Chapter 11
Waves
Waves
A wave is a oscillation/disturbance generated
from its source and travels over long distances.
zA wave transports energy but not matter.
z Examples:
Î Water waves: formed when you throw a
stone in water. Water moves up and down.
Î Sound waves (air moves back & forth)
ÎElectromagnetic waves: Light waves, Radio
waves, TV waves, X-rays etc. (what
moves?).
z
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¾All waves are produced by vibrating
(oscillating) sources.
¾If the source vibrates sinusoidally in
SHM, the wave will have sinusoidal
shape in space and time.
¾When waves travel through a medium,
the particles of the medium vibrate.
¾The manner in which medium particles
vibrate defines two types of waves:
Transverse and Longitudinal waves.
Types of Waves
z Transverse
and Longitudinal
1. Transverse Waves: The medium oscillates
perpendicular to the direction the wave is
moving.
ÎWater (more or less)
Î Electromagnetic waves: Light, radio, TV,
X-rays …)
ÎSlinky demo.
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Transverse Waves: The medium oscillates
perpendicular to the direction the wave is
moving.
Particles of medium
Wave velocity (v)
y
x
Transverse Waves…
Crest – highest point
Trough – Lowest point
Wavelength (λ) – distance from crest to next
crest. Or from a trough to next trough.
y
λ
Crest
Crest
trough
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trough
x
Oscillation of medium particles about their
equilibrium position with time.
Period (T) – Time for a medium particle to
oscillate through one cycle. Within this time,
the wave will have traveled through a
distance equal to one wavelength.
Amplitude (A) = maximum displacement from
equilibrium position.
T
y
A
¼T
½T
t
¾T T
Period (T) –Within one period (T), the wave
will have traveled through a distance equal
to one wavelength (λ)
Thus velocity of the wave v = distance/time
v = λ/T = λf
T
y
¼
T
y
A
½T ¾
T
t
T
Crest
λ
trough
Crest
trough
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x
2. Longitudinal Waves:
The medium oscillates in the same
direction as the wave is moving.
Examples:
ÎSound
ÎSlinky demo
Particles of medium
Wave velocity (v)
z Longitudinal:
The medium oscillates in
the same direction as the wave is moving
Particles of medium
Wave velocity (v)
Series of regions of compressions and
expansions (rarefactions) are formed.
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Longitudinal
One wavelength = distance from one region of
compression to the next region of
compression.
OR distance from one rarefaction to the next
rarefaction.
Time taken to make one wavelength is the
period T.
λ
Example
The speed of sound in air is a bit over 300
m/s, and the speed of light in air is about
300,000,000 m/s. Suppose we make a
sound wave and a light wave that both
have a wavelength of 3 meters. What is
the ratio of the frequency of the light
wave to that of the sound wave?
V = fλ
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Reflection of Waves
When a wave meets an obstacle in its
path, some of it will be reflected,
some will be absorbed (as heat
energy) and some will be transmitted
through the obstacle.
Medium 1
Reflected ray
Incident ray
vr, λr, fr
Medium 2
Transmitted (refracted) ray
vt, λt, ft
vi, λi, fi
Reflection of Waves
Reflection of waves – When waves
bounce off of an obstacle and take a
different path of propagation.
Eg. Echo – is reflection of sound waves.
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Reflection
When a traveling wave meets a
boundary, reflection occurs. part of
the wave travels backwards from
the boundary
Wave fronts: Representation of a whole
width of a wave crest.
Rays: Lines drawn perpendicular to the
wave fronts. Arrows point to direction
of travel.
Rays
Wave fronts
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When waves are reflected, they obey the
law of reflection:
Angle of incidence = angle of reflection.
θi = θr
Incident ray
Reflected ray
θi θ
r
Refraction
• When a wave crosses from one
medium to another, its speed and
direction changes.
• However, its frequency remains the
same.
• Will the wavelength be the same or
different across the media?
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• Recall: Wave velocity: v = λ/T = λf.
• In a given medium, the velocity (v) of a
wave stays constant.
• So for a wave traveling in the same
medium, if its frequency changes, the
wavelength will change accordingly to
maintain constant velocity.
• If the wave crosses one medium into a
different medium, its velocity (both
magnitude and direction) will change.
Refraction
• Velocity: vi, vt - different
• Frequency: fi, ft - same
• Wavelength: λi, λt - different
Medium 2
Medium 1
vr, λr, f
Reflected ray
Incident ray
Transmitted (refracted) ray
vt, λt, f
vi, λi, f
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In-phase
y
Wave 2
Wave 1
t
Two waves are in phase if they reach
identical positions at the same time.
Out of phase
y
Wave 2
t
Wave 3
Two waves are out of phase if they reach
identical positions at different times.
Wave 2 and wave 3 are out of phase by 180o
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Coherent Waves
Coherent Waves: two or more waves that
have the same frequency and keep the
same phase relationship between them.
y
Wave 3
Wave 1
t
Wave 2
Incoherent Waves
Incoherent waves: Waves for which the
phase relationship varies randomly.
y
Wave 1
Wave 2
t
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Interference and Superposition
Interference: What happens when 2 or
more waves pass through a point at
the same time.
Overall displacement = algebraic sum of
the separate displacements of each
wave – this is principle of
superposition.
Interference and Superposition
• When too waves overlap,
the amplitudes add.
• Constructive:
increases amplitude
• Destructive:
decreases amplitude
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Standing Waves
Interference of a wave and its reflection
creates standing wave.
Antinode
Node
Node – displacement is zero. Corresponds to
destructive interference.
Antinode – where cord moves with maximum
amplitude – Constructive interference.
Standing Waves
• Interference (superposition) of a wave and its
reflection creates standing wave.
•The two superimposed waves have equal
amplitudes, equal frequencies, equal wavelengths,
equal speeds but traveling opposite to each other.
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Resonant Modes of vibration
Fundamental Frequency
First Harmonic, n = 1
Second Harmonic
n=2
Third Harmonic
n=3
Fourth Harmonic
n=4
Resonant Modes of vibration
#
Wavelength Frequency Name
Loops (λ)
(f)
1
Fundamental
L = λ1/2
f1 = v/λ1
f1 = v/2L 1st Harmonic
λ1 = 2L/1
2
λ2=L
λ2 = 2L/2
3
L = 3λ3/2
λ3= 2L/3
4
L = 2λ4
λ4 = 2L/4
n
L = nλ/2
λn = 2L/n
f2 = v/λ2
f2 = 2v/2L
= 2f1
f3 = v/λ3
f3 = 3v/2L
= 3f1
f4 = v/λ4
f4 = 4v/2L
= 4f1
fn = v/λν
fn = nv/2L
= nf1
v = λf
2nd Harmonic
1st Overtone
3rd Harmonic
2nd Overtone
4th Harmonic
3rd Overtone
λn = 2L/n
fn = nv/(2L)
fn = nf1
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Example
A string with both ends fixed resonates in five
loops at a frequency of 525 Hz. If the
velocity of the waves in the string is 84 m/s
(a) How far apart are two adjacent nodes?
(b) What is its fundamental frequency?
(c) What is the length of the string?
Example
A standing wave is produced in a vibrating string as
shown below. If the length of the string is 1.5 m and
the frequency of the vibrating motor is 60 Hz, the
speed of the wave is
(A) 15 m/s
(B) 20 m/s
(C) 40 m/s
(D) 60 m/s
(D) 90 m/s
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