physics beyond 2000 chapter 9 wave phenomena. glossary wave: it is a periodic disturbance in a...
TRANSCRIPT
Physics Beyond 2000
Chapter 9
Wave Phenomena
Glossary
• Wave: It is a periodic disturbance in a medium or in space.
• There must be source(s) in periodic oscillation.
• There is a transfer of energy from the source.
Glossary
• Progressive waves (or travelling waves)
The wave pattern is moving forwards.
• Stationary waves (or standing waves)
The wave pattern remains at the same position.Progressive and stationary transverse waves:http://www.geocities.com/yklo00/2Waves.html
Stationary longitudinal wave:http://www.fed.cuhk.edu.hk/sci_lab/Simulations/phe/stlwaves.htm
Glossary
• Mechanical waves: with medium for the vibration.
• Electromagnetic waves: no medium is required. It is the electric field and magnetic field in vibration.
Glossary• Transverse wave: the direction of vibration
is parallel to the direction of travel of the wave.
• Longitudinal wave: the direction of vibration is perpendicular to the direction of travel of the wave.
http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
http://webphysics.davidson.edu/Applets/TaiwanUniv/waveType/waveType.html
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveType/waveType.html
Glossary
• Sinusodial wave:
• Square wave:
• Saw-toothed wave:
Observe these wave patterns on a CRO.
Progressive Waves
• The disturbance is propagating from a source to places.
• Energy is transferred from the source to places.
• Matter of medium is not transferred. It is vibrating.
http://members.nbci.com/surendranath/Applets.html
water ripples
http://www.freehandsource.com/_test/ripple.html
Ripple is an example of progressive wave that carriesenergy across the water surfaces.
http://www.colorado.edu/physics/2000/waves_particles/waves.html
water ripples
Ripple is an example of progressive wave that carriesenergy across the water surfaces.
ray
wavefront
source
Mechanical waves
• In need of a material medium for the propagation of wave.
• Examples are wave in a string, sound wave, water wave etc.
Electromagnetic waves
• They can travel through a vacuum.
• Speed of electromagnetic wave in vacuum is 3 × 108 ms-1.
• Electromagnetic spectrum:
http://www.colorado.edu/physics/2000/waves_particles/
Electromagnetic waves
• Electromagnetic waves are transverse waves.
• Electric field and magnetic field are changing periodically at right angle to each other and to the direction of propagation.
Oscillation of charges:http://www.colorado.edu/physics/2000/waves_particles/wavpart4.html
http://www.fed.cuhk.edu.hk/sci_lab/Simulations/phe/emwave.htm
Analytical Description of Progressive Waves
• Speed of propagation (c)
• Frequency (f)
• Wavelength (λ)
• Amplitude (a)
• Period (T)
Analytical Description of Progressive Waves
• T =
• c = f. λ =
f
1
t
swhere s the distance travelledin time t.
Example 1
• c = f. λ
Graphical Representation andPhase Relation
• Each particle in the wave is performing a simple harmonic motion, oscillating about its equilibrium position.
http://members.nbci.com/surendranath/Applets.html
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/wave/wave.html
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In phase (or phase difference = 0)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In phase (or phase difference = 0)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In phase (or phase difference = 0)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In phase (or phase difference = 0)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In antiphase (or phase difference = π)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In antiphase (or phase difference = π)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In antiphase (or phase difference = π)
Graphical Representation andPhase Relation
• There is a phase difference between any two particles.
• In antiphase (or phase difference = π)
Notation in the textbook• y = displacement of a vibrating particle.
• x = distance of the particle from the source.
y-x graph for a transverse wave (points along the path at a particular instant):
y
x0
c = speed of propagation
a
Notation in the textbooky-x graph for a transverse wave:
C
T
C: crestT: troughdirection of
propagation
Notation in the textbook• y = displacement of a vibrating particle.
• In transverse wave, y is positive if the particle is above the
equilibrium position;
y is negative if the particle is below the equilibrium position.
y > 0
y < 0
Equilibriumposition
Notation in the textbooky-x graph for a longitudinal wave:
C R
C:compressionR:rarefaction
Notation in the textbook• y = displacement of a vibrating particle.
• In longitudinal wave, y is positive if the particle displaces along the
direction of the travel of the wave;
y is negative if the particle displaces in the opposite direction of the travel of the wave.
wave direction
y > 0y < 0
Equilibriumposition
Equilibriumposition
Wave Speed and Speed of Particle
• Wave speed c =
• Speed of a particle vy =
dt
dx
dt
dy
The wave is moving forward with a constant speed c.
The particle is vibrating in a SHM with a changing speed.
Example 2
• Find the maximum speed of a vibrating particle.
• The particle is in SHM with
y = a.sin(t + o)
Phase Relationship
• For the motion of a vibrating particle in SHM,
y = a. sin(t + o)
with o the initial phase.
• In a wave, different particles have the different initial phase.
Phase difference between two points in a y-x graph
For two points with separation x in a wave with wavelength , their phase difference is
x
2
y
x
direction of propagation
0
x
in radian
P Q
Phase difference between two points in a y-x graphy
x
direction of propagation
0
x
Which point leads the other?
P Q
Phase difference between two points in a y-x graph
Compare points P and R.Which point leads the other?
y
x
direction of propagation
0
P
R
Phase difference between two points in a y-x graphy
x
direction of propagation
0
P
R
Draw the new wave pattern after a time t.The whole wave pattern moves to the right.
Phase difference between two points in a y-x graphy
x
direction of propagation
0
P
R
Point P will move back to the equilibrium positionfrom the crest after .Point R will move to the position of the crest fromthe equilibrium position.
4
T
Phase difference between two points in a y-x graphy
x
direction of propagation
0
P
R
After , P moves to the equilibrium position and R moves up to the position of the crest.So P leads R by
4
T
2
Phase difference between two waves in a y-x graphy
x
direction of propagation
0
Two waves y1 and y2 of the same frequency are moving to the right simultaneously.What is the phase difference between thesetwo waves?
y1 y2
Phase difference between two waves in a y-x graphy
x
direction of propagation
0
y1 y2
Measure the separation x of their crests ( choose x < /2 ).
x2
The phase difference is
x
Phase difference between two waves in a y-x graphy
x
direction of propagation
0
Choose a point P in front of the waves.Which wave has its crest reach the point first?
y1 y2
P
The first wave leads the second wave by .
Density variation along a longitudinal wave
• The centres of compression have the highest density and highest pressure (for gas).
C
Density variation along a longitudinal wave
• The centres of rarefaction have the lowest density and lowest pressure (for gas).
R
Density variation along a longitudinal wave
• The crest of the density/pressure leads that of displacement by /2.
x
x
Example 3
• Ripple in water is transverse wave.
• Hint: x2
The phase difference is
Example 4
• Which one leads?
• Keep the phase difference < .
Displacement-time graph
• Describe a particle of the wave at different time.
y
t0
a
-aT
o
Displacement-time graph
• For a particle performing simple harmonic motion,
y = a.sin(t + o)
• It is a sinusoidal wave.
y
t0
a
-aT
o
Displacement-time graph
• Other particles along the path are performing SHM at the same frequency but with a different phase o
y
t0
a
-aT
o
Displacement-time graph
• Two points P and Q are in the path of a wave.
y
t0
a
-aT
P Q
Displacement-time graph
• Wave P becomes crest earlier than wave Q . ( tP < tQ).
Wave P leads wave Q.
y
t0
a
-aT
P QtP tQ
Displacement-time graph
• Time difference t = tQ – tP
• Phase difference
y
t0
a
-aT
P QtP tQ
T
t 2
Example 5 y = a.sin(t + o)• yA = 0.1 sin (2t + /3) ψA = /3• yB = 0.1 sin (2t + /4) ψB = /4
phase of wave A ψA > phase of wave B ψB A leads B.
A
y
t
yA yB
B
0
Note: keep the phase difference < π
Traces on CRO• Use a double beam oscilloscope.• The two input waves must be of the same
frequency.• It is a y-t graph on the screen of the CRO.
Y1 Y2
to signal generator 1
to signal generator 2
y-x and y-t graphs• A typical example:
At t = 0,
At point P,
y
x0P
direction of propagation
y
t0
http://www.geocities.com/yklo00/Wave_yx-yt-graphs.htm
Wave speed of a mechanical wave
• c = mediumtheofpropertyinertial
mediumtheofpropertyelastic
Wave speed of a mechanical wave
• c =
• Longitudinal wave in a solid
mediumtheofpropertyinertial
mediumtheofpropertyelastic
E
c where E is the Young modulusof the solid and is the densityof the solid.
• Example 6
Wave speed of a mechanical wave
• c =
• Transverse wave in a string
mediumtheofpropertyinertial
mediumtheofpropertyelastic
T
c where T is the tension in the string and is the mass per unit length of the string.
• Example 7
Wave speed of a mechanical wave
• c =
• Sound wave in air
mediumtheofpropertyinertial
mediumtheofpropertyelastic
P
c where P is the air pressure, is the density of air and is a constant.
Sound wave in air
P
c where P is the air pressure, is the density of air and is a constant.
By gas law PV = nRT
TM
nRTP
Tc
Speed of sound in air isindependent of the pressure.It increases with temperature.
Measure speed of sound• Use a double beam oscilloscope.• Compare the phase difference between the
input from the signal generator and that from the microphone.
Y1 Y2
CRO Signal Generator
LoudspeakerMicrophone
Ruler
Measure speed of sound
• Place the microphone at a position such that both inputs are in phase.
Y1 Y2
CRO Signal Generator
LoudspeakerMicrophone
Ruler
Measure speed of sound
• Move the microphone towards the loudspeaker to locate the next position with both inputs in phase again.
Y1 Y2
CROSignal Generator
LoudspeakerMicrophone
Rulerx
Measure speed of sound• The distance x = wavelength • For accurate measurement, we usually
move the microphone more than one wavelength.
Y1 Y2
CROSignal Generator
LoudspeakerMicrophone
Rulerx
Measure speed of sound
• Find the frequency f from the signal generator or the period T from the time base of the CRO.
Y1 Y2
CROSignal Generator
LoudspeakerMicrophone
Rulerx
f
T
Measure speed of sound
• c = f. or c =
Y1 Y2
CROSignal Generator
LoudspeakerMicrophone
Rulerx
T
f
T
Water waves
• Ripple tank: for studying water wave
• Stroboscope: for freezing wave pattern.
Water waves
• Ripple tank
• Production of circular waves by a dipper.
Water waves
• Ripple tank
• Production of plane waves by a straight line.
Water waves
• Hand Stroboscope: for freezing the motion.
slit
Water waves
• Hand Stroboscope• Strobe frequency fs is the numb
er of slits passing the viewer per second.No. of slits on the hand strobe = nNo. of revolutions per second = m
fs = m.n
Freezing a periodic motion• A periodic motion with frequency fo and peri
od To.• To freeze the motion, - the strobe should rotate in the same directio
n as the periodic motion and - the strobe frequency should be
fs = fo, 2.fo, 3.fo,… or
fs = fo/2, fo/3, fo/4,…
Viewing a rotating arrow.
Freezing a periodic motion
• A periodic motion with frequency fo and period To.
• fs = 2.fo, 3.fo,… or Ts = To/2, To/3, … Though the motion is frozen, the observed p
attern becomes too dense. The motion is viewed before the arrow completes one cycle.
• Take fs = 2.fo as an example,
(Ts = To/2)Two arrows are seen.It is a double viewing.
Freezing a periodic motion• A periodic motion with frequency fo and peri
od To.• fs = fo/2, fo/3,…or Ts = 2.To, 3.To, …• The motion is viewed when the motion has c
ompleted more than one cycle.
• Take fs = fo/2 as an example
(Ts = 2.To)
Only one arrow is seen.It is viewed once every 2.To.
Freezing a periodic motion
• A periodic motion with frequency fo and period To.
• fs = fo or Ts = To
• The motion is viewed when the motion has just completed one cycle.
Only one arrow is seen.It is viewed once every To.
Example 8
• 4 arrows are seen.
• Think of the time and periods.
Find the frequency of motion
• Freeze the motion without increasing its density.
fs = fo, fo/2, fo/3,….
• Increase fs to a maximum so that it still can freeze the motion without increasing its density.
fs = f0
Find the frequency of water wave in a ripple tank
• Freeze the wave pattern without decreasing its wavelength.
fs = fo, fo/2, fo/3,….
• Increase fs to a maximum so that it still can freeze the wave pattern without decreasing its wavelength.
fs = f0
Example 9
• Freeze the wave pattern.
Typical Example: stroboscope
• 4 identical arrows on a disc which is rotating at 12 revolutions per second.
• Use a strobe with 12 slits.
• What are the possible strobe speeds at which 4 frozen arrows are seen?
Example 9
• Find the speed of water wave.
Forward and backward motions
• If fs is slightly larger than fo (Ts < To), the motion seems moving forward.
• The arrow rotates more than one cycle in Ts.
Ts Ts Ts
T0 + t
Forward and backward motions
• If fs is slightly less than fo (Ts > To), the motion seems moving backward.
• The arrow rotates less than one cycle in Ts.
Ts Ts Ts
T0 - t
General Wave Properties
• Wave properties:
• Reflection
• Refraction
• Diffraction
• Interference
• Water wave is an example in this section.
Huygen’s principle
• Suppose that there is a
straight wavefront.
• Take every point on the wavefront as a source of secondary waves.
• Draw the secondary waves.
• The new wavefront is the envelope of these secondary waves.
Straight wavefront:
c.t
Huygen’s principle
• Suppose that there is a curved wavefront.
• Take every point on the wavefront as a source of secondary waves.
• Draw the secondary waves.
• The new wavefront is the envelope of these secondary waves.
Curved wavefront:
ct
Reflection
• It is the return of wavefront when it encounters the boundary between two media.
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/propagation/propagation.html
Reflection
• It is the return of wavefront when it encounters the boundary between two media.
• Straight reflector and plane wave:
Reflection• It is the return of wavefront when it encount
ers the boundary between two media.
• Straight reflector and circular wavefront:
Radar and sonar
• Radar: RAdia Detection And Ranging
Radar and sonar
• Radar: RAdia Detection And Ranging
The station sends a pulse.
Radar and sonar
• Radar: RAdia Detection And Ranging
The station receives the pulse.
Radar and sonar
• Radar: RAdia Detection And Ranging
d
If the time difference is t, findthe distance d.
Radar and sonar
• Sonar: SOund Navigation And Ranging
seabed
Radar and sonar
• Sonar: SOund Navigation And Ranging
seabed
Send a pulse of ultrasound
Radar and sonar
• Sonar: SOund Navigation And Ranging
seabed
Receive the pulse of ultrasound
Refraction• It is the change of direction of wavefront as
it passes obliquely from one medium to another.
• The wave speeds must be different in the two media.
• The wavelength also changes.
• The frequency does not change.
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/propagation/propagation.html
http://www.fed.cuhk.edu.hk/sci_lab/download/project/Lightrefraction/LightRefract.html
Snell’s law
2
1
2
1
2
1
sin
sin
c
c
1
2
medium 1(wave with high speedand long wavelength)
medium 2(wave with low speedand short wavelength)
Normal
Snell’s law
2
1
2
1
2
1
sin
sin
c
c
1
2
medium 1(with high speed)
medium 2(with low speed)
NormalY
Y’ X
X’
Snell’s law
1
2
medium 1(with high speed)
medium 2(with low speed)
NormalY
Y’ X
X’
In time t, Y moves to Y’ with YY’ = c1.t.At the same time t, X moves to X’ with XX’ = c2.t
Snell’s law
1
2
medium 1(with high speed)
medium 2(with low speed)
NormalY
Y’ X
X’
Prove Snell’s law.Hint: Study triangles XYY’ and XX’Y.
Total internal reflection
Total internal reflection occurs when
• Wave moves from medium of low speed to another medium of high speed.
• The angle of incidence is equal to or larger than the critical angle. medium 2
(wave with high speed)
medium 1(wave with slow speed)
normal
Total internal reflection
• Note that there is not any wave into the second medium.
medium 2
medium 1
normal
Diffraction
• It is the spreading or bending of waves as they pass through a gap or round the edge of a barrier..
Diffraction• The extent of diffraction depends on the
relative sizes of the gap and the wavelength.
narrow gap wide gap
Interference
• Principle of superpositionThe total displacement of a point is equal to the algebraic sum of the individual displacements at that point.
Superposition of pulses:http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/wave/impulse.html
Superposition of continuous waves:http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveSuperposition/waveSuperposition.html
Interference of water waves
• Two dippers produce water waves of same frequency.
• The two dippers are in phase.
Interference of water waves
• Along lines of reinforcement (anti-nodal lines), the amplitude of each point is large.
• Along lines of cancellation (nodal lines), the amplitude of each point is almost zero.
• The interference pattern depends on the relative sizes of the source separation and wavelength.
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveInterference/waveInterference.html
Conditions to produce an interference pattern
• The two sources are coherent.• They produce waves of same frequency• The sources of waves maintain a constant
phase difference.
• The separation between the sources is only several wavelengths.
Path difference
• We always express the path of a wave train in terms of the number of wavelengths.
• A continuous wave travels from S1 to A. If the distance S1A = 10 m and the wavelength is 0.2 m. What is the path S1A in terms of the number of wavelengths?
S1 A10m
50λ
Path difference
• What is the path difference Δ of S1 and S2 at point A in terms of the number of wavelengths? The wavelength is 0.2 m.
S1 A
S2
10m
12m
Δ= |S1A – S2A|
Δ=10λ
Path difference and interference
The two coherent sources S1 and S2
are
Path difference at A = m. λ
where m = 0, 1, ...
Path difference at A= (m + ). λ where m = 0, 1, ...
in phase constructive interference at A
destructive interference at A
in anti-phase destructive interference at A
constructive interference at A
2
1
Determine the kind of interference
• Are the two sources coherent?• Check the phase difference between the two
sources.• Find the path difference at the point.• Determine the kind of interference at the point.
S1 A
S2
10m
12m
λ= 0.2m.S1 and S2 are coherent sources which are in phase.
Examples
• Coherent sources S1 and S2 are in phase.
• MN is the perpendicular bisector of S1S2.
• If there are constructive interference at points A, B and C, find their respective path difference in terms of λ.
S1
S2
M NA
B
C
Examples
• Coherent sources S1 and S2 are in phase.
• MN is the perpendicular bisector of S1S2.
• If there are destructive interference at points X and Y, find their respective path difference in terms of λ.
S1
S2
M NX
Y
Example 10
• Points with maximum sound are points with constructive interference.
• Points with minimum sound are points with destructive interference.
Energy and power of a mechanical wave
• Example: a transverse wave along a string with frequency and amplitude a.
Energy and power of a mechanical wave
• Example: a transverse wave along a string with frequency and amplitude a.
• Each particle on the string is performing simple harmonic motion.
• The energy of each particle is 22
2
1amUo
m
Energy and power of a mechanical wave
• The energy of each particle is
• If the mass per unit length of the string is ,
each wavelength of the string will carry energy
22
2
1amUo
m
22
2
1aE
Energy and power of a mechanical wave
• If the mass per unit length of the string is ,
each wavelength of the string will carry energy
• This is the amount of energy transferred in one period of time T.
m
22
2
1aE
Energy and power of a mechanical wave
• If the mass per unit length of the string is ,
each wavelength of the string will carry energy
• Power of the wave is P =
where c is the wave speed.
m
22
2
1aE
22
2
1ac
T
E
Energy and power of a mechanical wave
• If the mass per unit length of the string is ,
each wavelength of the string will carry energy
• Power of the wave is P =
where c is the wave speed.
m
22
2
1aE
22
2
1ac
T
E
Intensity
• Definition
Intensity I is defined as the power received per unit area.
• I a2.Power
AreaArea
PowerI
Intensity of spherical waves
•The wave spreads out from a point source.•The wave energy is distributed over a spherical area.
Point source
Intensity of spherical waves
• From a point source, the wave spreads out.• The wave energy is distributed over a spherical area.
Area = 4r2r
Intensity of spherical waves
•Assume that there is not any loss of energy (attenuation).•If the power of the source is Po, I =
Area = 4r2r
24 r
Po
I 2
1
rand
ra
1
Cylindrical waves• The wave spreads out from a line source.
• The wave energy is distributed on a cylindrical surface.
line source
Cylindrical waves• The wave spreads out from a line source.
• The wave energy is distributed on a cylindrical surface.
line source
Area A = 2πrhh
r
Cylindrical waves• Assume that there is not any loss of energy (attenuation).• If the power of the source is Po, I =
rh
Po2
line source
Area A = 2πrhh
r
rI
1 and
ra
1
Reflection andPhase Change
• Transverse wave
• Fixed at one end: phase change .
• Incident crest changes into trough on reflection.
• Incident trough changes into crest on reflection.
http://www.physics.nwu.edu/ugrad/vpl/waves/wavereflection.html
Reflection andPhase Change
• Transverse wave
• Free at one end : no phase change
• Incident crest is still a crest on reflection.
• Incident trough is still a trough on reflection.
http://www.physics.nwu.edu/ugrad/vpl/waves/wavereflection.html
Reflection andPhase Change
• Longitudinal wave
• Fixed at one end : phase change.
• Incident compression is still a compression on reflection.
• Incident rarefaction is still a rarefaction on reflection.
Reflection andPhase Change
• Longitudinal wave
• Fixed at one end : phase change. • Note that the displacement in the compressio
n of the incident pulse is to the right side (+ve) and the displacement in the compression of the reflected pulse is to the left side (-ve). There is a phase change
C C
incident pulse reflected pulse
Reflection andPhase Change
• Longitudinal wave
• Free at one end : no phase change
• Incident compression becomes a rarefaction on reflection.
• Incident rarefaction becomes a compression on reflection.
Stationary wave
• Principle of superposition is applied : wave meets wave.
• Energy is localized.
• The waveform is confined within some boundaries.
• The progressive wave still moves at speed c.
Stationary wave in a string
• Transverse wave.
• Incident wave and reflected wave meet.
• At some frequencies , nodes and antinodes are formed at some points of the string.
http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html
Stationary wave in a string
• Fundamental: the lowest frequency with a node at either side and an antinode in the middle.
• Overtones: some higher frequencies with node at either side and some antinodes between the ends.
http://members.nbci.com/_XMCM/surendranath/Harmonics/Harmonics.html
Frequency of stationary waves in a string
• When a stationary wave is formed, the length of the string must be a whole number of half-wavelength .
2
2
n where n = 1, 2, …
Frequency of stationary waves in a string
2
n where n = 1, 2, …
and
From
show that the possible frequencies Tn
fn 2
T
c where T is the tension and is the mass per unit length of the string
Note of a vibrating string
• The air is also set into vibration by the vibrating string.
• The frequency of air is the same as that of the vibrating string.
• If the frequency is within audible frequency range, a note may be heard.
Properties of stationary wave
• Waves are travelling in both directions but energy is confined in the boundaries.
• All parts of the string (except at the nodes) are vibrating with the same frequency but with different amplitudes.
http://members.nbci.com/_XMCM/surendranath/Harmonics/Harmonics.html
Properties of stationary wave
• Within one loop, all points are moving in phase.
• Points between adjacent loops are in anti-phase.
http://members.nbci.com/_XMCM/surendranath/Harmonics/Harmonics.html
Properties of stationary wave
• If the frequency of the stationary wave is equal to the natural frequency of the string, resonance occurs.
• Distance between adjacent anti-nodes is
http://members.nbci.com/_XMCM/surendranath/Harmonics/Harmonics.html
2
Measuring the wavelength of microwave
• A stationary wave is formed between the transmitter and the reflector.
• The probe shows a large current at positions of nodes and small current at positions of antinodes.
to microammeterto power supply
reflector
probe transmitter
Measuring the wavelength of microwave
to microammeterto power supply
reflector
probe transmitter
Move the probe to detect the positions of the nodesCount the number of nodes = n.Measure the distance moved = d 2
.
nd
N N N NA A A A
Formation of stationary waves
• It is a result of superposition of two waves with the same frequency and amplitude and are travelling in the same medium in opposite directions.
http://www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveSuperposition/waveSuperposition.html
Standing Longitudinal Wave
• http://www.fed.cuhk.edu.hk/sci_lab/Simulations/phe/stlwaves.htm
Beats
• Two waves with slightly different frequencies propagating in the same direction overlap with each other.
• The resultant amplitude varies periodically.
http://webphysics.ph.msstate.edu/jc/library/15-11/index.html
Beats• Two loudspeakers emit sound waves of slightly
different frequencies.
• The microphone receives both sound.
• The beat is seen on the CRO.
to CROto signal generator 1 to signal generator 2
loudspeaker 1 loudspeaker 2microphone
Formation of beats
• Two waves with frequency f1 and f2 respectively with f1 slightly higher than f2.
• Take them as rotating vectors with angular velocity 1 and 2 with 1 > 2.
t = 0 =2f
Formation of beats
• At t = 0, the two vectors coincide.
time = 0 =2f
Formation of beats
• At t = 0, the two vectors coincide.
• Vector 1 rotates faster and leads vector 2.
=2ftime = 0
2
1time = t
Formation of beats• Find the time Tb when vector 1 catches up
with vector 2. (Hint: vector 1 runs one cycle more than vector 2.)
=2ftime = 0
At time = Tb , they coincideagain.
Formation of beats
=2ftime = 0
At time = Tb , they coincideagain.
1Tb - 2Tb = 221
1
ffTb
Tb is the period of the beat.
Formation of beats
=2ftime = 0
At time = Tb , they coincideagain.
Tb is the period of the beat.
The beat frequency fb = f1 – f2
Frequency of the resultant wave
• The resultant wave frequency is
221 ff
f
Example 11
• Checking frequency.
• e.g. fine tuning a piano.
Analytical treatment of beats
• The equations of the two waves reaching the same position:
y1 = a.sin(2f1t) and
y2 = a.sin(2f2t)
• The equation of the resultant wave is
y = y1 + y2
).2
.2sin()..2
.2cos(.2 2112 tff
tff
a
Analytical treatment of beats
).2
.2sin()..2
.2cos(.2 2112 tff
tff
ay
A varying amplitude frequency ofthe resultantwave
Analytical treatment of beats
).2
.2sin()..2
.2cos(.2 2112 tff
tff
ay
A varying amplitude
The amplitude varies at frequency:
|2
| 12 fff A
Analytical treatment of beats
).2
.2sin()..2
.2cos(.2 2112 tff
tff
ay
A varying amplitude
The amplitude varies at frequency:
|2
| 12 fff A
As our ear respondsto intensity a2,the number of maximum heard per second= 2.fA
|| 12 ff
Polarization
• A wave is unpolarized if the vibration is not confined to any direction.
• An unpolarized wave can be made to be plane-polarized by a process called polarization.
• Only transverse wave can be polarized.
Polarizer and analyzer
unpolarizedwave
polarizer
polarizedwave
•An unpolarized transverse wave passes through a polarizer and becomes polarized.•The plane of vibration of the wave is parallel to the slit of the polarizer.•The intensity of the polarized wave is half that of the unpolarized wave.
Polarizer and analyzer
• Use an analyzer to change the polarization of the incident wave.
incidentwave
emergingwave
analyzer
Polarizer and analyzer• Use an analyzer to change the polarization of the
incident wave.• If the intensity of the emerging wave is unchanged,
the plane of polarization of the wave is parallel to the analyzer.
analyzer
unpolarizedwave
polarizer
incident wave is polarized
emerging wave
Polarizer and analyzer• Use an analyzer to change the polarization of the
incident wave.• If the intensity of the emerging wave is zero, the
plane of polarization of the wave is perpendicular to the analyzer.
unpolarizedwave
polarizer
incident wave is polarized
analyzer
no emerging wave
Polarizer and analyzer• Use an analyzer to change the polarization of the
incident wave.• If the intensity of the emerging wave is reduced,
the plane of polarization of the wave neither parallel nor perpendicular to the analyzer.
unpolarizedwave
polarizer
incident wave is polarized
analyzer
emerging wave with lessintensity
Polarizer and analyzer• If a is the amplitude of the polarized wave emergi
ng from the 1st slit and the 2nd slit makes an angle to the 1st slit,
the amplitude a’ of the wave emerging from the 2n
d slit is a’ = a.cos
unpolarizedwave
polarizer
incident wave is polarized
analyzer
emerging wave with lessintensity
a a.cos
Polarizer and analyzer
• The amplitude a’ of the wave emerging from the 2nd slit is a’ = a.cos
a
directionof 2nd slit
a’
Polarizer and analyzer• The amplitude a’ of the wave emerging from the 2
nd slit is a’ = a.cos • The intensity of the emerging wave = Io.cos2 wh
ere Io is the intensity of wave from the 1st slit.
unpolarizedwave
polarizer
incident wave is polarized
analyzer
emerging wave with lessintensity
Io Io.cos2
Polaroid filters
• Light can be polarized by using a polaroid filter.
• Use two polaroid filters to analyze the polarization of light.