Wave Motion
A wave is, in general, a disturbance that moves through
a medium.
A wave carries energy from one location to another
without transporting the material of the medium.
Examples of mechanical waves include water waves,
waves on a string, and sound waves.
What is the medium?
Slinky
Water
Transverse and Longitudinal Waves
There are two types of mechanical waves:
Transverse waves: The particles of the medium vibrate
up and down (perpendicular to the wave).
Longitudinal waves: The particles in the medium vibrate
along the same direction as the wave (parallel). The
medium undergoes a series of expansions and
compressions. The expansions are when the coils are
far apart and compressions are when they are when the
coil is close together.
Expansions and compressions are the analogs of the
crests and troughs of a transverse wave.
Longitudinal waves:
PARTS OF A WAVE
Amplitude: maximum displacement from equilibrium
Crest: Top part of the wave
Trough: Bottom part of the wave
Wavelength: Length from crest to crest
or trough to trough
Wave Motion
Wave velocity v is the
velocity with which the
wave crest is propagating.
Wave velocity v depends
on the medium.
A wave crest travels one
wavelength in one period:
On a string with tension
FT and mass per unit length of
the string (linear density) m/L
the velocity (m/s) of the wave is:
9.1 Measurements show that the wavelength of a sound wave in a
certain material is 18.0 cm. The frequency of the wave is 1900 Hz.
What is the speed of the sound wave?
λ = 0.18 m
f = 1900 Hz
v=λf
= 0.18 (1900)
= 342 m/s
9.2 A horizontal cord 5.00 m long has a mass of 1.45 g.
a. What must be the tension in the cord if the wavelength of a
120 Hz wave is 60 cm?
L=5m
m = 1.45x10-3 kg
f = 120 Hz
λ = 0.6 m
v=λf
= 0.6 (120)
= 72 m/s
FT
v
m/ L
2
v m (72)21.45 x103
FT 

5
L
= 1.5 N
b. How large a mass must be hung from its end to give it this
tension?
FT = FG = mg
FT 1.5
m

= 0.153 kg
g 9.8
REFLECTION
A wave front of a wave striking a straight barrier follows
the law of reflection. This law states that: "The angle of
incidence equals the angle of reflection".
The angle of incidence is the angle between the
direction of the incoming wave and a normal drawn to
the surface. The angle of reflection is the angle between
the direction of the reflected wave and the normal to the
surface.
BEHAVIOR OF WAVES
When a wave travels from one medium to another, the
wave is both reflected and transmitted.
When a wave passes into a new medium, its speed
changes. The wave must have the same frequency in the
new medium as in the old medium, thus, the wavelength
adjusts.
PARTIAL REFLECTION
When a wave travels from one medium to another, the
wave is both reflected and transmitted.
When a wave passes into a new medium, its speed
changes. The wave must have the same frequency in the
new medium as in the old medium. Thus, the wavelength
adjusts.
REFRACTION
A water wave traveling from deep water into shallow
water changes direction. This phenomenon is known as
refraction.
Interference: What happens when two waves pass
through the same region?
When two crests overlap it is called constructive
interference. The resultant displacement is larger then
the individual ones.
When a crest and a trough interfere, it is called
destructive interference. The resultant displacement is
smaller.
DIFFRACTION
Waves tend to bend as they pass an obstacle. For
example, water waves passing a log in a river will tend to
bend around the log after they pass. This bending is
known as diffraction. The amount of diffraction depends
on the wavelength and the size of the obstacle.
STANDING WAVES IN
STRINGS
At certain frequencies
standing waves can be
produced in which the
waves seem to be standing
still rather than traveling.
This means that the string
is vibrating as a whole.
This is called resonance
and the frequencies at
which standing waves
occur are called resonant
frequencies or harmonics.
A standing wave is produced by the superposition of
two periodic waves having identical frequencies and
amplitudes which are traveling in opposite directions. In
stringed musical instruments, the standing wave is
produced by waves reflecting off a fixed end and
interfering with oncoming waves as they travel back
through the medium.
Second Harmonic
Third Harmonic
The points of
destructive
interference (no
vibration) are
called nodes, and
the points of
constructive
interference
(maximum
amplitude of
vibration) are
called antinodes.
When a wave reflects off of something, it can interfere with its
own reflection. The interference is alternately constructive or
destructive as the two waves move past each other. This
creates a standing wave.
There can be more than one frequency for standing
waves.
Frequencies at which standing waves can be produced
are called the natural (or resonant) frequencies.
When a string on an instrument is plucked, vibrations,
that is, waves, travel back and forth through the medium
being reflected at each fixed end. Certain sized waves
can survive on the medium. These certain sized waves
will not cancel each other out as they reflect back upon
themselves. These certain sized waves are called the
harmonics of the vibration. They are standing waves.
That is, they produce patterns which do not move.
Standing Waves in Strings
A string can be fixed in both sides, like a guitar or piano
string.When the string is plucked, many frequency
waves will travel in both directions. Most will interfere
randomly and die away. Only those with resonant
frequencies will persist.
Since the ends are fixed, they will be the nodes.
The wavelengths of the standing waves have a simple
relation to the length of the string.
The lowest frequency called the fundamental frequency
has only one antinode. That corresponds to half a
wavelength:
The other natural frequencies are called overtones. They
are also called harmonics and they are integer multiples
of the fundamental.
The fundamental is called the first harmonic.
The next frequency has two antinodes and is called the
second harmonic.
9.3 A metal string is under a tension of 88.2 N. Its length is 50 cm
and its mass is 0.500 g.
a. Find the velocity of the waves on the string.
m = 5x10-4 kg
FT = 88.2 N
L = 0.5 m
FT
88.2
v

m/ L
5 x104 / 0.5
= 297 m/s
b. Determine the frequencies of its fundamental, first overtone and
second overtone.
Fundamental:
L = 1/2 λ
λ = 2L
= 2(0.5)
=1m
f 
v

297

= 297 Hz
1
First overtone
fn = n f'
f2 = 2(297)
= 594 Hz
Second overtone
fn = n f'
f3 = 3(297)
= 891 Hz
9.4 A string 2.0 m long is driven by a 240 Hz vibrator at its end. The
string resonates in four segments. What is the speed of the waves
on the string?
L=2m
f = 240 Hz
4 segments: 4th Harmonic
L = 4/2 λ = 2 λ
λ = ½ L = ½ (2) = 1 m
v=fλ
= 240 (1)
= 240 m/s
9.5 A banjo string 30 cm long resonates in its fundamental to a
frequency of 256 Hz. What is the speed of the waves on the string?
L = 0.3 m
f = 256 Hz
Fundamental: L = ½ λ
λ = 2L = 2(0.3) = 0.6 m
v=fλ
= 256 (0.6)
= 154 m/s
9.6 A string vibrates in five segments to a frequency of 460 Hz.
a. What is its fundamental frequency?
f5 = 460 Hz
fn = n f'
f n 460
f 

= 92 Hz
5
n
'
b. What frequency will cause it to vibrate in three segments?
fn = n f'
f3 = 3(92)
= 276 Hz
SOUND WAVES
Sound is a longitudinal wave produced by a vibration
that travels away from the source through solids,
liquids, or gases, but not through a vacuum. The speed
of sound is independent of the pressure, frequency, and
wavelength of the sound. However, the speed of sound
in a gas is proportional to the temperature T. The
following equation is useful in determining the speed of
sound in air
v = 331 + 0.6 T
Units: m/s
Where 331 is the speed of sound in m/s at 0°C,
and T is the temperature in °C
There are two main characteristic of sound: pitch and
loudness.
Pitch refers to how high (or low) the sound is. It is
measured by the frequency.
We hear frequencies in the range of 20 Hz to 20,000 Hz.
This is called the audible range.
Frequencies above this range are called ultrasonic.
Sound waves whose frequency is lower than the audible
range are called infrasonic.
INTERFERENCE OF SOUND WAVES: BEATS
If two sources are close in frequency, the sound from
them interferes and what we hear is an alternating
sound level. The level rise and falls. If the alternating
sound is regular, it is called beats.
The beat frequency equals the difference in
frequencies between the sources.
This is a way to tune musical instruments. Compare a
tuning fork to a note and tune until the beats
disappear.
CI
Constructive
Interference
DI
Destructive
Interference
HUMAN EAR:
The ear consists of three basic parts:
Outer ear: serves to collect and channel
sound to the middle ear.
Middle ear: serves to transform the
energy of a sound wave into the
internal vibrations of the bone
structure of the middle ear and
transform these vibrations into a
compressional wave in the inner ear.
Inner ear: serves to transform the energy of a
compressional wave within the inner ear fluid into
nerve impulses which can be transmitted to the brain.
Hearing:
A compression forces the eardrum inward and an
expansion forces the eardrum outward, thus vibrating
the eardrum at the same frequency of the sound wave.
SOURCES OF SOUND
Sound comes from a vibrating object. If an object
vibrates with frequency and intensity within the audible
range, it produces sound we can hear.
MUSICAL INSTRUMENTS
- String Instruments: guitar,
violin and piano
-Wind Instruments:
Open Pipe: flute and some
organ pipes
Closed Pipe: clarinet,oboe
and saxophone
STRING INSTRUMENTS
The sounds produced by vibrating strings are not very
loud. Many stringed instruments make use of a
sounding board or box, sometimes called a resonator, to
amplify the sounds produced. The strings on a piano are
attached to a sounding board while for guitar strings a
sound box is used. When the string is plucked and
begins to vibrate, the sounding board or box begins to
vibrate as well. Since the board or box has a greater area
in contact with the air, it tends to amplify the sounds.
On a guitar or a violin, the length of the
strings are the same, but their mass per
length is different. That changes the
velocity and so the frequency changes.
WIND INSTRUMENTS
Wind instruments produce sound from the vibrations of
standing waves in columns of air inside a pipe or a tube.
In a string, the ends are nodes. In air columns the ends
can be either nodes or antinodes.
Open pipe
Half Closed Pipe
HARMONICS
a) For open pipe
The overtones will be multiples of the fundamental
b) For closed pipe
The overtones will be the odd multiples of the
fundamental
9.7 A saxophone plays a tune in the key of B-flat. The saxophone
has a third harmonic frequency of 466.2 Hz when the speed of
sound in air is 331 m/s. What is the length of the pipe that makes
up the saxophone?
n=3
f3 = 466.2 Hz
v = 331 m/s
f' = f3/n
= 466.2/3
= 155.4 Hz
Saxophone is closed so: L= 1/4 λ
1
 1  v  331 = 0.53 m
L   
4(155.4)
4
4 f
9.8 An organ pipe that is open at both ends has a fundamental
frequency of 370.0 Hz when the speed of sound in air is 331 m/s.
What is the length of this pipe?
f' = 370 Hz
v = 331 m/s
L= 1/2 λ
331
1
1
v
 

= 0.447 m
L   
2(370)
2
2 f
9.9 What is the fundamental frequency of a viola string that is
35.0 cm long when the speed of waves on this string is 346 m/s?
L = 0.35 m
v = 346 m/s
L= 1/2 λ and λ = 2L
v
346
= 494.2 Hz

f  
 2 L 2(0.35)
v
9.10 A pipe that is open at both ends has a fundamental frequency
of 125 Hz. If the pipe is 1.32 m long, what is the speed of the waves
in the pipe?
f' = 125 Hz
L = 1.32 m
L= 1/2 λ and
λ = 2L
v=λf
=2 L f
= 2(1.32)(125)
= 330 m/s
9.11 A pipe that is closed on one end has a seventh harmonic
frequency of 466.2 Hz. If the pipe is 1.53 m long, what is the speed
of the waves in the pipe?
n=7
f7 = 466.2 Hz
L = 1.53 m
L= 7/4 λ and λ = 4/7 L
 4L 
 41(1.53) 
v f 
  466.2 
 = 407.6 m/s
7


7


DOPPLER EFFECT
When a source of sound waves and a listener approach
one another, the pitch of the sound is increased as
compared to the frequency heard if they remain at rest. If
the source and the listener recede from one another, the
frequency is decreased. This phenomenon is known as
the Doppler effect.
Light waves also exhibit the
Doppler effect. The spectra of
stars that are receding from us is
shifted toward the longer
wavelengths of light. This is
known as the red shift.
Measurement of the red shift allows
astrophysicists to calculate the
speed at which stars are moving
away. Since almost all stars and
galaxies exhibit a red shift, it is
believed that the universe is
expanding.
SHOCK WAVES AND THE SONIC BOOM
When the speed of a source of sound exceeds the speed of
sound, the sound waves in front of the source tend to overlap
and constructively interfere. The superposition of the waves
produce an extremely large amplitude wave called a shock
wave.
The shock wave contains a great
deal of energy. When the shock
wave passes a listener, this energy
is heard as a sonic boom.
The sonic boom is heard only for a
fraction of a second; however, it
sounds as if an explosion has
occurred and can cause damage.