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196

The waves we have considered so far in Chapter 15,

Chapter 16andChapter 17have beenprogressive

waves; they start from a source and travel outwards. A

second important class of waves is stationarywaves

(standing waves). These can be observed as follows.

Use a long spring or a slinky spring. A long rope or

piece of rubber tubing will also do. Lay it on the oor

and x one end rmly. Move the other end from side to

side so that transverse waves travel along the length of

the spring and reect off the xed end (Figure 18.1). If

you adjust the frequency of the shaking, you should beable to achieve a stable pattern like one of those shown

in Figure 18.2. Alter the frequency in order to achieve

one of the other patterns.

You should notice that you have to movethe end

of the spring with just the right frequency to get one

of these interesting patterns. The pattern disappears

when the frequency of the shaking of the free end of

the spring is slightly increased or decreased.

Stationary waves

Chapter 18

Nodes and antinodesWhat you have observed is a stationary wave on the

long spring. There are points along the spring that

remain (almost) motionless while points on either

side are oscillating with the greatest amplitude. The

points that do not move are called the nodesand the

points where the spring oscillates with maximum

amplitude are called the antinodes. At the same time,

it is clear that the wave prole is not travelling along

the length of the spring. Hence we call it a stationary

wave or a standing wave.

fixed

end

free

end

A

mplitude

Amplitude

Amplitude

Distance

Distance

Distance

antinode

node

Figure 18.2 Different stationary wave patterns are possible, depending on the frequency of vibration.

Figure 18.1 A slinky spring is used to generate a

stationary wave pattern.

e-Learning

Objectives
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resultant

wave moving to right

wave moving to left

wave moving to right

wave moving to left

resultant wave

Key

NN

A A A AA

N N N N

Snapshots

of the

waves

over a

time

of one

period, T.

T= period of wave

Distance

s

s

s

s

t = 0

t = T

Displacement

x

x

t =

profile at t = and

profile at t =

profile att

=0

andT

x

x

t = T2

T

4

t = 3T4

3T4

T

4

T

2

2

2

We normally represent a stationary wave by

drawing the shape of the spring in its two extreme

positions (Figure 18.3a). The spring appears as a

series of loops, separated by nodes. In this diagram,

point A is moving downwards. At the same time,

point B in the next loop is moving upwards. The

phase difference between points A and B is 180.

Hence the sections of spring in adjacent loops are

always moving in antiphase; they are half a cycle out

of phase with one another.

a

b

B

A

Distance

Amplitude

Amplitude

Distance

Figure 18.3 The xed ends of a long spring must be

nodes in the stationary wave pattern.

Formation of stationary wavesImagine a string stretched between two xed points,

for example a guitar string. Pulling the middle of

the string and then releasing it produces a stationary

wave. There is a node at each of the xed ends and an

antinode in the middle. Releasing the string produces

two progressive waves travelling in opposite directions.

These are reected at the xed ends. The reected

waves combine to produce the stationary wave.

Figure 18.1shows how a stationary wave can

be set up using a long spring. A stationary wave

is formed whenever two progressive waves of

the same amplitude and wavelength, travelling in

oppositedirections, superimpose. Figure 18.4uses a

displacements against distancex graph to illustrate

the formation of a stationary wave along a long

spring (or a stretched length of string).

At time t=0, the progressive waves travelling tothe left and right are in phase. The waves combine

constructivelygiving amplitude twice that of

each wave.

After a time equal to one quarter of a period

(t= T4), each wave travels a distance of one quarter

of a wavelength to the left or right. Consequently,

the two waves are in antiphase (phase difference

= 180). The waves combine destructivelygiving

zero displacement.

Figure 18.4 The blue-coloured wave is movingto the left and the red-coloured wave to the right.

Theprinciple of superpositionof waves is used to

determine the resultant displacement. The prole of

the long spring is shown in green.

Chapter 18: Stationary waves

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198

After a time equal to one half of a period (t= T2 ),the two waves are back in phase again. They once

again combine constructively.

After a time equal to three quarters of a period(t= 3T

4), the waves are in antiphase again. They

combine destructivelywith the resultant wave

showing zero displacement.

After a time equal to one whole period (t= T), thewaves combine constructively. The prole of the

slinky spring is as it was at t= 0.

This cycle repeats itself, with the long spring showing

nodes and antinodes along its length. The separation

between adjacent nodes or antinodes tells us about the

progressive waves that produce the stationary wave.

A closer inspection of the graphs in Figure 18.4

shows that the separation between adjacent nodes

or antinodes is related to the wavelengthof theprogressive wave. The important conclusions are:

separation between two adjacent nodes

(or antinodes) =

2

separation between adjacent node and antinode =

4

The wavelengthof anyprogressive wave can be

determined from the separation between neighbouring

nodes or antinodes of the resulting standing wave

pattern. (This is =2 .) This can then be used to

determine either the speed v of the progressive wave

or its frequencyfby using the wave equation:

v =f

It is worth noting that a stationary wave does not

travel and therefore has no speed. It does not transfer

energy between two points like a progressive wave.

Table 18.1shows some of the key features of a

progressive wave and its stationary wave.

Progressive

wave

Stationary

wave

wavelength

frequency f f

speed v zero

Table 18.1 A summary of progressive and

stationary waves.

SAQ

1 A stationary (standing) wave is set up on a

vibrating spring. Adjacent nodes are separated by

25 cm. Determine:

a the wavelength of the stationary

wave

b the distance from a node to an

adjacent antinode.

Observing stationary waves

Stretched strings

A string is attached at one end to a vibration

generator, driven by a signal generator (Figure

18.5). The other end hangs over a pulley and

weights maintain the tension in the string. When thesignal generator is switched on, the string vibrates

with small amplitude. However, by adjusting the

frequency, it is possible to produce stationary waves

whose amplitude is much larger.

pulley

vibration

generator

weights

signalgenerator

Figure 18.5 Meldes experiment for investigating

stationary waves on a string.

The pulley end of the string is unable to vibrate; this

is a node. Similarly, the end attached to the vibrator

is only able to move a small amount, and this is also

a node. As the frequency is increased, it is possible

to observe one loop (one antinode),two loops, three

loops and more. Figure 18.6shows a vibrating string

where the frequency of the vibrator has been set to

produce two loops.

Hint

Answer

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A ashing stroboscope is useful to reveal the motion

of the string at these frequencies, which look blurred

to the eye. The frequency of vibration is set so that

there are two loops along the string; the frequency of

the stroboscope is set so that it almost matches thatof the vibrations. Now we can see the string moving

in slow motion, and it is easy to see the opposite

movements of the two adjacent loops.

This experiment is known asMeldes experiment,

and it can be extended to investigate the effect of

changing the length of the string, the tension in the

string and the thickness of the string.

SAQ

2 Look at the stationary (standing) wave on the

string in Figure 18.6. The length of the vibratingsection of the string is 60cm.

a Determine the wavelength of the stationary

wave and the separation of the two

neighbouring antinodes.

The frequency of vibration is increased until a

stationary wave with three antinodes appears on

the string.

b Sketch a stationary wave

pattern to illustrate the

appearance of the string.

c What is the wavelength of this

stationary wave?

Microwaves

Start by directing the microwave transmitter at a

metal plate, which reects the microwaves back

towards the source (Figure 18.7). Move the probe

Figure 18.6 When a stationary wave is established,

one half of the string moves upwards as the other half

moves downwards. In this photograph, the string is

moving too fast to observe the effect.

receiver around in the space between the transmitter

and the reector and you will observe positions of

high and low intensity. This is because a stationary

wave is set up between the transmitter and the

sheet; the positions of high and low intensity are the

antinodes and nodes respectively.

If the probe is moved along the direct line from

the transmitter to the plate, the wavelength of the

microwaves can be determined from the distance

between the nodes. Knowing that microwaves travel

at the speed of light c (3.0108ms1), we can then

determine their frequencyfusing the wave equation

c=f .

reflecting

sheet

probe

meter

microwave

transmitter

Figure 18.7 A stationary wave is created whenmicrowaves are reected from the metal sheet.

SAQ

3 a Draw a stationary wave pattern for the

microwave experiment above. Clearly show

whether there is a node or an antinode at the

reecting sheet.

b The separation of two adjacent

points of high intensity is found

to be 14mm. Calculate the

wavelength and frequency

of the microwaves.

Hint

Answer

Hint

Answer

Extension

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200

Sound waves in air columns

A glass tube (open at both ends) is clamped so that

one end dips into a cylinder of water; by adjusting its

height in the clamp, you can change the length of the

column of air in the tube (Figure 18.8). When you hold

a vibrating tuning fork above the open end, the aircolumn may be forced to vibrate, and the note of the

tuning

fork

air

water

4

Figure 18.8 A stationary wave is created in the air in

the tube when the length of the air column is adjusted

to the correct length.

tuning fork sounds much louder. This is an example

of a phenomenon called resonance. The experiment

described here is known as the resonance tube.

For resonance to occur, the length of the air

column must be just right. The air at the bottom of the

tube is unable to vibrate, so this point must be a node.

The air at the open end of the tube can vibrate most

freely, so this is an antinode. Hence the length of

the air column must be one-quarter of a wavelength

(Figure 18.9a). (Alternatively, the length of the air

column could be set to equal three-quarters of a

wavelength see Figure 18.9b.)

SAQ

4 Explain how two sets of identical but oppositely

travelling waves are established in the

microwave and air columnexperiments described above.

Stationary waves and musical instruments

The production of different notes by musical

instruments often depends on the creation of stationary

waves (Figure 18.10). For a stringed instrument

such as a guitar, the two ends of a string are xed, so

nodes must be established at these points. When the

string is plucked half-way along its length, it vibrates

with an antinode at its midpoint. This is known asthefundamental mode of vibrationof the string. The

fundamental frequency is the minimumfrequencyof

a standing wave for a given system or arrangement.antinode

node

antinode

node

a b

4

34

Figure 18.9 Stationary wave patterns for air in a tube

with one end closed.

Figure 18.10 When a guitar string is plucked, the

vibrations of the strings continue for some time

afterwards. Here you can clearly see a node close to

the end of each string.

Answer

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Similarly, the air column inside a wind instrument is

caused to vibrate by blowing, and the note that is heard

depends on a stationary wave being established. By

changing the length of the air column, as in a trombone,

the note can be changed. Alternatively, holes can be

uncovered so that the air can vibrate more freely, giving

a different pattern of nodes and antinodes.

In practice, the sounds that are produced are

made up of several different stationary waves

having different patterns of nodes and antinodes.

For example, a guitar string may vibrate with two

antinodes along its length. This gives a note having

twice the frequency of the fundamental, and is

described as a harmonicof the fundamental. The

musicians skill is in stimulating the string or air

column to produce a desired mixture of frequencies.

= L

L= length of string

fundamental = 2L f0

2f0

= L 3f0

wavelength frequency

second harmonic

third harmonic

A

A A

AA A

N N

N NN

N NNN23

Figure 18.11 Some of the possible stationary waves for a xed string of lengthL.

The frequency of the harmonics is a multiple of the fundamental frequencyf0.

The frequency of a harmonic is always a multiple

of the fundamental frequency. The diagrams show

some of the modes of vibrations for a xed length of

string (Figure 18.11) and an air column in a tube of a

given length that is closed at one end (Figure 18.12).

Determining the wavelength andspeed of soundSince we know that adjacent nodes (or antinodes) of

a stationary wave are separated by half a wavelength,

we can use this fact to determine the wavelengthof

a progressive wave. If we also know the frequency

fof the waves, we can nd their speed vusing the

wave equation v=f .

L= length

of air

column

fundamental secondharmonic

thirdharmonic

= 4L

f0

=

3f0

wavelength

frequency

N

A

N

A

N

A

N

A

N

A

N

A

4L3

=

5f0

4L5

Figure 18.12 Some of the possible stationary waves for an air column, closed at one end.

The frequency of each harmonic is an odd multiple of the fundamental frequencyf0.

Extension

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Figure 18.13 Kundts dust tube can be used to

determine the speed of sound.

One approach uses Kundts dust tube (Figure

18.13). A loudspeaker sends sound waves along the

inside of a tube. The sound is reected at the closed

end. When a stationary wave is established, the dust

(ne powder) at the antinodes vibrates violently.

It tends to accumulate at the nodes, where the

movement of the air is zero. Hence the positions of

the nodes and antinodes can be clearly seen.

signal

generator

loudspeaker

glass tube

closed end

dust piles

up at nodesA

N

A

N

A

N

A

An alternative method is shown in Figure 18.14; this

is the same arrangement as used for microwaves. The

loudspeaker produces sound waves, and these are

reected from the vertical board. The microphone

detects the stationary sound wave in the space

between the speaker and the board, and its output is

displayed on the oscilloscope. It is simplest to turn

off the time base of the oscilloscope, so that the spot

no longer moves across the screen. The spot moves

up and down the screen, and the height of the vertical

trace gives a measure of the intensity of the sound.

By moving the microphone along the line between

the speaker and the board, it is easy to detect nodes

and antinodes. For maximum accuracy, we do not

measure the separation of adjacent nodes; it is better

to measure the distance across several nodes.

The resonance tube experiment (Figure 18.8) canalso be used to determine the wavelength and speed

of sound with a high degree of accuracy. However,

to do this, it is necessary to take account of a

systematic error in the experiment, as discussed in the

Eliminating errors section onpage 203.

SAQ

5 a For the arrangement shown in Figure 18.14,

suggest why it is easier to determine accurately

the position of a node rather than an antinode.

b Explain why it is better to measurethe distanceacross

several nodes.

6 For sound waves of frequency 2500Hz, it is found

that two nodes are separated by 20cm, with three

antinodes between them.

a Determine the wavelength of these sound

waves.

b Use the wave equation v=f to determine

the speed of sound

in air.

oscilloscope

loudspeaker

microphoneto signalgenerator(2 kHz)

reflecting

board

Figure 18.14 A stationary sound wave is established

between the loudspeaker and the board.


202

Answer

Answer

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The resonance tube experiment illustrates an

interesting way in which one type of experimental

error can be reduced or even eliminated.

Look at the representation of the stationary

waves in the tubes shown in Figure 18.9. In each

case, the antinode at the top of the tube is shown

extending slightly beyond the open end of the

tube. This is because experiment shows that the air

slightly beyond the end of the tube vibrates as part

of the stationary wave. This is shown more clearly

in Figure 18.15.

c

4

43

Figure 18.15 The antinode at the open end of a

resonance tube is formed at a distance cbeyond the

open end of the tube.

for the shorter tube,

4= l1+c

for the longer tube,3

4= l2+c

Subtracting the rst equation from the second

equation gives:

3

4

4= (l2+c)(l1+c)

Simplifying gives:

2= l2l1

and hence= 2(l2l1)

So, although we do not know the value of c, we canmake two measurements (l1and l2) and obtain an

accurate value of. (You may be able to see from

Figure 18.15that the difference in lengths of the

two tubes is indeed equal to half a wavelength.)

The end-correction cis an example of a

systematic error. When we measure the length l

of the tube, we are measuring a length which is

consistently less than the quantity we really need to

know (l+c). However, by understanding how the

systematic error affects the results, we have been

able to remove it from our measurements.Other examples of systematic errors in physics

include:

meters and other instruments which have beenincorrectly zeroed (so that they give a reading

when the correct value is zero)

meters and other instruments which have beenincorrectly calibrated (so that, for example, all

readings are consistently reduced by a factor of,

say, 1.0%).

Eliminating errors

The antinode is at a distance cbeyond the end

of the tube, where cis called the end-correction.

Unfortunately, we do not know the value of c.

It cannot be measured directly. However, we

can write:

SAQ

7 In a resonance tube experiment, resonance is

obtained for sound waves of frequency 630Hz

when the length of the air column is 12.6cm and

again when it is 38.8cm. Determine:

a the wavelength of the sound waves causing

resonance

b the end-correction for this tube

c the speed of sound in air. Answer

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204

Summary

Stationary waves are formed when two identical waves travelling in opposite directions meet andsuperimpose. This usually happens when one wave is a reection of the other.

A stationary wave has a characteristic pattern of nodes and antinodes.

A node is a point where the amplitude is always zero. An antinode is a point of maximum amplitude. Adjacent nodes (or antinodes) are separated by a distance equal to half a wavelength. We can use the wave equation v =f to determine the speed vor the frequencyf of a progressive wave.

The wavelengthis found using the nodes or antinodes of the stationary wave pattern.

Questions

1 The diagram shows a stretched wire held

horizontally between supports 0.50 m apart.

When the wire is plucked at its centre, a

standing wave is formed and the wire vibrates

in its fundamental mode (lowest frequency).

a Explain how the standing wave is formed. [2]

b Draw the fundamental mode of vibration of the wire. Label the position of any

nodes with the letter N and any antinodes with the letter A. [2]

c What is the wavelength of this standing wave? [1]

OCR Physics AS (2823) January 2006 [Total 5]

2 The diagram shows an arrangement where

microwaves leave a transmitter T and move

in a direction TP which is perpendicular to

a metal plate P.

a When a microwave detector D is slowly

moved from T towards P the pattern of the

signal strength received by D is high, low, high, low etc.

Explain:

why these maxima and minima of intensity occur

how you would measure the wavelength of the microwaves how you would determine their frequency. [6] b Describe how you could test whether the microwaves leaving the transmitter are

plane polarised. [2]

OCR Physics AS (2823) June 2004 [Total 8]

Answer

continued

D

T P

transmitter

0.50m

Glossary

Hint

Answer

Hint

Answer

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3 a In standing waves, there are nodesand antinodes. Explain what is meant by:

i a node [1]

ii an antinode. [1]

b The diagram shows a long glass tube within

which standing waves can be set up. A vibrating tuning fork is placed above the

glass tube and the length of the air column is

adjusted, by raising or lowering the tube in

the water, until a sound is heard.

tuning fork

0.32 m

water

air column

long glass tube

A B

State how you would set up a standing wave on the string. [1]

b The standing wave vibrates in its fundamental mode, i.e. the lowest frequency at

which a standing wave can be formed. Draw this standing wave. [1]

c Figure 2shows the appearance of another standing wave formed on the same string.

Figure 1

A B

Figure 2

The distance between A and B is 1.8m. Use Figure 2to calculate

i the distance between neighbouring nodes [1]

ii the wavelength of the standing wave. [1]


i The standing wave formed in the air column is the fundamental (the lowest

frequency). Make a copy of the diagram and show on it the position of a node

label as N, and an antinode label as A. [2]

ii When the fundamental wave is heard, the length of the air column is 0.32m.

Determine the wavelength of the standing wave formed. [1]

iii The speed of sound in air is 330ms1. Calculate the frequency of the

tuning fork. [3]


4 a Figure 1shows a string stretched between two points A and B.

Hint

Answer

Hint

Hint

Answer

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COAS_P1_Ch18_it