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TAMPINES JUNIOR COLLEGE JC1 PHYSICS LECTURE 2009

Content 1. What is a wave? 2. Types of waves 3. Description of waves 4. Graphical representations of transverse and longitudinal waves 5. General wave equation 6. Wave speed equation v = f λ 7. Intensity of a wave 8. Polarisation 9. Determination of frequency and wavelength of sound Learning Outcomes Candidates should be able to: (a) show an understanding and use the terms displacement, amplitude, phase

difference, period, frequency, wavelength and speed. (b) deduce, from the definitions of speed, frequency and wavelength, the

equation v = f λ. (c) recall and use the equation v = f λ. (d) show an understanding that energy is transferred due to a progressive

wave. (e) recall and use the relationship, intensity (amplitude)2. (f) analyse and interpret graphical representations of transverse and

longitudinal waves. (g) show an understanding that polarisation is a phenomenon associated with

transverse waves. (h) determine the frequency of sound using a calibrated c.r.o. (i) determine the wavelength of sound using stationary waves.

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1. What is a wave?

A wave involves a disturbance from an equilibrium position and the disturbance travels from one region of space to another. A wave is said to be propagated because the pattern of the disturbance occurring at one place at one time is repeated at a later time at a different place.

The source of any wave is a vibration or an oscillation. For example, you

can produce waves on a rope by fixing one end to a wall and moving the other end up and down. These up and down movements make up the vibrations or oscillations.

Einstein and Infield made these remarks about wave phenomena: A bit of gossip starting in Washington reaches New York very quickly, even though not a single individual who takes part in spreading it travels between these two cities. There are two quite different motions involved, that of the rumor, Washington to New York, and that of the persons who spread the rumor. The wind, passing over a field of grain, sets up a wave which spreads out across the whole field. Here again we must distinguish between the motion of the wave and the motion of the separate plants, which only undergoes small oscillations.... The particles constituting the medium perform only small vibration, but the whole motion is that of a progressive wave. The essentially new thing here is that for the first time we consider the motion of something which is not matter, but energy propagated through matter.1

1 A. Einstein and L. Infeld, The Evolution of Physics, New York, Simon and Schuster, 1961.

down

up

Wave doesn't transfer matter, but transfers energy and momentum

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Electric field

Magnetic field

2. Types of Waves

In order for waves to propagate, there must be a material substance called a medium. The medium provides some physical connection or mechanism through which a disturbance at one point can influence adjacent points, thus allowing wave motion.

Waves that require a medium to travel are known as mechanical waves.

An exception to mechanical waves is the electromagnetic waves, which do not require a medium to propagate and can travel through vacuum.

Electromagnetic (EM) waves

The EM spectrum, as shown on the left, covers a vast range of different waves, all of which travel with the same speed of 3 108 m s-1 in a vacuum. The extent of the EM spectrum covers radio waves, microwaves, infra-red, visible light, ultraviolet, X-rays, and gamma rays. The EM spectrum is continuous, i.e. no gaps of non-existent frequencies. There is also no sharp boundary between one type of radiation and the next. An EM wave consists of oscillating electric and magnetic fields which are perpendicular to one another as shown in the figure below. The 2 fields vary sinusoidally and are always in phase with one another.

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All EM waves may be distinguished by their wavelengths or frequencies, but the frequency is more fundamental because it does not change when the waves go from one medium to another. The frequency depends only on the source. ------------------------------------------------------------------------------------------------------------

Waves are classified as progressive (or traveling), as opposed to stationary (or standing), if their waveforms move (or progress).

All waves in its “natural” state are progressive.

Stationary waves are formed by the superposition of two waves.

Progressive waves can be classified as either traverse and longitudinal,

depending on the kind of vibration or oscillation that the underlying media is undergoing.

Transverse Wave

A transverse wave is propagated by vibrations that are perpendicular to the direction of wave travel. (E.g. rope waves, water waves, EM waves)

Crest: Maximum point on the transverse wave. Trough: Minimum point on the transverse wave.

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Longitudinal Waves

A longitudinal wave is propagated by vibrations that are parallel to the direction of wave travel. (E.g. sound waves, waves produced in a vertical oscillating spring under tension)

Compression: Regions of high or maximum pressure, where wave

particles of the medium are momentarily packed closer than usual.

Rarefaction: Regions of low or minimum pressure, where wave

particles of the medium are momentarily packed further apart than usual.

Transverse and Longitudinal Waves Applet http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=30

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Learning Checks 1. A progressive wave transfers _______ from its point of origin to other

points further away. a) molecules b) energy c) matter d) force

2. A transverse wave propagates along a string. The particles in the string

move a) perpendicular to the direction of propagation. b) parallel to the direction of propagation. c) depends on the initial disturbance d) not covered in the reading assignment

3. Which one of the following is an example of longitudinal waves? a) electromagnetic waves b) rope waves c) ocean waves d) sound waves produced by a string

4. Arrange the following in order of increasing frequency: yellow light, infra-red, ultraviolet, x-rays, - rays, microwaves, blue light.

Microwaves, infra-red, yellow light, blue light, ultraviolet, x-rays, - rays

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3. Description of Waves

Waves can be produced by many different sources, such as sound waves produced by different musical instruments. Thus waves can come in different waveforms or wave profiles.

The waveform or wave profile of the simplest wave, i.e. the disturbance

oscillates periodically with a fixed frequency and wavelength, takes the shape of a sinusoidal curve.

Although many waves in real life are not sinusoidal, it turns out that more

complex waveforms can always be constructed by mixing a series of sinusoidal waves in appropriate proportions.

In our study of waves, we shall limit ourselves to sinusoidal waves of the

form y = A sin θ.

displacement, y

distance/m

A

displacement, y

time/s

T

0 0

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The following terms are used to describe wave motion.

1. Displacement, y refers to the direction and distance the particles in a wave are displaced/moved from their equilibrium position.

2. Amplitude, A is the maximum displacement of a particle in a wave

from the equilibrium position. It is also the distance from its equilibrium position to the crest or trough of the wave motion. The total swing from a crest to a trough is twice the amplitude.

3. Wavelength, λ is the distance between successive crests or the

distance between successive troughs or distance between two successive points in phase on a wave. (Unit: m)

4. Wavefront is an imaginary line on a wave system joining a set of

points which are all in phase. To represent waves, usually a series of wavefronts separated by are shown.

5. Period, T is the time taken for a particle to complete one vibration

or oscillation. It is also the time taken for the wave to travel a distance of one wavelength. (Unit: s)

6. Frequency, f is the number of complete vibrations or cycles that pass through a given point per unit time. (Unit: s-1 or Hz)

.

7. Speed of a wave, v is the speed in which the crests of the wave move. It is also the speed at which energy is transmitted. This is different from the speed in which individual particles move. (Unit: m s-1)

8. Phase, refers to the stage that a wave particle has reached within

the complete cycle of a wave, expressed in terms of a fraction of the cycle.

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9. Phase difference, Δ between two waves of the same frequency or between 2 points in a wave can be thought of as a delay or advance in the start of one wave’s cycle with respect to another, expressed in terms of a fraction of the cycle or in terms of an angle (where 2π rad or 360o represents one cycle).

When the phase difference between two waves or 2 points of the

same wave is an integer multiple of 2π rad or 360o, including zero, the waves are said to be in phase. When the phase difference between two waves or 2 points of the same wave is an odd integer multiple of π rad or 180o, the waves are said to be anti-phase.

Example

Consider the displacement-distance graph, (i) What is the phase difference between P1 and P2? A /2 B 3/2

C D 2

(ii) What is the phase difference between P3 and P5? A /2 B 3/2

C D 2

(iii) Which 2 points are in phase? A P1 and P2 B P3 and P5

C P2 and P4 D P2 and P5

(iv) Which 2 points are anti-phase? A P1 and P2 B P3 and P5

C P2 and P4 D P2 and P5

distance / m

P2

y0

displacement / m

-y0

P1

P3

P5

P4

2 0

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Other than 2π rad or 360o, one wave cycle can also be represented by one wavelength λ or one period T, hence we can form the following relation:

T

tx

2

To find phase difference of two points in a wave, we can use:

2x

2T

t

Example Given that the wavelength of a progressive wave is 4.0 m and x represents the distance of wave travel, calculate i) the phase angles at x = 0.50 m and 1.60 m, ii) the phase difference between any two points which are 1.1 m apart on the

wave.

i) Use

2x

80.020.4

60.12

25.020.4

50.02

x

x

ii) Use

2x

55.020.4

1.12

x

Note that part ii) agrees with part i).

displacement

distance/m

x

P

displacement

time/s

T

t

P

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4. Graphical Representations of Transverse and Longitudinal Waves

Graphical Representations of Transverse Waves The following graphs show the variation of the displacement of a transverse wave with distance along the wave at a particular time t. (Considers ALL particles) The graph below shows the variation of the displacement of the same transverse wave with time at a particular point along the wave. (Considers only ONE particle)

T 2

T 4

T 4

3T

A

A

displacement

time t

t = 0

t = T/4

t = T/2

t = 3T/4

t = T

Displacement-distance graphs of a transverse wave traveling to the right

(Particle Motion) (Wave Motion)

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Graphical Representations of Longitudinal Waves

The following graphs show the variation of the displacement of a longitudinal wave with distance along the wave at a particular time t. (Considers ALL particles)

c r c r

r c r

c r c r

c r c

r c r c

(Particle Motion) (Wave Motion)

Displacement-distance graphs of a longitudinal wave traveling to the right

c r cr

r c r

c r cr

c r c

r c rc

t = 0

t = T/4

t = T/2

t = 3T/4

t = T

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The graph below shows the variation of the displacement of the same longitudinal wave with time at a particular point along the wave. (Considers only ONE particle) Comparing longitudinal displacement-distance graph with pressure-distance graph

At which positions in the pressure – distance graph are the pressure variation the greatest?

T 2

T 4

T 4

3T

A

A

displacement

time t

particles' movement

Compression Rarefaction Compression

particles' relative movement

Fig. 5.1 Pressure – Distance graph

Pressure

Patm

Patm - P

Patm + P

P

Displacement Displacement – Distance graph

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Learning Checks 1 A wave pulse is moving, as illustrated, with uniform speed v along a rope.

Which of the graphs below correctly shows the relation between the displacement s of point P and time t?

2 A wave pulse is moving, as illustrated, with uniform speed v along a rope. Which of the graphs below correctly shows the relation between the displacement s of point P and time t?

t

s

s

t

t

s

s

t

A B

C D

vP

v

P

s

t

A

s

t

B

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5. General Wave Equation In Mathematics, you learnt that a simple sine graph such as the one shown below can be represented by the equation y = sin . The y values vary between -1 and 1, and the curve cuts the x-axis at specific values 0, 180, 360 etc measured in degrees of angle.

Now, what happens if you see a sine-shaped curve on a piece of string with the following dimensions?

What is the equation that describes the curve now? Is it still y = sin ? The y values now should vary between 2 while the x-axis is no longer an angle but a distance. Does sin (6.0 cm) mean anything? Answer is 'no', sin (6.0 cm) is meaningless in mathematics. Therefore, we need to 'transform' or 'convert' or 'map' distance into an angle for the sine function to make sense. Since the wavelength of 6.0 cm corresponds to one cycle, thus we can convert distance to angle in this way:

6.0 cm 360 o

x cm o x cm corresponds to degrees

To do the 'conversion', we use the proportion formula = ox360

6. Hence instead

of y = sin, the equation that describes the wave on the string above is

y = 2 sin( ox360

6) where 2 and x are measured in cm.

, angle

y

0

1

-1

90 o 180 o 270 o 360 o

2.0 cm

6.0 cm 6.0 cm

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Now, you can check if the equation works for say x = 3 cm. Putting x = 3 cm will give y = 2 sin(180o) which is zero as desired. Check for yourself that for any other values of x, the equation will give the correct y values in cm.

More generally, for any sine wave with a wavelength and a maximum value of A, the equation will become:

y = A sin ( ox360

)

If angle is in radians (more common), then the equation is:

y = A sin (

2x )

In addition, you may have learnt in your Math that

equation is y = A sin (

2x )

if curve shifts to the right by Δx,

where Δx corresponds to .

And

equation is y = A sin (

2x )

if curve shifts to the left by Δx,

where Δx corresponds to .

Example

A progressive wave has amplitude 8.0 m and wavelength 4.0 m. Given that x represents distance of wave travel and x = 0.50 m, deduce the displacement y.

mx

Ay 7.5)20.4

50.0sin(0.8)2sin(

Mapping distance x to angle: 2

ydisplacement

x distanceΔx

y = A sin (

2x )

A

ydisplacement

x distance

Δxy = A sin (

2

x )A

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6. Wave Speed Equation From the basic definition of speed,

time

distancespeed

Since for one complete wave cycle the distance traveled is the wavelength and the time is the period,

period

wavelengthspeed

But frequency

1period

Therefore, speed = frequency x wavelength Note: Wave velocity is often referred to as the phase velocity. It must be distinguished from the velocity of a particle of the medium. Example 1 A radio station transmits a signal with a frequency of 100 MHz. If the velocity of the radio waves through the air is 3 x 108 m s-1 what is the wavelength of the signal?

v = f λ

λ = v / f λ = 3 x 108 / 1 x 108 = 3 m Example 2 A sound wave of frequency 400 Hz is traveling in air at a speed of 320 m s-1. What is the difference in phase between two points on the wave 0.2 m apart in the direction of travel?

Phase difference, = x

2

v = f 320 = 400 = 0.80 m Therefore, = (0.2/0.8) x 2 = 1.6 rad

v = f λ

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7. Intensity of a Wave Energy of a Wave The amount of energy carried by a wave is related to the amplitude of the wave. A high energy wave is characterized by high amplitudes; a low energy wave is characterized by low amplitudes.

The energy transported by a wave is proportional to the square of the amplitude A of the wave.

E A2

Until now, we have been looking mostly at waves traveling in a straight line. In reality, many waves are not confined to traveling in a line. For instance, sound waves from a speaker spread out in many directions. Assuming the source is radiating energy at a certain fixed rate, when the waves are further away from the source, the energy must be spread out over a larger area and hence when the

waves are further away, there is less energy in each particle's vibrations. Even for a wave along a string where the energy does not spread out, the vibrations will in reality still get lesser further away from the source. This is due to attenuation or energy lost as heat. When a wave travels along a string, the stretching and relaxing of the string causes heat to be produced and so the wave looks more like:

distance

displacement

Energy in a wave drops with distance if the wave

Energy in a wave drops with distance also because

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Intensity of a Wave The intensity of a wave is defined as the rate of energy flow through a unit area perpendicular to the direction of travel of the wave. (Unit: W m-2)

tArea

kA

t

EPI

2

Area

1

Area

The expression assumes no energy lost as heat. It can be seen that at the area concerned, the intensity I of a wave is proportional to the square of the amplitude A of a wave.

For sound, intensity is a measure of loudness. For light, it is a measure of brightness.

[Optional] Consider a small spherical object oscillating and producing spherical wavefronts that moves outward from the source. Assuming that the power emitted by the source is uniformly distributed over a spherical surface of area 4r2 at any distance r from the source, the intensity at that distance r will be,

24Area r

PPI

Thus, 2

1

rI (an inverse square law)

I A2

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Learning Checks 1. An ocean wave has an amplitude of 2.5 m. Weather conditions suddenly

change such that the amplitude is increased to 5.0 m. The intensity of the wave is A halved B doubled C quadrupled D remains the same

2. A sound wave has amplitude A at a certain position. At another point

further from the source, due to attenuation and spreading, the amplitude is halved. The new intensity of the wave is A ½ of the previous value B twice of the previous value C ¼ of the previous value D 4 times of the previous value

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8. Polarisation As a result of the transverse nature of the vibration, transverse waves have an additional property which is not possessed by longitudinal waves. Recall that the movement of particles in a transverse wave is at right angles to the direction of wave travel. In three dimensions (3-D), this also means that there are many possible directions for particle movement as the figure below shows. Frequently, particle movement in a transverse wave takes place in many different directions. The wave is then said to be unpolarised. For example, light for a hot object, such as the Sun or a filament lamp, is unpolarised because it is emitted totally at random from the atoms of the hot object. Polarisation is the process of limiting the vibration of the particles of a wave to one direction or one plane only. Polarisation can only be achieved with transverse waves. Longitudinal waves do not exhibit polarisation, because for these waves the direction of oscillation is along the direction of travel. Polarising Rope

Fig. 1 shows a transverse wave on a rope that is polarised in a vertical plane. Fig. 2 shows a transverse wave on a rope that is polarised in a horizontal plane. If the rope is waved randomly in all directions, the resulting wave is said to be unpolarised.

Direction of particle displacement (many possibilities)

Direction of wave travel

Fig. 1 Fig. 2

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Polarising Light In the case of light, polarisers or polarising filters made of transparent plastic sheets can be used to produce polarised light. Polarisation of Light Applet http://micro.magnet.fsu.edu/primer/java/scienceopticsu/polarizedlight/filters/index.html

Polarising Microwave For microwaves, they usually come out of a transmitter already polarised. Otherwise, a polariser made of metal grill can be used to polarise them.

Applications Examples of applications of polarisation include polaroid filters on the cameras, polarised sunglasses, and polarising lamps.

Unpolarised light (many directions)

Polariser with axis vertical

Polarised light (only one direction)

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Quick Check 5 1. We know that electromagnetic waves are ________ because they can be

polarised. _______ cannot be polarised, and so must be __________. A longitudinal, water wave, transverse

B transverse, water wave, longitudinal C longitudinal, sound, transverse D transverse, sound, longitudinal

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9. Determination of Frequency and Wavelength of Sound 1) Determination of Frequency of Sound Using a Calibrated C.R.O The first thing to know about a Cathode Ray Oscilloscope (CRO) is that it is always measuring an electrical input. Therefore if it is used to measure some feature of a sound wave, it must be connected to a transducer of some sort in order to convert the sound energy into electrical energy. A microphone is an appropriate transducer to pick up the sound wave and feed the wave signal into the y-plates of the CRO. A waveform can thus be displayed on the screen. The scales of the x- and y-plates can be adjusted the same way you would adjust the scales for your graphs in order to get a sufficiently large image on the screen. For example, if a waveform is displayed as shown and the sensitivities are as follows: y-plates: 5 V cm-1 x-plates: 100 s cm-1 From the display, one is able to calculate the period of the wave and hence the frequency of sound. Determine the period (3 small squares for one wave cycle), T = 3 100 s = 300 s or 3.0 10-4 s Hence, the frequency f = 1/T = 1/3.0 10-4 = 3.3 103 Hz This method works only when the sound waves have a single frequency. That is usually not the case for most sources of sound such as that from a human or an instrument. Note: Although the wave may look on the screen as if it were a transverse wave, it in reality is a longitudinal wave. The trace on the screen is actually a graph of displacement against time.

1 cm

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2) Determination of Wavelength of Sound Using Stationary Waves Standing Waves (to be covered in detail in next chapter on Superposition) Standing waves are formed from two equal amplitude progressive waves (can be either transverse or longitudinal) traveling in opposite directions and overlapping (superimposing) upon each other. The resultant wave remains located between the two sources of origin. The setup below shows a vibrator creating transverse waves which travel to the right, get reflected from the right fixed end and superimpose with the leftward waves to form stationary waves.

Since stationary waves are formed from 2 equal amplitude waves traveling in opposite directions, they do not have a net energy transfer in either direction. Another characteristic of standing waves is that parts of the wave have zero amplitude. These points are called nodes. Halfway between nodes are points where the amplitude is a maximum. These points are called antinodes. Determining Wavelength of Sound The apparatus below is used to set up stationary sound waves between the loudspeaker and the reflector.

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The loudspeaker connected to an audio-frequency generator sends a sound wave of known frequency f towards the reflector. The superposition of the incident wave and the reflected wave produces a stationary wave between the loudspeaker and the reflector. A microphone connected to the CRO is moved along the straight line between the loudspeaker and the reflector and the CRO will display the signal. What is being discovered? The CRO will display a waveform of the same frequency as that of the sound, but the amplitude of the displayed waveform will change from maximum to minimum when the microphone is moved from an anti-node to a node.

This means that the air molecules at the anti-nodes will vibrate with the most amplitude while those at the nodes do not vibrate. Hence the membrane in the microphone will also vibrate with the greatest amplitude at the anti-nodes and stay almost stationary at the nodes (‘almost’ because of imperfect stationary wave). The distance between 2 consecutive nodes or anti-nodes is equal to /2. Therefore if we take note of the distance moved by the microphone when the amplitude on the CRO changes from maximum to minimum, we would be able to find the wavelength of the sound wave.

Near a node Near an anti-node

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Worked Examples Example 1 The figures below show two graphs which refer to the same wave

Calculate the speed of the wave.

N90/III/3 (part) 1st graph: 3½ sine waves correspond to 17 ms Hence, period T = (17/3.5) ms Frequency f = 1/T

= (3.5/17) kHz 2nd graph: = 1.75 m Therefore, v = f = (3.5/17) 1000 1.75 = 360 m s-1 Example 2 A sound wave of amplitude 0.20 mm has an intensity of 3.0 W m-2. What will be the intensity of a sound wave of the same frequency which has an amplitude of 0.40 mm? Since I A2,

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22

2

2

2

1

2

1

2

mW12

20.0

40.0

3

I

I

A

A

I

I

Example 3 (Optional) A small source of sound radiates energy equally in all directions. At a particular frequency the intensity of the sound 1.0 m from the source is 1.0 x 10-5 W m-2, corresponding to an amplitude of oscillation on the air molecules of 70 m (1 m or micrometer = 1 x 10-6 m). Assuming that the sound is propagated without energy loss, find a) the intensity of the sound and b) the amplitude of oscillation of the air molecules at a distance of 5.0 m from the source.

(J83/I/4) Solution (a) Since I 1/r2,

27-2

2

5-2

2

2

1

1

2

mW10 x 4.0

50.0

10.0

10 x 1.0

I

I

r

r

I

I

(b) Since I A2,

m10x 4.1

10 x 7010 x 1.0

10 x 4.0

5-2

2

6-2

5-

7-

2

1

2

1

2

A

A

A

A

I

I

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Example 4 A source of sound of frequency 2500 Hz is placed several metres from a plane wall in a large chamber containing a gas. A microphone, connected to a cathode-ray oscilloscope, is used to detect nodes and anti-nodes along the line XY between the source and the wall.

The microphone is moved from one node through 20 anti-nodes to another node, a distance of 1.90 m. What is the speed of sound in the gas?

(J95/I/10) Each loop of node or anti-node = /2 The microphone moved through 20 loops of nodes, therefore 20(/2) = 1.9 = 0.19 m v = f

= 2500(0.19) = 475 m s-1

Example 5 Which one of the following effects provides direct experimental evidence that light is a transverse, rather than a longitudinal, wave motion?

a) Light is diffracted by a narrow slit. b) Two coherent light waves can be made to interfere. c) The intensity of light from a point source falls off inversely as the square of

the distance from the source. d) Light is refracted by a glass prism. e) Light can be polarised by reflection at a water surface.

(J94/I/10; J78/I/11) Answer: e Longitudinal waves do not exhibit polarisation. Therefore if light can be polarised, it must be a transverse wave.