 3rdharmonic… there is no such thing as
a 2nd harmonic for closed ended pipes. In fact
all of the Harmonics in closed ended Pipes are
going to be odd numbers.
Closed Ended Pipes
Harm.
#
# of Waves
in Column
# of
Nodes
# of
Antinodes
LengthWavelength
Relationship
1
1/4
1
1
λ= (4/1)*L
3
3/4
2
2
λ= (4/3)*L
5
5/4
3
3
λ= (4/5)*L
7/4
4
4
λ= (4/7)*L
9/4
5
5
λ= (4/9)*L
7
9
Check your Understanding
The speed of sound waves in air is 340 m/s. Determine the fundamental
frequency (1st harmonic) of a closed-end air column that has a length of
76.5 cm.
Since this is the First Harmonic:
  4L
  4  0.765m 
  3.06m
m
340
v
s
Now for the Frequency: f 


3.06m
You could also have just

111Hz
v
used: f 
4L
Open Ended Pipes
I know you’re probably thinking that there
couldn’t possibly be any more stuff to learn about
Resonance, but we still have to do Open
Ended Pipes. Thankfully, they’re not that hard,
and since you already have the basics
for closed pipes it should go pretty easy for you.
• The fundamental (first harmonic) for an open ended pipe needs to be
an antinode at both ends, since the air can move at both ends.
• So this is the smallest wave we can fit into a open ended pipe: ½ of a
wavelength
L
Fundamental
First harmonic
 = 2L
Open Ended Pipes
So the basis for drawing the standing wave patterns for air
columns is that vibrational antinodes will be present at any
open end and vibrational nodes will be present at any closed
end. If this principle is applied to open-end air columns, then
the pattern for the fundamental frequency (the lowest
frequency and longest wavelength pattern) will have
antinodes at the two open ends and a single node in between.
For this reason, the standing wave pattern for the fundamental
frequencies) for an open-end air column looks like the
diagrams below.
Open Ended Pipes
The relationships between the standing wave pattern for a
given harmonic and the length-wavelength relationships for
open ended tubes are summarized in the table below.
# of
Harmonic
Waves in
#
Column
# of
Nodes
# of
Antinodes
LengthWavelength
Relationship
1
1/2
1
2
Wavelength = (2/1)*L
2
1 or 2/2
2
3
Wavelength = (2/2)*L
3
3/2
3
4
Wavelength = (2/3)*L
4
2 or 4/2
4
5
Wavelength = (2/4)*L
5
5/2
5
6
Wavelength = (2/5)*L
Check your Understanding
A 4 m long organ pipe (open at both ends) produces a musical note at
its fundamental frequency.
a) Determine the wavelength of the note produced.
b) What is the frequency of the pipe given the speed of sound is 346
m/s?
Since this is the First Harmonic:
  2L
  2  4m 
  8m
Now for the Frequency:
f 
v

m
346
Did you notice, that if we made
the pipe longer,
s
 be bigger, and since
the wavelength would
8m
wavelength and frequency
are inversely related,
that means the frequency would be smaller.
 43.3Hz
Check your Understanding
Example: An organ pipe that is open at both ends has a
fundamental frequency of 382 Hz at 0.0 °C.
a) What is the fundamental frequency for this pipe at 20.0 °C?
b) How long is this organ pipe?
At Tc = 0.0 °C, the speed of sound is 331 m/s.
At Tc = 20.0 °C, the speed of sound is 343 m/s.
v
v

The fundamental frequency is: f1 
1 2 L
Speed of sound: v  f 
v
v
f1 

1 2 L
v0
f0 
2L
v0
2L 
f0
v20
f20 
2L
v0 v20

f 0 f20
v20
2L 
f20
m
m
343
s 
s
382 Hz
f20
331
m

 343   382 Hz 
s

f20 
 396 Hz
331Hz
How long is this organ pipe?
2L 
v0
f0
v0
L
2 f0
m
331
s
L
2  382 Hz 
 0.433m
Fixed-end standing waves (violin string)
Fundamental
First harmonic
 = 2L
If the string
vibrates in more
than one
segment, the
resulting modes
of vibration are
called overtones
2
n  L
n
First Overtone
Second harmonic
=L
Second Overtone
Third harmonic
 = 2L/3
12.7 Beats
When two waves with nearly the same frequency are
superimposed, the result is a pulsation called beats.
Beats
• “Beats is the word physicists use to
describe the characteristic loud-soft
pattern that characterizes two nearly (but
not exactly) matched frequencies.
• Musicians call this “being out of tune”.
Two waves of
different
frequency
Superposition
of the above
waves
The beat frequency is f  f1  f2
Beats
If the beat frequency exceeds about 15 Hz, the ear will
perceive two different tones instead of beats.
What word best describes this to
physicists?
Amplitude
Answer: beats
What word best describes this to
musicians?
Amplitude
Answer: bad intonation
(being out of tune)
Check Your Understanding
A tuning fork with a frequency of 256 Hz is sounded together
with a note played on a piano. 12 beats are heard in 4
seconds. What is the frequency of the piano note?
12beats
Number of Beats

Beat Frequency:
4s
Total Time
 3 Hz
Since Beat Frequency also equals: f1  f2
Then: 3 Hz  256 Hz  f2
f2  253 Hz or 259 Hz
Without further information,
there is no way to know
which answer is correct...
Echolocation
Sound waves can be sent out from a transmitter of some sort;
they will reflect off any objects they encounter and can be
received back at their source. The time interval between
emission and reception can be used to build up a picture of the
scene.
Example A boat is using sonar to detect the bottom of a
freshwater lake. If the echo from a sonar signal is heard
0.540 s after it is emitted, how deep is the lake? Assume
the lake’s temperature is uniform and at 25 C.
Example 1: A boat is using sonar to detect the bottom of a
freshwater lake. If the echo from a sonar signal is heard
0.540 s after it is emitted, how deep is the lake? Assume
the lake’s temperature is uniform and at 25 C.
The signal travels two times the depth of the lake so
the one-way travel time is 0.270 s. From table 12.1,
the speed of sound in freshwater is 1493 m/s.
depth  vt
 1493 m/s  0.270 s 
 403 m
The Speed of Sound Waves
The speed of sound in different materials can be determined
as follows:
v
B
In thin solid rods v 
Y
In fluids


B is the bulk modulus of the
fluid and  its density.
Y is the Young’s modulus of
the solid and  its density.
The bulk modulus (B) of a substance measures the
substance's resistance to uniform compression. While Young’s
modulus (K) measures the Stiffness or elasticity of a solid
The Speed of Sound Waves
In ideal gases v  v0
T
T0
Here v0 is the speed at a temperature T0 (in kelvin) and v
is the speed at some other temperature T (also in kelvin).
For air, a useful approximation to the above expression is
v   331  0.606TC  m/s
where Tc is the air temperature in C.
Materials that have a high restoring force (stiffer) will have a
higher sound speed.
Materials that are denser (more inertia) will have a lower
sound speed.
Example 1:
A copper alloy has a Young’s Modulus of 1.11011 Pa and a
density of 8.92103 kg/m3. What is the speed of sound in a
thin rod made of this alloy?
v
1.1  1011 Pa

 3500 m/s
3
3

8.9  10 kg/m
Y
The speed of sound in this alloy is slightly less than the
value quoted for copper (3560 m/s) in table 12.1.
Example 2
Bats emit ultrasonic sound waves with a frequency as high as
1.0105 Hz. What is the wavelength of such a wave in air of
temperature 15.0 C?
v   331  0.606TC  m/s
The speed of sound in
air of this temperature
is 340 m/s.
v
340 m/s
3
 

3
.
4

10
m
5
f 1.0  10 Hz