GENERAL PROPERTIES OF
OCILLATIONS AND WAVES
Waves
• Definition:
• A wave is a propagation of a disturbance through a
medium without any net displacement of the medium
Altivenatively
A wave can simple be described as a disturbance that
travels through a medium from one location to another
location
Several forms exist:
Simple Harmonic Motion(SHM)
Categories of Waves
Transverse Waves
Longitudinal Waves
Categories of Waves
Surface Waves
Electromagnetic Waves
Mechanical Waves
Description of a Wave
Crest, Trough, Wavelength, Amplitude, Frequency,
Period, Intensity, Phase Angle, Phase Velocity
General Wave Equation
v 2
2
t
x
2
2
is the wave amplitude, v and the characteristic
phase velocity.
Equation of a travelling wave
y ( x, t ) = Acos ( kx - w t - j )
y ( x, t ) = Asin ( kx - w t - j )
y ( x, t ) = Asin ( kx - w t ) Simple form this, moving to the right
Superposition of Waves
• The principle of superposition may be applied to waves whenever
two (or more) waves travelling through the same medium at the
same time.
• The waves pass through each other without being disturbed. The net
displacement of the medium at any point in space or time is simply
the sum of the individual wave displacements.
Two sine waves travelling in the same direction:
Constructive and Destructive Interference
Consider two similar travelling waves in the same direction. Principle
of superposition gives
y x, t A sin kx t A sin kx t
and using the trigonometrically identity
B A A B
sin A sin B 2 cos
sin
2 2
gives
2 A cos sin kx t
2
2
• The superposition of two or more waves is called interference
Constructive interference:
These two waves are in phase.
Their crests are aligned.
Their superposition produces a
wave with amplitude 2a
Destructive interference:
These two waves are out of phase.
The crests of one are aligned with
the troughs of the other.
Their superposition produces a
wave with zero amplitude
Phase – The term phase is used to describe the relative position of
the wave crests as they meet.
a. In phase – results in an
amplitude that is the sum of
both crests.
b. Out of phase – occurs when the
crests do not meet together
and the waves partially or
completely cancel each other.
This complete cancellation
occurs when a trough of one
wave meets the crest of another.
Two sine waves travelling in opposite directions
create a standing wave
Consider two oppositelly travelling waves
x, t A sin kx t
x, t A sin kx t
Adding these gives
x, t Asin kx t sin kx t
x, t Asin kx cos t cos kx sin t sin kx cos t cos kx sin t
x, t Asin kx cos t sin kx cos t
x, t 2 A sin kx cos t
Formation of Standing waves
A standing wave pattern is a vibrational pattern created within
a medium when the vibrational frequency of the source causes
reflected waves from one end of the medium to interfere with
incident waves from the source.
This interference occurs in such a manner that specific points
along the medium appear to be standing still.
A careful study of the standing wave patterns reveal a clear
mathematical relationship between the wavelength of the wave which
produces the pattern and the length of the medium in which the
pattern is displayed.
Formation of Standing Waves
For the nth harmonic
n
L
2
Two sine (or cosine) waves with different
frequencies: Beats
Consider two sine waves
y x, t A sin k1 x 1t
y x, t A sin k 2 x 2 t
Application of Principle of Superposition gives
y x, t A sin k1 x 1t A sin k 2 x 2 t
1 2 k1 k 2 1 2
k1 k 2
x
t sin
x
t
= 2 A cos
2
2
2
2
Beat Frequency
f 1 2 f1 f 2
Beat
f beat f1 f 2
• The group velocity of a wave is the velocity with
which the overall shape of the wave's amplitudes
—known as the modulation or envelope of the
wave —propagates through space.
• The group velocity vg is defined by the equation
¶w
vg =
¶k
Interference, Diffraction, and Polarization
Key Question:
What are some ways
light behaves like a
wave?
Interference, Diffraction, and Polarization
• In 1807, Thomas Young
(1773-1829) did the most
convincing experiment
demonstrating that light
is a wave.
• A beam of light fell on a
pair of parallel, very thin
slits in a piece of metal.
A pattern of alternating bright
• After passing through the and dark bands formed is called an
interference pattern.
slits, the light fell on a
screen.
Interference
One hole
diffraction
• Diffraction is the bending of waves that occurs
when a wave passes through a narrow
opening
• If a wave falls on a barrier that has an opening
of dimension similar to the wavelength, the
wave will flair into the region beyond
Dark fringes occur at :
w sin m
Diffraction by a single slit
Dark fringes
Bright fringes
Predict pattern for two holes
Condition for bright fringes:
Δ= nλ = d sin where n = 0, 1, 2, 3 …
Condition for dark fringes:
Δ = ½λ, 3/2 λ, 5/2λ or
where n = 0, 1, 2, 3
1 22n 1
Diffraction gratings
A diffraction grating is a precise array of tiny engraved lines,
each of which allows light through.
The spectrum produced is a mixture of many different
wavelengths of light.
BASICS OF DIFFRACTION
• Single slit interference
P– 1st maximum
Q– 1st secondary maximum
θ = nλ/d
Intensity of the beam is governed by
I = I0 { sin β / β }2
Where β = (π / λ) d sin θ
Doppler Effect
DEFINITION
• First explained in 1842 by
Christian Doppler, the
Doppler Effect is the shift
in frequency and wavelength
of waves which results from
a source moving with
respect to the medium, a
receiver moving with
respect to the medium, or
even a moving medium.
• In other words, the Doppler
Effect refers to the change
in pitch of a sound due to
the motion either of the
source or of the listener.
STATIONARY SOUND SOURCE
•
The picture to the right
shows a stationary sound
source. Sound waves are
produced at a constant
frequency f0, and the
wavefronts propagate
symmetrically away from
the source at a constant
speed v, which is the speed
of sound in the medium. The
distance between
wavefronts is the
wavelength. All observers
will hear the same
frequency, which will be
equal to the actual
frequency of the source.
The Doppler Effect
As a wave source approaches, an observer
encounters waves with a higher frequency. As the
wave source moves away, an observer encounters
waves with a lower frequency.
THE DOPPLER EFFECT
CHARACTERISTICS OF THE DOPPLER
EFFECT
• If two objects are approaching each other, or if
an initial object is approaching a second standing
object, the pitch is higher
• If two objects are moving apart, or if an initial
object is moving apart from a second standing
object, the pitch is lower
SOURCE MOVING WITH vsource < vsound
•
The picture shows the same sound
source is radiating sound waves at a
constant frequency in the same
medium. However, now the sound
source is moving to the right with a
speed vs = 0.7 v (Mach 0.7). The
wavefronts are produced with the
same frequency as before. However,
since the source is moving, the
center of each new wavefront is
now slightly displaced to the right.
As a result, the wavefronts begin to
bunch up on the right side (in front
of) and spread further apart on the
left side (behind) of the source. An
observer in front of the source will
hear a higher frequency f ´ > f0, and
an observer behind the source will
hear a lower frequency f ´ < f0.
The Doppler Effect
This apparent change in frequency due to the motion of the
source (or receiver) is called the Doppler effect.
The greater the speed of the source, the greater will be the
Doppler effect.
DOPPLER EFFECT
Doppler Effect is the apparent change in the observed frequency of a
wave, as of sound or light, occurring when the source and observer
are in motion relative to each other, with the frequency increasing
when the source and observer approach each other and decreasing
when they move apart.
Source moving away from stationary observer
v
f o
f
v vs
Observer hears sound of longer wavelength, lower
frequency and lower pitch
The Doppler Effect
Sound
The Doppler effect causes the changing pitch of a siren.
When a firetruck approaches, the pitch sounds higher than normal
because the sound wave crests arrive more frequently.
When the firetruck passes and moves away, you hear a drop in pitch
because the wave crests are arriving less frequently.
Note: The change in loudness is not the
Doppler Effect! It is the shift in frequency!
DOPPLER EFFECT
Source moving towards from stationary observer
v
f o
f
v vs
Observer hears sound of shorter wavelength, higher
frequency and higher pitch
DOPPLER EFFECT
Observer moving towards from stationary source
v vr
f
fo
v
Observer hears sound of shorter wavelength, higher
frequency and higher pitch
DOPPLER EFFECT
Observer moving away from stationary source
v vr
f
fo
v
Observer hears sound of longer wavelength, lower
frequency and lower pitch
EQUATION
• The perceived frequency (f ´) is related to the
actual frequency (f0) and the relative speeds of
the source (vs), observer (vo), and the speed (v) of
waves in the medium by an equation:
OTHER USES OF THE DOPPLER
EFFECT
• Although first discovered for sound waves, the
Doppler effect holds true for all types of waves
including light (and other electromagnetic waves).
The Doppler effect for light waves is usually
described in terms of colors rather than
frequency. A red shift occurs when the source and
observer are moving away from each other, and a
blue shift occurs when the source and observer
are moving towards each other. The red shift of
light from remote galaxies is proof that the
universe is expanding.
The Doppler Effect
Light
The Doppler effect also occurs for light.
• When a light source approaches, there is an increase in its
measured frequency.
• When it recedes, there is a decrease in its frequency.
(This is an important slide)
The Doppler Effect
Increasing frequency is called a blue shift, because the increase is
toward the high-frequency, or blue, end of the spectrum.
Decreasing frequency is called a red shift, referring to the lowfrequency, or red, end of the color spectrum.
Distant galaxies show a red shift in their light. A measurement of
this shift enables
astronomers to calculate their
speeds of recession. The red shift
Is also a piece of evidence for the
Big Bang theory.
Doppler Effect in Light
Redshift
Occurs whenever a light source moves away from the observer,
corresponding to the Doppler shift that changes the perceived
frequency of sound waves. Observer sees a longer wavelength
observed emitted
z
emitted
z>0
for redshift
z<0
for blueshift
Bel and Decibel
Bel
It is a measurement unit used when comparing two sound
intensities. Whenever the intensity of sound is increased by a
factor of 10, the increase in intensity is said to be 1 bel. The
dynamic range of audibility of the human ear is 12 bels or 120
decibels
Decibel (dB)
The sound intensity I may be expressed in decibels above the
standard threshold of hearing Io . The expression is
I
1 dB 10 log
I0
0
