Physics Notes Chapter 12 - Waves

CHAPTER TWELVE: WAVES

Waves & Vibrations

WAVE is any oscillation where energy is transferred in the direction of the wave travel without the large scale/long distance permanent displacement of matter.

MECHANICAL WAVES

When waves progress through a substance, the particles of the substance vibrate in a certain way which makes nearby particles vibrate in the same way, and so on.

e.g. sound waves, seismic waves, waves on a string, water waves

ELECTROMAGNETIC WAVES

EM waves are vibrating electric and magnetic fields that progress through space without the need for a substance.

e.g. Infrared Radiation, Heat, Light, Ultraviolet Radiation, X-Rays

TRANSVERSE

Waves in which the direction of vibration is perpendicular to the direction in which the wave travels.

e.g. waves on a string, EM waves, secondary seismic waves

LONGITUDINAL

Waves in which the direction of vibration of the particles is parallel to (along) the direction in which the wave travels. You get a series of compressions and rarefactions.

e.g. sound waves, primary seismic waves

TRAVELLING (PROGRESSIVE) WAVES

Waves which travel through a substance (or through space if electromagnetic). Energy is transferred along the wave.

STANDING (STATIONARY) WAVES

Wave pattern with nodes and antinodes formed when two or more progressive waves of the same frequency and amplitude pass through each other. Energy is not transferred along the wave.

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Transverse Wave: Displacement-Distance Graph

Key Terms: Displacement: distance and direction from the equilibrium point Amplitude: maximum displacement from equilibrium Wavelength: distance between 2 adjacent particles which have the same phase Frequency: number of oscillations per second by the source / number of waves passing a point per

second. Unit = Hertz (Hz) Period: time for one complete oscillation

Period(t )= 1Frequency

You can only show the period on a Displacement-Time graph (as shown on the left)

Speed:The higher the frequency of the wave, the shorter its wavelength. For waves of frequency (f) and

wavelength (λ)Wave Speed (c )=Frequency ( f ) xWavelength¿λ)

c=fλ

Note: The symbol (c) is used as a general symbol for the speed of waves. In the data booklet, (c) refers to the speed of electromagnetic waves in free space (speed of light).

Phase Difference:PHASE DIFFERENCE between two vibrating particles is the fraction of a cycle between the

vibrations of the two particles.

Phase Difference is measured either in degrees or radians, where

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Amplitude

Amplitude

Wavelength

λ

One Complete Wave

Peak

Trough

Displacement

Distance along wave

1 cycle = 360° = 2 radiansπFor two points at distance (d) apart along a wave of wavelength ¿ )λ

Phase Difference∈radians=2 π dλ

The diagram below shows three successive snapshots of the particles of a transverse wave progressing from left to right across

the diagram. Particles O, P, Q, R and S are spaced 14

of a wavelength apart.

(The Table shows the phase difference between O and each of the other particles)

Describing Waves with Graphs

To study the motion of the vibrating particles on a wave, we draw graphs. Two types of graphs are commonly used:

- Displacement-Distance graphs - Displacement-Time graphs

Displacement-Distance graphs:A displacement-distance graph is also called a displacement-position graph. It shows the

displacement of the particles at various positions at a certain time. Although it looks like a photograph of a transverse wave, it can be used to describe BOTH a

transverse and a longitudinal wave (Figures a. and b.)

3Fig b. Displacement-distance graph of

a longitudinal wave at t=0.

P Q R SDistance from O in terms of wavelength ( )λ

14λ

12λ

34λ λ

Phase Difference relative to O (radians)

12π π

32π 2π

From a displacement-distance graph, we can directly read the following information: - Amplitude - Wavelength - Locations of crests and troughs (for a transverse wave), or compressions and

rarefactions (for a longitudinal wave) The displacement-distance graph is especially useful to study a longitudinal wave. For example, look at the displacement of compression: it is zero unlike the crest. This is easy to notice on the graphs.

Fig c. Time series of Displacement-Distance graphs of a wave.

Using a series of displacement-distance graphs at various times, we can see the motion of the wave (Figure c.)

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By comparing the changes in these graphs, we can deduce the travelling speed and direction of the wave, as well as the time-varying directions of the motion of the vibrating particles. Displacement-Time graphs:

Unlike a displacement-position graph, a displacement-time graph describes the displacement of ONE particle at various times at a certain position.

Figure d. shows how the displacements of particles P, Q, and R in Figure c. vary with time. Each particle has its own displacement-time graph.

Fig d. Displacement-time graphs for particles at different positions.

On the contrary, using a number of displacement-time graphs at various positions, we can construct back the displacement-distance graph of the wave at a certain time.

From a displacement-time graph, we can directly read the following information: - Amplitude - Period (and hence, the frequency) - Direction of motion of the particle at various times

If we have a snapshot of the wave too, we can deduce the motion of the wave (i.e. its travelling speed and direction)

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Polarisation

POLARISATION is the restriction of transverse waves using a filter (mechanical/ electronic) so that they have different amplitudes in different planes.

Unpolarised: No restrictions on vibrations (normal behaviour)Plane (Linear) Polarised: Vibrations limited to one plane. (The waves are vertically plane polarised)

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Period (T)

Amplitude (a)

P

Q

Wavelength ( )λ

Unpolarised Polarised

Plane of Polarisation: The plane in which the medium oscillates and the energy is propagated. (For Electromagnetic waves it is defined as the plane in which the electric field oscillates)

Polarisation can be used to distinguish between longitudinal and transverse waves since only transverse waves show polarisation effects. (Longitudinal waves propagate parallel to the direction of motion so if polarised, you will effectively block the wave).

Note: If polarised wave is passed through another filter at 90° to the plane of polarisation, the wave is stopped

Polarisation with LightMost light sources are unpolarised light. This means that the light waves oscillate

equally in all planes perpendicular to the direction of propagation. Unpolarised light can be made polarised by passing it through a polarised filter.

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A Polaroid filter only lets one plane of oscillation through and absorbs the others so that the light emerging from it is plane polarised.

Analysing the Polarisation of LightThe light to be analysed is passed through a Polaroid sheet (Polariser) and the transmitted light is viewed through another Polaroid sheet (Analyser).

The analyser is rotated (in the plane perpendicular to the lights velocity) while viewing the transmitted light:

AXES PARALLELThe light from the Polaroid is plane polarised, so the transmitted intensity will be a maximum when the analyser is parallel to the plane of polarisation.

AXES PERPENDICULAR Intensity will be zero when analyser is at right angles to polariser.

In a 360° rotation of the analyser the intensity will vary continuously and have TWO MAXIMA and TWO ZEROS.

If unpolarised light is used and the Polaroid rotated 360° there will be no change in intensity. There will always be a plane of oscillation parallel to the Polaroid’s transmission direction.

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1. parallel polaroid filters 2. crossed polaroid filters 3. rotation by intermediate filter

1. parallel polaroid filters 2. crossed polaroid filters 3. rotation by intermediate filter

Detecting PolarisationThe orientation of the detector must match the plane of

oscillation of the wave. If the detector is perpendicular to the plane of polarisation then no reading will be given. Waves used during TV transmission are plane polarised so the aerial must have the correct orientation to achieve optimum direction.

Example: The diagram below shows an arrangement used to investigate the properties of microwaves. When the transmitter T was rotated through 90° about the straight line XY, the receiver signal decreased to zero. Explain why this happened and state the property of microwaves responsible for this effect.

- Microwaves from the transmitter are polarised (or vibrate in a certain plane or direction)- Rotating the transmitter rotates the plane of polarisation of the microwaves- Received signal becomes zero when receiver is perpendicular to the plane of polarisation of the

microwaves

Applications Polaroid Sunglasses: Only let light polarised in one direction and absorb the rest.

Intensity is reduced. Also, reflected light and sunlight scattered from particles in the atmosphere is partially polarised. If the Polaroid in the sunglasses is in the correct orientation it can absorb the reflected light and greatly reduce glare.

Stress Patterns : Certain materials (especially polymers) rotate the plane of polarised light as it passes through them. The amount of rotation depends upon the stress the material is under. If the light is viewed through crossed Polaroids, coloured fringes are seen that are most concentrated at areas of greatest stress.

Wave PropertiesWavefronts

WAVEFRONTS are lines of constant phase (e.g. wavecrests) that are at right angles to the direction of wave travel/energy propagation.

Source 2D 3D

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Reflection- No transmission- No absorption Reflection

Angle of Incidence = Angle of Reflection

RefractionREFRACTION is the change of direction of a wave when it crosses a boundary where its

speed changes. This change in velocity means that the wavelength of the wave must change, since frequency is constant and v=fλ

DiffractionDIFFRACTION is the spreading of waves on passing through a gap or near an edge.

Every point on the wave acts as an independent emitter.

Satellite TV dishes in Europe need to point South, since the satellites orbit the Earth directly above the equator. The bigger the dish, the stronger the signal it can receive, because more radio waves are reflected by the dish onto the aerial. But a bigger dish reflects the radio waves to a smaller focus, because it diffracts the waves less. The dish therefore needs to be aligned more carefully than a smaller dish, otherwise it will not focus the radio waves onto the aerial.

Interference

THE PRINCIPLE OF SUPERPOSITION states that when two waves meet, the total displacement at a point is equal to the sum of the individual displacements at that point.

Crest + Crest SupercrestCrest + Trough Zero (They superpose to produce maximum negative

displacement – not a minimum)Trough + Trough Supertrough

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Interference is a characteristic of waves. It occurs when two or more waves arrive at the same point at the same time. The signals superpose to give a combined displacement which can be bigger or smaller than the individual displacements depending upon their relative phases.

The difference is best observed when the sources S1 & S2 are COHERENT. So the resultant displacement at point P is the sum of the separate displacements due to the two waves S1 & S2 at P.

COHERENT waves are waves with a constant phase difference. (They don’t have to be in phase for this to be true). They will have the same frequency and wavelength. Constructive Interference: occurs when waves arrive at a point in phase with each other, producing a maximum disturbance in the medium.

Destructive Interference: occurs when two waves arrive at a point exactly out of phase with each other, producing minimum disturbance in the medium. (The waves are out of phase by half a wavelength, 180° or radians)π

Superposition : will occur whether waves are coherent or not. (However, if the waves are coherent, they will interfere to produce a fixed pattern – as above)The same rules apply - the resultant displacement at any point is always the sum of the separate displacements of the wave at that point:

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Examples:1) What is the phase difference between A and B (shown below)?

-Zero Phase Difference- 2 phase difference or π- 4 phase difference π etc.

2) Sketch the Superposition Pattern of A and B.

3) What is the phase difference between C and D (shown below)?

One oscillation is equivalent to 360°. Therefore:

14wavelength=1

4x 360 °=90 °

34wavelength=3

4x 360 °=270 °

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5) Sketch a diagram showing two waves of equal amplitude with a phase difference

equal to 13

of a wavelength.

The diagram below shows three particles in a medium that is transmitting a sound wave. Particles A and C are separated by one wavelength and particle B is halfway between them when no sound is being transmitted.

(a) Name the type of wave that is involved in the transmission of this sound. (1)Longitudinal Wave

(b) At one instant particle A is displaced to the point A’ indicated by the tip of the arrow on the diagram. Show on the diagram the displacements of particles B and C at the same instant. Label the position B’ and C’, respectively. (2)

Distance AC is one wavelength meaning that A and C move in phase. Distance AB is half a wavelength, consequently B moves 180° out of phase with A and C. If there is no absorption (or no spreading out of the wave) the displacements will have the same magnitude. If there is absorption (or spreading out) these magnitudes will decrease along the wave.

(c) Explain briefly how energy is transmitted along the wave. (2)Particles in the medium are made to vibrate longitudinally, these cause nearby particles to vibrate in the same direction.

Stationary Waves

STATIONARY WAVES are a wave pattern with nodes and antinodes formed when two or more progressive waves of the same frequency and amplitude pass through each other.

They can be formed by: - 1 source being reflected - 2 sources in opposite directions (with the same frequency)

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B’ C’

NODES – positions of zero amplitude (no vibration)ANTINODES – positions of maximum amplitudeThe wave profile of a stationary wave does not move. In a stationary wave there is no transfer of energy along the wave.

Particles within each loop are vibrating in phase. Particles in adjacent loops are π or 180° out of phase (There are only 2 phases).

Any two points have phase difference:Phase Difference=nπ

(n) is the number of nodes between the two points

For a stationary wave each point has a different amplitude. Wavelength (¿ frequency )of standingwave=Wavelength (¿ frequency )of incident (¿ reflected )waves

Distance between adjacent antinodes (¿nodes )= λ2

( 12of a wavelength)

Distancebetweennode∧antinode= λ4

(14of awav elength¿

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Explanation:Consider a snapshot of two progressive waves passing through each other:-When they are in phase, they reinforce each other to produce a large wave- A quarter of a cycle later, the two waves have each moved one-quarter of a wavelength in opposite directions. They are now in anti-phase so they cancel each other out.- After a further quarter cycle, the two waves are back in phase. The resultant is again a large wave as at the beginning, except reversed.

Stationary Wave Cycle:

At each antinode, the particles move from

maximum amplitude, to zero, to maximum amplitude the other way, back to zero, then to first displacement.

All points between two nodes move in phase with each other. They all reach maximum displacement together.

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t = 0

t = T4

t = T2

t = 3T4

t = T

Sometimes stationary waves are shown as being incomplete. (i.e. there are no complete nodes) This is because as the original wave is reflected off a surface, it loses some of its energy and so doesn’t not have exactly the same wavelength and frequency of the original wave. So when the two waves deconstructively interfere they have an amplitude of non-zero because the waves are not exactly the same and so cannot form a perfect node.

t = 0

t = T4

t = T2

t = 3T4

t = T

This is an example of a stationary wave cycle with 4 nodes.

The pattern at the lowest possible frequency has a node at each end, with maximum displacement at the mid-point (antinode).

Wave profile of the first stationary wave occurs at the FUNDAMENTAL FREQUENCY (f0)(1st Harmonic) Because the length (L) of the vibrating section of the string is between adjacent

nodes, and the distance between adjacent nodes is 12π the wavelength of the waves that form

this pattern, the FUNDAMENTAL WAVELENGTH λ0 = 2L

Therefore, the FUNDAMENTAL FREQUENCY f0 = cλ0

= c2L

c = speed of progressive waves on the wire

At higher frequencies, other patterns appear, these are referred to as OVERTONES.

First Overtone(2nd Harmonic)

f = 2 x f0

2c2L

=

cL

Second Overtone(3rd Harmonic)

f = 3 x f03 c2L

Third Overtone(4th Harmonic)

f = 4 x f0

4c2L

=

2cL

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Fourth Overtone(5th Harmonic)

f = 5 x f05 c2L

Standing waves make it easy to determine wavelength (twice the distance between adjacent nodes)

Examples:

An elastic string of length 2m is clamped at both ends. Near one end a vibration generator is loosely attached. The vibrator oscillates with fixed amplitude but with variable frequency. A graph of maximum amplitude along the string against frequency of vibration is shown.

1) Sketch the instantaneous appearance of the string at 15Hz.(15Hz is the lowest frequency at which a standing wave can be produced – must be fundamental frequency)

2) Sketch the instantaneous appearance of the string at 30Hz.(30Hz will produce a wave of half the length of the 15Hz string)

3) Sketch the instantaneous appearance of the string at 45Hz. (45Hz will produce a wave of one-third of the length of the 15Hz string)

4) What is the wavelength of the waves on the string at these frequencies?At 15Hz = 2 x length of string = 2 x 2m = 4mλAt 30Hz = length of string = 2mλ

At 45Hz = λ23

x length of string = 23

x 2 = 1.33m

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5) What is the speed of travel of the waves along the string?v = fλv = 15Hz x 4m = 60ms-1

Examples of Stationary Waves

Sound in a pipe: Sound resonates at certain frequencies in an air-filled tube or pipe. Longitudinal stationary waves are established at certain frequencies only, due to interference between the incident and reflected progressive waves.

There will be an ANTI-NODE, at the open end, NODE at the closed end.

ANTI-NODE at each end (wave reflected by free air at end of tube)

Using Microwaves: Microwaves from a transmitted are directed normally at a metal plate, which reflects the microwaves back towards the transmitter. When a detector is

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moved along the line between the transmitted and the metal plate, the detector signal is found to be zero at equally spaced positions along the line.

The reflected waves and the waves from the transmitter form a stationary wave pattern. The positions where no signal is detected are where nodes occur. They are spaced at intervals of half a wavelength.

Example:

Standing (Stationary) Waves Progressive WavesSimilarities:

Both caused by vibrationBoth have maxima and minima

Both use v=fλDifferences:Energy not transferred along the wave Energy is transferred along the waveParticles have different amplitudes (Varies from zero at the nodes to a maximum at the antinodes)

Particles have same amplitude

Adjacent particles are in phase Adjacent particles are not in phaseAll particles (except at the nodes) vibrate at the same frequency

All particles vibrate at the same frequency

Phase Difference between two particles is equal to nπ (where n is the number of nodes between the two particles)

Phase Difference between two particles is equal

to 2π dλ

(where d=distance apart and is the λ

wavelength)

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Making MusicA guitar produces sound when its strings vibrate as a result of being plucked. When a

stretched string or wire vibrates, its pattern of vibration is a mix of its fundamental mode of vibration and the overtones.

The sound produced is the same mix of frequencies which change with time as the pattern of vibration changes. A spectrum analyser can be used to show how the intensity of a sound varies with frequency and with time. Combined with a sound synthesiser, the original sound can be altered by amplifying or suppressing different frequency ranges.The pitch of a note corresponds to frequency, so the pitch of a note from a stretched string can be altered by changing the tension of the string or by altering its length.

You can increase the pitch by:- Shortening the length- Raising the tension

You can lower the pitch by:- Increasing the length- Lowering the tension

By changing the length or altering the tension, a vibrating string or wire can be tuned to the same pitch as a vibrating tuning fork. However, the sound from a vibrating string includes the overtone frequencies as well as the fundamental frequency, whereas a tuning fork vibrates only as its fundamental frequency. The wire is then tuned when its fundamental frequency is the same as the tuning fork frequency.

A simple visual check is to balance a small piece of paper on the wire at its centre and place the base of the vibrating tuning fork on one end of the wire. If the wire is tuned to the tuning fork, it will vibrate in its fundamental mode and the piece of paper will fall off.

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