physics beyond 2000 chapter 9 wave phenomena. glossary wave: it is a periodic disturbance in a...

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.

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 =

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

• There is a phase difference between any two particles.

• In phase (or phase difference = 0)

• 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):

c = speed of propagation

Notation in the textbooky-x graph for a transverse wave:

C: crestT: troughdirection of

propagation

• 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.

Equilibriumposition

Notation in the textbooky-x graph for a longitudinal wave:

C:compressionR:rarefaction

• 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

Wave Speed and Speed of Particle

• Wave speed c =

• Speed of a particle vy =

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

direction of propagation

in radian

Phase difference between two points in a y-x graphy

Which point leads the other?

Phase difference between two points in a y-x graph

Compare points P and R.Which point leads the other?

Draw the new wave pattern after a time t.The whole wave pattern moves to the right.

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.

After , P moves to the equilibrium position and R moves up to the position of the crest.So P leads R by

Phase difference between two waves in a y-x graphy

Two waves y1 and y2 of the same frequency are moving to the right simultaneously.What is the phase difference between thesetwo waves?

Measure the separation x of their crests ( choose x < /2 ).

The phase difference is

Choose a point P in front of the waves.Which wave has its crest reach the point first?

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).

• The centres of rarefaction have the lowest density and lowest pressure (for gas).

• The crest of the density/pressure leads that of displacement by /2.

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.

• For a particle performing simple harmonic motion,

y = a.sin(t + o)

• It is a sinusoidal wave.

• Other particles along the path are performing SHM at the same frequency but with a different phase o

• Two points P and Q are in the path of a wave.

• Wave P becomes crest earlier than wave Q . ( tP < tQ).

Wave P leads wave Q.

P QtP tQ

• Time difference t = tQ – tP

• Phase difference

P QtP tQ

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.

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.

to signal generator 1

to signal generator 2

y-x and y-t graphs• A typical example:

At t = 0,

At point P,

http://www.geocities.com/yklo00/Wave_yx-yt-graphs.htm

Wave speed of a mechanical wave

• c = mediumtheofpropertyinertial

mediumtheofpropertyelastic

• c =

• Longitudinal wave in a solid

mediumtheofpropertyinertial

c where E is the Young modulusof the solid and is the densityof the solid.

• Example 6

• c =

• Transverse wave in a string

c where T is the tension in the string and is the mass per unit length of the string.

• Example 7

• c =

• Sound wave in air

c where P is the air pressure, is the density of air and is a constant.

Sound wave in air

c where P is the air pressure, is the density of air and is a constant.

By gas law PV = nRT

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.

CRO Signal Generator

LoudspeakerMicrophone

Measure speed of sound

• Place the microphone at a position such that both inputs are in phase.

CRO Signal Generator

• Move the microphone towards the loudspeaker to locate the next position with both inputs in phase again.

CROSignal Generator

Rulerx

Measure speed of sound• The distance x = wavelength • For accurate measurement, we usually

move the microphone more than one wavelength.

CROSignal Generator

Rulerx

• Find the frequency f from the signal generator or the period T from the time base of the CRO.

CROSignal Generator

Rulerx

• c = f. or c =

CROSignal Generator

Rulerx

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.

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:

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:

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

The station sends a pulse.

Radar and sonar

The station receives the pulse.

Radar and sonar

If the time difference is t, findthe distance d.

Radar and sonar

• Sonar: SOund Navigation And Ranging

seabed

Radar and sonar

seabed

Send a pulse of ultrasound

Radar and sonar

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

medium 1(wave with high speedand long wavelength)

medium 2(wave with low speedand short wavelength)

Normal

Snell’s law

medium 1(with high speed)

medium 2(with low speed)

NormalY

Y’ X

Snell’s law

NormalY

Y’ 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

NormalY

Y’ 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

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.

Δ= |S1A – S2A|

Δ=10λ

Path difference and interference

The two coherent sources S1 and S2

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

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.

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.

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.

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.

• Each particle on the string is performing simple harmonic motion.

• The energy of each particle is 22

• The energy of each particle is

• 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.

• Power of the wave is P =

where c is the wave speed.

• Power of the wave is P =

where c is the wave speed.

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

• From a point source, the wave spreads out.• The wave energy is distributed over a spherical area.

Area = 4r2r

•Assume that there is not any loss of energy (attenuation).•If the power of the source is Po, I =

Area = 4r2r

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

Cylindrical waves• Assume that there is not any loss of energy (attenuation).• If the power of the source is Po, I =

line source

Area A = 2πrhh

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

• 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

• Longitudinal wave

• Fixed at one end : phase change.

• Incident compression is still a compression on reflection.

• Incident rarefaction is still a rarefaction on reflection.

• 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

incident pulse reflected pulse

• 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 .

n where n = 1, 2, …

Frequency of stationary waves in a string

n where n = 1, 2, …

show that the possible frequencies Tn

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.

• Within one loop, all points are moving in phase.

• Points between adjacent loops are in anti-phase.

• If the frequency of the stationary wave is equal to the natural frequency of the string, resonance occurs.

• Distance between adjacent anti-nodes is

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

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

• 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

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

1Tb - 2Tb = 221

Tb is the period of the beat.

Formation of beats

=2ftime = 0

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

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

.2sin()..2

.2cos(.2 2112 tff

.2sin()..2

.2cos(.2 2112 tff

A varying amplitude frequency ofthe resultantwave

.2sin()..2

.2cos(.2 2112 tff

A varying amplitude

The amplitude varies at frequency:

| 12 fff A

.2sin()..2

.2cos(.2 2112 tff

A varying amplitude

The amplitude varies at frequency:

| 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.

• 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

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

analyzer

no emerging wave

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

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

analyzer

a a.cos

• The amplitude a’ of the wave emerging from the 2nd slit is a’ = a.cos

directionof 2nd slit

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

analyzer

Io Io.cos2

Polaroid filters

• Light can be polarized by using a polaroid filter.

• Use two polaroid filters to analyze the polarization of light.

physics beyond 2000 chapter 9 wave phenomena. glossary wave: it is a periodic disturbance in a...

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