1
A stationary wave is formed on a stretched string. Discuss the formation of this wave.
Your answer should include:
•
•
•
an explanation of how the stationary wave is formed
a description of the features of the stationary wave
a description of the processes that produce these features.
The quality of your written communication will be assessed in your answer.
(Total 6 marks)
2
In the first diagram, PQ is a stretched string of length 0.34 m. When it is plucked, it vibrates at a
frequency of 440 Hz.
(a)
(i)
On the second diagram, draw the fundamental mode of vibration for the string.
(1)
(ii)
State the wavelength of the standing wave produced when the string is plucked.
...............................................................................................................
(1)
(iii)
The same string is lightly touched at its midpoint and is plucked at the point X. Draw
the standing wave produced on the diagram below.
(1)
(b)
The tension of the string is increased. State the effect this has on the fundamental
frequency of vibration of the string.
........................................................................................................................
(1)
(Total 4 marks)
Page 1 of 56
3
(a)
When an earthquake occurs longitudinal waves (P waves) and transverse waves (S
waves) are produced in the Earth’s crust. The P waves travel faster than the S waves. A
station, whose task is to detect and locate the position of earthquakes, is at a distance d
from the point where the earthquake originates (the epicentre).
The speed of P waves is 7.5 km s–1 and that of S waves is 5.0 km s–1 . For a particular
earthquake the station detects the P wave 1.5 s before the S wave.
(i)
Write down expressions for the time it takes each wave to travel the distance d from
the epicentre to the station.
Time for P waves ..................................................................................
Time for S waves ...................................................................................
(1)
(ii)
Determine the distance of the epicentre from the station.
(2)
(b)
The earthquake can set up resonant vibrations in bridges causing them to collapse. The
diagram below shows one such bridge. The modes of vibration of the bridge are similar to
those of a stretched string.
(i)
Explain how a stationary wave is set up in a stretched string.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
(2)
(ii)
The velocity of transverse waves along the bridge is 180 m s–1. Determine the
frequency of the vibrations produced by an earthquake that would cause the central
span of the bridge to resonate at its fundamental frequency (first harmonic).
(3)
Page 2 of 56
(iii)
A designer assumes the highest frequency produced by an earthquake is 1.5 times
the fundamental frequency and decides to modify the bridge by building an extra
support midway between the two existing supports.
Explain whether this modification would eliminate resonant vibrations caused by an
earthquake.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
(2)
(Total 10 marks)
4
State two factors that affect the fundamental frequency of a vibrating stretched string.
Factor 1 ...................................................................
Factor 2 ...................................................................
(Total 2 marks)
5
The drawing below shows a standing wave set up on a wire of length 0.87 m. The wire is vibrated
at a frequency of 120 Hz.
(a)
Calculate the speed of transverse waves along the wire.
(3)
(b)
Show that the fundamental frequency of the wire is 40 Hz.
(2)
(Total 5 marks)
Page 3 of 56
6
The equation for the speed, v, of a transverse wave along a stretched string is:
where T is the tension in the string and μ is the mass per unit length of the string.
(a)
State the quantities that would need to be measured in order to calculate a single value for
the speed of the wave using the equation. Name a suitable measuring instrument for each
quantity.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(4)
(b)
The apparatus shown in the diagram below could be used to measure a value for v.
Explain how this apparatus may be used to calculate an accurate value of the speed of the
transverse wave along the string.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(4)
Page 4 of 56
(c)
With the signal generator in the diagram below set at 152 Hz, 10 loops fit the vibrating
length of the string exactly. The string is of length 2.0 m and the mass on the end of it is
0.72 kg.
the Earth’s gravitation field strength, g = 9.8 N kg–1
Calculate the mass of the string.
Mass = ....................................................
(5)
(Total 13 marks)
7
Figure 1 shows a stretched string driven by a vibrator. The right-hand end of the string is fixed to
a wall. A stationary wave is produced on the string; the string vibrates in two loops.
Figure 1
(a)
State the physical conditions that are necessary for a stationary wave to form on the string.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(b)
Explain how you know that the wave on the string is transverse.
........................................................................................................................
(1)
(c)
Compare the amplitude and phase of the oscillations of points A and B on the string.
Amplitude ........................................................................................................
Phase ..............................................................................................................
(2)
Page 5 of 56
(d)
The length of the string is 1.2 m and the speed of the transverse wave on the string is
6.2 m s–1.
Calculate the vibration frequency of the vibrator.
Vibration frequency ...........................
(3)
(e)
The frequency of the vibrator is tripled.
Sketch the new shape of the stationary wave on Figure 2.
Figure 2
(ii)
Show on your diagram three points P, Q and R that oscillate in phase.
(2)
(Total 11 marks)
Page 6 of 56
8
(a)
State the conditions necessary for a stationary wave to be produced.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(b)
The diagram shows a stationary wave on a stretched guitar string of length 0.62 m.
The speed of transverse waves along the string is 320 m s–1. Calculate the frequency of the
note being played.
Frequency ..........................................
(3)
(Total 6 marks)
Page 7 of 56
9
(a)
Explain how a stationary wave is produced when a stretched string is plucked.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(b)
(i)
On Figure 1, draw the fundamental mode of vibration of a stretched string. Label any
nodes with a letter N and any antinodes with a letter A.
Figure 1
(2)
(ii)
On Figure 2, draw the fourth harmonic (third overtone) for the stretched string. Label
any nodes with a letter N and any antinodes with a letter A.
Figure 2
(2)
(c)
The fundamental frequency of vibration, f, of a string is given by:
f=
where
T = the tension in the string
l = the length of the string
µ = the mass per unit length of the string
A string has a tension of 180 N and a length of 0.70 m.
(i)
What would need to be done to the length of the string in order to double the
frequency?
...............................................................................................................
(1)
Page 8 of 56
(ii)
What would need to be done to the tension of the string in order to double the
frequency?
...............................................................................................................
(2)
(Total 10 marks)
10
Figure 1 shows a violin string. One way to produce a musical note is to pull the centre of the
string to one side and then release it quickly.
Figure 1
(a)
Draw on Figure 1 the fundamental standing wave that will appear on the string when the
note is sounding.
(1)
(b)
(i)
Sketch on Figure 2 the standing wave that corresponds to a frequency of three times
that of the fundamental.
Figure 2
(ii)
State the name given to points on the standing wave where there is no vibration of
the string.
...............................................................................................................
(2)
Page 9 of 56
(c)
Children often learn to play the violin on a small instrument with shorter strings. These
shorter strings have to produce the same fundamental frequencies as those on the full-size
instrument. State two ways in which this can be achieved.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(2)
(Total 5 marks)
11
The figure below shows a graph of displacement against time for two waves A and B. These
waves meet in phase and add to form a resultant wave.
(a)
State the amplitude of the resultant wave
........................................................................................................................
(1)
Page 10 of 56
(b)
Calculate the ratio
intensity of wave B : intensity of wave A.
(2)
(Total 3 marks)
12
The figure below shows the appearance of a stationary wave on a stretched string at one instant
in time. In the position shown each part of the string has its maximum displacement. The arrow at
W shows the direction in which the point W is about to move.
(a)
(i)
Mark clearly on the diagram the directions in which points X, Y and Z are about to move.
(ii)
State the conditions necessary for a stationary wave to be produced on the string.
...............................................................................................................
...............................................................................................................
(4)
(b)
In the figure above, the frequency of vibration is 120 Hz. Calculate the frequency of the
fundamental vibration for this string.
frequency of the fundamental vibration...................................................
(2)
(Total 6 marks)
Page 11 of 56
13
Which one of the following statements about stationary waves is true?
A
Particles between adjacent nodes all have the same amplitude.
B
Particles between adjacent nodes are out of phase with each other.
C
Particles immediately on either side of a node are moving in opposite directions.
D
There is minimum disturbance of the medium at an antinode.
(Total 1 mark)
14
Explain the differences between an undamped progressive transverse wave and a stationary
transverse wave, in terms of (a) amplitude, (b) phase and (c) energy transfer.
(a)
amplitude
progressive wave ...........................................................................................
........................................................................................................................
stationary wave ..............................................................................................
........................................................................................................................
(b)
phase
progressive wave ...........................................................................................
........................................................................................................................
stationary wave ..............................................................................................
........................................................................................................................
(c)
energy transfer
progressive wave ...........................................................................................
........................................................................................................................
stationary wave ..............................................................................................
........................................................................................................................
(Total 5 marks)
15
frequency of vibration = 50 Hz
The diagram above shows a stationary wave on a stretched string at a time t = 0. Which one of
the diagrams, A to D, correctly shows the position of the string at a time t = 0.010 s?
Page 12 of 56
A
B
C
D
(Total 1 mark)
16
The graph shows the variation of displacement of the particles with distance along a stationary
transverse wave at time t = 0 when the displacement of the particles is greatest. The period of
the vibrations causing the wave is 0.040 s.
(a)
Using the same axes,
(i)
draw the appearance of the wave at t = 0.010 s, labelling this graph B,
(ii)
draw the appearance of the wave at t = 0.020 s, labelling this graph C,
(iii)
show an antinode labelled A and a node labelled N.
(3)
Page 13 of 56
(b)
(i)
Describe the motion of the particle at V, giving its frequency and amplitude.
...............................................................................................................
...............................................................................................................
(ii)
State the amplitude of the particle at W and its phase relations with the particle at V
and the particle at Z.
...............................................................................................................
...............................................................................................................
...............................................................................................................
(6)
(Total 9 marks)
17
Which one of the following statements about stationary waves is true?
A
Particles between adjacent nodes all have the same amplitude.
B
Particles between adjacent nodes are out of phase with each other.
C
Particles immediately on either side of a node are moving in opposite directions.
D
There is a minimum disturbance of the medium at an antinode.
(Total 1 mark)
18
A uniform wire fixed at both ends is vibrating in its fundamental mode. Which one of the following
statements is not correct for all the vibrating particles?
A
They vibrate in phase.
B
They vibrate with the same amplitude.
C
They vibrate with the same frequency.
D
They vibrate at right angles to the wire.
(Total 1 mark)
19
Stationary waves are set up on a length of rope fixed at both ends. Which one of the following
statements is true?
A
Between adjacent nodes, particles of the rope vibrate in phase with each other.
B
The mid point of the rope is always stationary.
C
Nodes need not necessarily be present at each end of the rope.
D
Particles of the rope at adjacent antinodes always move in the same direction.
(Total 1 mark)
Page 14 of 56
20
Which line, A to D, in the table gives a correct difference between a progressive wave and a
stationary wave?
progressive wave
stationary wave
all the particles vibrate
some of the particles do not
vibrate
B
none of the particles vibrate
with the same amplitude
all the particles vibrate with
the same amplitude
C
all the particles vibrate in
phase with each other
none of the particles vibrate in
phase with each other
D
some of the particles do not
vibrate
all the particles vibrate in
phase with each other
A
(Total 1 mark)
21
Figure 1 represents a stationary wave formed on a steel string fixed at P and Q when it is
plucked at its centre.
Figure 1
(a)
Explain why a stationary wave is formed on the string.
......................................................................................................................
......................................................................................................................
......................................................................................................................
......................................................................................................................
......................................................................................................................
(3)
(b)
(i)
The stationary wave in Figure 1 has a frequency of 150 Hz. The string PQ has a
length of 1.2 m.
Calculate the wave speed of the waves forming the stationary wave.
Answer ........................... m s–1
(2)
Page 15 of 56
(ii)
On Figure 2, draw the stationary wave that would be formed on the string at the
same tension if it was made to vibrate at a frequency of 450 Hz.
Figure 2
(2)
(Total 7 marks)
22
A stationary wave is formed by two identical waves of frequency 300 Hz travelling in opposite
directions along the same line. If the distance between adjacent nodes is 0.60 m, what is the
speed of each wave?
A
180 m s−1
B
250 m s−1+
C
360 m s−1
D
500 m s−1
(Total 1 mark)
23
A microwave transmitter is used to direct microwaves of wavelength 30 mm along a line XY. A
metal plate is positioned at right angles to XY with its mid-point on the line, as shown.
When a detector is moved gradually along XY, its reading alternates between maxima and
minima. Which one of the following statements is not correct?
A
The distance between two minima could be 15 mm.
B
The distance between two maxima could be 30 mm.
C
The distance between a minimum and a maximum could be 30 mm.
D
The distance between a minimum and a maximum could be 37.5 mm.
(Total 1 mark)
Page 16 of 56
24
In testing a particular type of guitar string, a string is stretched and vibrated for a long period of
time using a mechanical vibrator as shown in Figure 1. The right-hand end of the string is fixed.
A stationary wave is produced on the string; the string vibrates in two loops.
Figure 1
(a)
State the conditions that are necessary for a stationary wave to form on the string.
......................................................................................................................
......................................................................................................................
......................................................................................................................
......................................................................................................................
(3)
(b)
Explain how you know that the wave on the string is transverse.
......................................................................................................................
(1)
(c)
Compare the amplitude and phase of the oscillations of points A and B on the string.
Amplitude ....................................................................................................
Phase ..........................................................................................................
(2)
(d)
The length of the string is 1.2 m and the speed of the transverse wave on the string is
6.2 m s–1.
Calculate the vibration frequency of the vibrator in Hz.
Vibration frequency ...........................Hz
(3)
Page 17 of 56
(e)
(i)
The frequency of the vibrator is tripled.
Sketch the new shape of the stationary wave on Figure 2.
Figure 2
(ii)
Show on your diagram three points P, Q and R that oscillate in phase.
(2)
(Total 11 marks)
25
(a)
(i)
A piano string has a tension of 681 N. It vibrates with a fundamental frequency (first
harmonic) of 92.5 Hz and has a mass per unit length of 1.87 × 10–2 kg m–1.
Calculate the length of the string.
.............................................................................................................
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.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
length of string ...................................... m
(3)
Page 18 of 56
(ii)
The figure below shows a string stretched between fixed ends.
Draw onto the figure the first overtone (second harmonic) mode of vibration.
(1)
(iii)
State how you could make a string on a stringed instrument vibrate in this mode of
vibration.
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
.............................................................................................................
(2)
Page 19 of 56
(b)
Describe how you would investigate the variation of the fundamental frequency (first
harmonic) of a string with its length.
State which variable(s) you would need to control and how you would do so.
You may wish to assist your account by drawing a diagram.
......................................................................................................................
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......................................................................................................................
......................................................................................................................
......................................................................................................................
(4)
(Total 10 marks)
Page 20 of 56
26
Figure 1 shows a side view of a string on a guitar. The string cannot move at either of the two
bridges when it is vibrating. When vibrating in its fundamental mode the frequency of the sound
produced is 108 Hz.
(a)
(i)
On Figure 1, sketch the stationary wave produced when the string is vibrating in its
fundamental mode.
Figure 1
(1)
(ii)
Calculate the wavelength of the fundamental mode of vibration.
answer = ........................................... m
(2)
(iii)
Calculate the speed of a progressive wave on this string.
answer = ...................................... m s–1
(2)
Page 21 of 56
(b)
While tuning the guitar, the guitarist produces an overtone that has a node 0.16 m from
bridge A.
(i)
On Figure 2, sketch the stationary wave produced and label all nodes that are
present.
Figure 2
(2)
(ii)
Calculate the frequency of the overtone.
answer = ...................................... Hz
(1)
(c)
The guitarist needs to raise the fundamental frequency of vibration of this string.
State one way in which this can be achieved.
......................................................................................................................
......................................................................................................................
(1)
(Total 9 marks)
Page 22 of 56
27
(a)
State two differences between stationary waves and progressive waves.
first difference ................................................................................................
........................................................................................................................
........................................................................................................................
second difference ..........................................................................................
........................................................................................................................
........................................................................................................................
(2)
(b)
A violin string has a length of 327 mm and produces a note of frequency 440 Hz.
Calculate the frequency of the note produced when the same string is shortened or
“stopped” to a length of 219 mm and the tension remains constant.
frequency ................................................. Hz
(2)
(Total 4 marks)
Page 23 of 56
28
When a note is played on a violin, the sound it produces consists of the fundamental and many
overtones.
Figure 1 shows the shape of the string for a stationary wave that corresponds to one of these
overtones. The positions of maximum and zero displacement for one overtone are shown. Points
A and B are fixed. Points X, Y and Z are points on the string.
Figure 1
(a)
(i)
Describe the motion of point X.
...............................................................................................................
...............................................................................................................
...............................................................................................................
...............................................................................................................
(2)
(ii)
State the phase relationship between
X and Y .................................................................................................
X and Z .................................................................................................
(2)
(b)
The frequency of this overtone is 780 Hz.
(i)
Show that the speed of a progressive wave on this string is about 125 ms–1.
(2)
(ii)
Calculate the time taken for the string at point Z to move from maximum displacement
back to zero displacement.
answer = ................................... s
(3)
Page 24 of 56
(c)
The violinist presses on the string at C to shorten the part of the string that vibrates.
Figure 2 shows the string between C and B vibrating in its fundamental mode. The length
of the whole string is 320 mm and the distance between C and B is 240 mm.
Figure 2
(i)
State the name given to the point on the wave midway between C and B.
...............................................................................................................
(1)
(ii)
Calculate the wavelength of this stationary wave.
answer = ................................... m
(2)
(iii)
Calculate the frequency of this fundamental mode. The speed of the progressive
wave remains at 125 ms–1.
answer = .................................Hz
(1)
(Total 13 marks)
Page 25 of 56
29
The figure below shows a continuous progressive wave on a rope. There is a knot in the rope.
(a)
Define the amplitude of a wave.
........................................................................................................................
........................................................................................................................
(2)
(b)
The wave travels to the right.
Describe how the vertical displacement of the knot varies over the next complete cycle.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(c)
A continuous wave of the same amplitude and frequency moves along the rope from the
right and passes through the first wave. The knot becomes motionless.
Explain how this could happen.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(3)
(Total 8 marks)
Page 26 of 56
30
Which of the following is correct for a stationary wave?
A
Between two nodes the amplitude of the wave is constant.
B
The two waves producing the stationary wave must always be 180° out of
phase.
C
The separation of the nodes for the second harmonic is double the
separation of nodes for the first harmonic.
D
Between two nodes all parts of the wave vibrate in phase.
(Total 1 mark)
Page 27 of 56
31
Figure 1 and Figure 2 show a version of Quincke’s tube, which is used to demonstrate
interference of sound waves.
Figure 1
Figure 2
A loudspeaker at X produces sound waves of one frequency. The sound waves enter the tube
and the sound energy is divided equally before travelling along the fixed and movable tubes. The
two waves superpose and are detected by a microphone at Y.
(a)
The movable tube is adjusted so that d1 = d2 and the waves travel the same distance from
X to Y, as shown in Figure 1. As the movable tube is slowly pulled out as shown in Figure
2, the sound detected at Y gets quieter and then louder.
Explain the variation in the loudness of the sound at Y as the movable tube is slowly pulled
out.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(4)
Page 28 of 56
(b)
The tube starts in the position shown in Figure 1.
Calculate the minimum distance moved by the movable tube for the sound detected at Y to
be at its quietest.
frequency of sound from loud speaker = 800 Hz
speed of sound in air = 340 m s–1
minimum distance moved = ............................... m
(3)
(c)
Quincke’s tube can be used to determine the speed of sound.
State and explain the measurements you would make to obtain a value for the speed of
sound using Quincke’s tube and a sound source of known frequency.
........................................................................................................................
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........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(4)
(Total 11 marks)
32
(a)
Musical concert pitch has a frequency of 440 Hz.
A correctly tuned A-string on a guitar has a first harmonic (fundamental frequency) two
octaves below concert pitch.
Determine the first harmonic of the correctly tuned A-string.
frequency................................................. Hz
(1)
Page 29 of 56
(b)
Describe how a note of frequency 440 Hz can be produced using the correctly tuned
A-string of a guitar.
........................................................................................................................
........................................................................................................................
(1)
(c)
Describe the effect heard when notes of frequency 440 Hz and 430 Hz of similar amplitude
are sounded together.
........................................................................................................................
........................................................................................................................
........................................................................................................................
........................................................................................................................
(2)
(Total 4 marks)
Page 30 of 56
Mark schemes
Page 31 of 56
1
The student’s writing should be legible and the spelling, punctuation and grammar should
be sufficiently accurate for the meaning to be clear.
The student’s answer will be assessed holistically. The answer will be assigned to one of
three levels according to the following criteria.
Answers may cover some of the following points:
•
(1) a wave and its reflection / waves travelling in opposite directions
meet / interact / overlap / cross / pass through etc
•
•
point (1) must be stated together i.e it should not be necessary to
search the whole script to find the two parts namely the directions of
the waves and their meeting
(2) same wavelength (or frequency)
(3) node − point of minimum or no disturbance
•
points (3) may come from a diagram but only if the node is written in
full and the y-axis is labelled amplitude or displacement
(4) antinode − is a point of maximum amplitude
•
•
•
•
point (4) may come from a diagram but only if the antinode is
written in full and the y-axis is labelled amplitude or displacement
(5) node - two waves (always) cancel / destructive interference / 180° phase
difference / in antiphase [out of phase is not enough] (of the two waves at the
node) [not peak meets trough]
(6) antinode − reinforcement / constructive interference occurs /
(displacements) in phase
(7) mention of superposition [not superimpose] of the two waves
(8) energy is not transferred (along in a standing wave).
if any point made appears to be contradicted elsewhere the point is
lost − no bod’s
High Level (Good to excellent): 5 or 6 marks
The information conveyed by the answer is clearly organised, logical and coherent, using
appropriate specialist vocabulary correctly. The form and style of writing is appropriate to
answer the question.
6 marks: points (1) AND (2) with 4 other points which must include point (4) or the passage
must indicate that the wave is oscillating at an antinode
5 marks: points (1) AND (2) with any three other points
although point (1) may not be given as a mark the script can be
searched to see if its meaning has been conveyed as a whole
before restricting the mark and not allowing 5 or 6 marks
Intermediate Level (Modest to adequate): 3 or 4 marks
The information conveyed by the answer may be less well organised and not fully coherent.
There is less use of specialist vocabulary, or specialist vocabulary may be used incorrectly.
The form and style of writing is less appropriate.
4 marks: (1) OR (2) AND any three other points
3 marks: any three points
Low Level (Poor to limited): 1 or 2 marks
The information conveyed by the answer is poorly organised and may not be relevant or
Page 32 of 56
coherent. There is little correct use of specialist vocabulary.
The form and style of writing may be only partly appropriate.
2 marks: any two points
1 marks: any point or a reference is made to both nodes and antinodes
[6]
2
(a)
(i)
one loop shown
B1
(1)
(ii)
0.68 m
B1
(1)
(iii)
two equal length loops shown
B1
(1)
(b)
frequency increases
B1
(1)
[4]
3
(a)
(i)
d / 7.5 and d / 5.0 (denominator may be in m s–1)
or d / 7.5 and d / 7.5 + 1.5
or d / 5.0 – .5 and d / 5.0
B1
(1)
(ii)
d / 7.5 + 1.5 = d / 5.0
C1
22.5 (22 – 23) km
A1
(2)
(b)
(i)
interference / superposition of waves (condone waves superimpose) of:
same frequency travelling in opposite directions
or an incident and a reflected wave
B1
idea of a resonant length
eg length of string is a whole number of half wavelengths of the wave
or length such as to produce nodes and antinodes
or fixed ends are nodes
B1
(2)
Page 33 of 56
(ii)
wavelength of fundamental = 64 m
C1
v = fλ
C1
2.8 Hz
A1
(3)
(iii)
(natural / fundamental) frequency of oscillation of the new spans
= 2 × (ii) (5.6 Hz) or twice original frequency
or wavelength is half the original wavelength(= 32 m)
M1
clear link and conclusion shown between the new natural frequency of the
spans
and the max frequency of the earthquake
examples:
second calculation plus conclusion that resonant vibrations would
not take place
or calculation and comparison of the wavelength of the
earthquake wave travelling along the bridge and the resonant
wavelength (42 m and 32 m)
A1
(2)
[10]
4
Two of
B1
Mass or mass/unit length
B1
Tension
Length
Temperature
[2]
Page 34 of 56
5
(a)
λ = 0.58(m) or (2 / 3 × 0.87)
C1
c = fλ or substituted values
C1
69.6 (70) m s–1
A1
(3)
(b)
λ = 0.87 × 2 or λ = 1.74 or in formula
M1
69.6 / 1.74 or 70 / 1.74 = 40.2
A1
(2)
or
The drawing shows third harmonic (second overtone)
M1
so 120 = 3 × f0 so f0 = 40 Hz
do not allow just 120 / 3
A1
[5]
6
(a)
tension – newtonmeter
B2
or tension – from mass on balance
B1
and – multiply by g
B1
mass – balance / scales
B1
length – rule / tape / ruler
B1
(4)
(b)
frequency read from signal generator when standing wave produced / use of strobe
etc.
B1
measure λ using several loops or full length of string
B1
node → node / each loop = λ / 2
B1
use of c = fλ
B1
(4)
Page 35 of 56
(c)
λ = 0.40 (m)
C1
c = 60.8 (m
s–1)
e.c.f. from λ
C1
T = 7.06 (N)
C1
μ = 1.9(1) ×
10–3
(kg
m–1)
c.a.o.
A1
m = 2 × μ value (= 3.8 ×
10–3
kg or equivalent unit) e.c.f. s.f.p. applied only at this
answer
B1
(5)
[13]
7
(a)
Reflection implied /2 waves in opposite directions/fixed
end (not ends)
B1
Similar amplitude/little energy loss at wall
B1
frequency constant or same frequency/wavelength or correct
wavelength condition specified
B1
3
(b)
displacement perpendicular to rest/average/mean position
of string
or string displacement perpendicular to energy propagation
direction OWTTE
B1
1
(c)
A larger than B
B1
A180° /π out of phase with B OWTTE
B1
2
Page 36 of 56
(d)
λ = 1.2
B1
c = f λ; allow e.c.f. from wrong λ
M1
f = 6.2/1.2 = 5.2 Hz
A1
3
(e)
(i)
diagram correct (6 loops)
B1
1
(ii)
Q and R correctly in phase with P; must be a
position where vt occurs
B1
1
[11]
8
(a)
superposition (of progressive waves)
B1
incident wave and reflected wave/wave reflected through
180O/waves travelling in opposite directions
B1
same frequency/wavelength
B1
in same medium.
B1
Any 3 out of 4 points
3
Page 37 of 56
(b)
f = c/λ
C1
λ = 1.24
C1
f = 258 Hz
A1
e.g. f = 512 gets 1 mark
3
[6]
9
(a)
idea that there are waves in opposite directions
B1
because of reflection at end of string
B1
the two waves interfere with each other/superimpose
B1
3
(b)
(i)
one loop
M1
with N at each end and A in the middle
A1
2
(ii)
4 approximately even loops
B1
all nodes and antinodes correctly marked for their
number of loops
B1
2
Page 38 of 56
(c)
(i)
length halved/0.35 (m)
B1
1
(ii)
tension greater
C1
T = 720 (N)/increased by a factor of 4
A1
2
[10]
10
(a)
single loop/half of sine wave shown between fixed points
B1
1
(b)
(i)
3 loops shown
B1
(ii)
node
B1
2
(c)
greater mass per unit length of string/thicker string
B1
less tension/loosen string/slacken string
B1
2
[5]
Page 39 of 56
11
(a)
4 mm
B1
1
(b)
3:1; 3/1
C1
9 or 9:1
A1
2
[3]
12
(a)
(i)
Z down
B1
X and Y up
B1
(ii)
any two of:
same frequency/wavelength not ‘it has same frequency’
moving in opposite directions,
reflected at end of string,
same/similar amplitude
integer no of ½ wavelengths between walls
B2
4
(b)
indicates f is 3 times fundamental in some way or that
length is 3λ/2
C1
40 Hz
A1
2
[6]
13
C
[1]
Page 40 of 56
14
amplitude:
each point along wave (1)
has same amplitude for progressive wave
but varies for stationary wave (1)
phase:
progressive wave, adjacent points vibrate with different phase (1)
stationary wave, between nodes all particles vibrate in phase
[or there are only two phases] (1)
energy transfer:
progressive wave, energy is transferred through space (1)
stationary wave, energy is not transferred through space (1)
[5]
15
16
C
[1]
(a)
(i)
B line along distance axis (1)
(ii)
C negative sine wave starting at O (1)
(iii)
A, N (1)
(3)
(b)
(i)
s.h.m. [or particle stationary] (1)
amplitude = 20 mm (1)
= 25 Hz or s–1 (1)
(ii)
10 mm (1)
W, V phase difference π [or antiphase or 180°] (1)
W, Z in phase (1)
(6)
[9]
17
18
19
20
C
[1]
B
[1]
A
[1]
A
[1]
Page 41 of 56
21
(a)
(progressive waves travel from centre) to ends and reflect (1)
two (progressive) waves travel in opposite directions along the string (1)
waves have the same frequency (or wavelength) (1)
waves have the same (or similar) amplitude (1)
superposition (accept ‘interference’) (1)
max 3
(b)
(i)
wavelength (= 2 × PQ = 2 × 1.20 m) = 2.4 m (1)
speed (= wavelength × frequency = 2.4 × 150) = 360 m s–1 (1)
(answer only gets both marks)
(ii)
diagram to show three ‘loops’ (1) and of equal length and
good shape (1) (or loop of one third length (1))
4
[7]
22
23
24
C
[1]
C
[1]
(a)
reflection implied/2 waves in opposite directions/fixed end (not ends) (1)
similar amplitude/little energy loss at wall (1)
frequency constant or same frequency/wavelength or correct wavelength
condition specified (1)
3
(b)
displacement perpendicular to rest/average/mean position of string
or string displacement perpendicular to energy propagation
direction owtte (1)
1
(c)
A larger than B (1)
A 180° (or π rad) out of phase with B (owtte) (1)
2
Page 42 of 56
(d)
λ = 1.2 (1)
c = f λ; allow e.c.f from wrong λ (1)
f = 6.2/1.2 = 5.2 Hz (1)
3
(e)
(i)
diagram correct (6 loops) (1)
(ii)
Q and R correctly in phase with P; must be a position where
movement occurs (1)
2
[11]
25
(a)
(i)
rearrangement of f =
correct subs l =
to give l =
C1
or 92.5 =
C1
1.0(3) (m) condone sf
A1
3
(ii)
2 loops roughly equal
B1
1
(iii)
(lightly) stop (in centre)
B1
pluck or bow
B1 2
Page 43 of 56
(b)
keeps tension or mass per unit length constant
B1
way of measuring frequency or producing vibration of known f
B1
way of measuring length (at resonance)
B1
use of suitable graph (f vs 1/l or l vs 1/f) to display results
B1
marks may be awarded for information seen on diagram
4
[10]
26
(a)
(i)
one ‘loop’ (accept single line only, accept single dashed line)
+ nodes at each bridge (± length of arrowhead)
+ antinode at centre (1)
1
(ii)
λ0 = 2L or λ = 0.64 × 2 (1)
= 1.3 (m) (1) (1.28)
2
(iii)
(c = f λ) = 108 × (a)(ii) (1)
= 138 to 140(.4) (m s–1) (1) ecf from (a) (ii)
2
(b)
(i)
four antinodes (1) (single or double line)
first node on 0.16 m (within width of arrowhead)
+ middle node between the decimal point and the centre of the
‘m’ in ‘0.64 m’
+ middle 3 nodes labelled ‘N’, ‘n’ or ‘node’
(1)
2
Page 44 of 56
(ii)
(4 f0 =) 430 (Hz) (1) (432)
or use of f =
gives 430 to 440 Hz
correct answer only, no ecf
1
(c)
decrease the length/increase tension/tighten string (1)
1
[9]
27
(a)
max 2 from
in progressive waves, all points have the same amplitude (in turn),
in stationary waves, they do not
B1
in stationary waves, points between nodes are in phase, in progressive
waves, all points within one wavelength are out of phase with
each other
B1
in stationary waves, there is no energy transfer along the wave,
in progressive waves, there is
B1
stationary waves have nodes and antinodes but progressive waves do not
B1
where there are single relevant statements but no clear comparison
between stationary and compressive waves, award 1 mark for
two such statements
2
Page 45 of 56
(b)
f α 1/l orƒ=
or fl = const
C1
657/660 (Hz)
A1
2
[4]
28
(a)
(i)
oscillates / vibrates
(allow goes up and down / side to side / etc, repeatedly, continuously, etc)
about equilibrium position / perpendicularly to central line
2
(ii)
X and Y: antiphase / 180 (degrees out of phase) / п (radians out of phase)
X and Z: in phase / zero (degrees) / 2п (radians)
2
(b)
(i)
v = fλ
= 780 x 0.32 / 2 or 780 x 0.16 OR 780 x 320 / 2 or 780 x 160
THIS IS AN INDEPENDENT MARK
= 124.8
(m s–1) correct 4 sig fig answer must be seen
2
(ii)
¼ cycle
T = 1 / 780 OR = 1.28 × 10–3
0.25 × 1.28 × 10–3
= 3.2 × 10–4 (s)
Allow correct alternative approach using distance of 0.04m
travelled by progressive wave in ¼ cycle divided by speed.
0.04 /125
= 3.2 × 10–4 (s)
3
(c)
(i)
antinode
1
Page 46 of 56
(ii)
2 x 0.240
= 0.48 m
‘480m’ gets 1 mark out of 2
2
(iii)
(f = v/λ = 124.8 or 125 / 0.48 ) = 260 (Hz) ecf from cii
1
[13]
29
(a)
the maximum displacement (of the wave or medium)
from the equilibrium position
accept ‘rest position’, ‘undisturbed position’, ‘mean position’
2
(b)
(vertically) downwards (¼ cycle to maximum negative displacement)
then upwards (¼ cycle to equilibrium position and ¼ cycle to maximum
positive displacement)
down (¼ cycle) to equilibrium position/zero displacement and correct
reference to either maximum positive or negative displacement or correct
reference to fractions of the cycle
candidate who correctly describes the motion of a knot 180 degrees out of
phase with the one shown can gain maximum two marks
(ie knot initially moving upwards)
3
(c)
max 3 from
stationary wave formed
by superposition or interference (of two progressive waves)
knot is at a node
waves (always) cancel where the knot is
allow ‘standing wave’
3
[8]
30
D
[1]
Page 47 of 56
31
(a)
Initially the path difference is zero/the two waves are in phase when they meet/the
(resultant) displacement is a maximum ✓
Alternative:
Constructive interference occurs when the path difference is a
whole number of wavelengths and the waves are in phase
1
As the movable tube is pulled out, the path difference increases and the two waves
are no longer in phase, so the displacement and loudness decrease ✓
Destructive interference occurs when the path difference is an odd
number of half wavelengths and the waves are in antiphase
1
When the path difference is one half wavelength, the two are in antiphase and sound
is at its quietest. ✓
Initially the path difference is zero and the sound is loud
1
As the path difference continues to increase, the two waves become more in phase
and the sound gets louder again. ✓
As the pipe is pulled out the path difference gradually increases,
changing the phase relationship and hence the loudness of the
sound
1
(b)
Use of wavelength = speed / frequency
The first mark is for calculating the wavelength
1
To give: 340 / 800 = 0.425 m ✓
Path difference = one half wavelength = 0.21 m ✓
The second mark is for relating the wavelength to the path
difference
Path difference = 2 (d2 – d1) = 2 (distance moved by movable tube)
1
Distance moved by movable tube = 0.10 m. ✓
The final mark is for relating this to the distance moved by the tube
and working out the final answer.
1
(c)
Start with d1 = d2
(Alternative mark scheme involving changing frequency and
measuring to first min for each one can gain equal credit)
Measure distance moved by movable tube for each successive minima and maxima✓
Start with d1 = d2
Measure distance moved by movable tube for first minimum.
1
Page 48 of 56
Each change in distance is equal to one quarter wavelength. ✓
Distance is equal to one quarter wavelength
1
Continue until tube is at greatest distance or repeat readings for decreasing distance
back to starting point. ✓
Repeat for different measured frequencies.
1
Use speed = frequency x wavelength ✓
Use speed = frequency x wavelength)
1
[11]
32
(a)
110 Hz
B1
1
(b)
(Use finger on the fret so that) a ¼ length of the string is used to sound the note or
hold string down on 24th fret
B1
1
(c)
Mention or description of beats or description of rising and falling amplitude / louder
and quieter
Regular rising and falling of loudness owtte
B1
B1
Beat frequency 10(.0Hz) Allow beat frequency = 430 - 420
2
[4]
Page 49 of 56
Examiner reports
1
4
5
6
Almost all students made a good effort at answering this question and almost all of those knew
that standing waves are constructed from two waves. This being the case it was appropriate that
this question was the basis of the quality of written communication assessment in this
examination. Weaker students often spent too long setting the scene. They gave details of the
apparatus and explained how the string was plucked or vibrated before the bullet points were
addressed. Often at this stage these were answered with very brief responses that gave very
little detail. The middle ability group of students fared much better. They could describe what
nodes and antinodes were and how they came about in terms of the interference of two waves.
What was often missing was the fact that the two waves that superpose have the same
frequency or wavelength. Many of this group and a large percentage of the top ability group
understood that an antinode was a maximum of the motion but they referred to the maximum
displacement rather than the maximum amplitude. A couple of points separated this top group
from the middle students as well as the quality of the structure of their writing and spelling. First
they referred to the waves superposing unlike the majority who thought the waves superimposed
on each other. Secondly, they sometimes included a point about the lack of energy transmission
in a standing wave.
There were many correct answers to this question. Incorrect responses were usually couched in
terms of the amplitude of the wire and the frequency of the signal.
(a)
The majority of candidates were able to calculate the speed of the transverse wave,
correctly calculating the wavelength and multiplying their value by the frequency.
(b)
Candidates going on to use their value of speed were usually successful. However those
candidates trying to argue in terms of fundamental and third harmonic (or second overtone)
were often unconvincing and rarely gained credit for their answers.
(a)
The majority of candidates were able to suggest what quantities needed to be measured in
order to calculate the speed of the waves however many of these failed to give suitable
measuring instruments with which to make the measurements.
Surprising few candidates suggested using a newton meter in order to measure the tension
in the string.
(b)
This was often too loosely answered with insufficient detail of how multiple loops should be
used in order to measure an accurate value for the wavelength. Again many candidates
failed to explain that the calibrated signal generator would give a value for the frequency of
the wave.
(c)
This was often the part most successfully answered by candidates. Although many gave a
value for the wavelength of 0.20 m, with error carried forward this still gave access to the
remaining four marks. Only the strongest candidates tended to multiply the value for μ by
the 2 m that gave the total mass of the string.
Page 50 of 56
7
8
Candidates showed a misunderstanding of the concept of a standing wave throughout the whole
question.
(a)
Many ‘physical’ conditions were encountered in scripts, most of them spurious and
incorrect. Candidates seemed unable to distinguish between a standing and a travelling
wave. Responses here indicated that an understanding of reflection, or two waves in
motion, or the need for appropriate boundary conditions, or even the requirement for
constraints on frequency were rare.
(b)
General misunderstandings carried through to this part also where statements intended to
show an understanding of why the wave was stationary were poor and unfocussed.
(c)
A sizeable minority were unable to compare the amplitude of A and B correctly. Comments
such as ‘A and B are not quite in phase’ indicated further misunderstandings of exactly
what the wave is doing.
(d)
The calculation was carried out well even by those who could not translate the diagram into
a correct wavelength (2.4 m and 0.6 m were both popular distractors).
(e)
(i)
Many candidates were able to draw a correct new shape for the stationary wave.
Those who failed usually drew three ‘loops’ rather than six.
(ii)
The crucial point in the question was that the three points had to oscillate in phase.
Those candidates who identified nodal displacements points penalised themselves
accordingly.
Part (a) was generally not answered well; in part because the conditions were not known, but
also as result of poor writing skills.
Failure to realise that the wavelength was twice the length of the guitar string was a very
common error in part (b). Of the candidates who did get the calculation right, many lost a mark
for quoting too many significant figures in the final answer.
9
(a)
Most candidates had some idea of the formation of a standing wave. They were unclear
about where reflection occurred or failed to mention the superposition of the two waves.
Some candidates thought that the two waves were always 180° out of phase.
(b)
(i)
This was usually correct.
(ii)
This was not done quite as well as some candidates misidentified the fourth harmonic
and some failed to label all of the nodes.
Page 51 of 56
(c)
10
This part was usually done correctly.
(ii)
A large number of candidates correctly stated that the tension should be quadrupled.
A few simply stated that it should be increased and some others thought it should be
doubled. Both of these responses gained partial credit.
(a)
Almost all candidates were able to draw the pattern of the fundamental standing wave on a
stretched string.
(b)
(i)
The drawn standing wave patterns for a frequency three times higher were good.
(ii)
The use of the word ‘node’ as a descriptor for the point of no displacement was
recognised well, only a handful were wrong.
(c)
11
(i)
Again, many were able to translate their knowledge of the factors influencing the frequency
of the string to this slightly unusual case of the miniature violin. They fluently and concisely
described appropriate changes to the mass per unit length of the string (increase) and to
the tension (decrease). Some, however, lost marks because they simply stated the factors
rather than the direction in which they needed to change. The question clearly implied a
requirement for this and this was recognised by the vast majority of candidates.
Many candidates were able to complete the task of adding together two amplitudes from the
graph. For the minority, 2 mm rather than 4 mm was a very common error. Graph labels provided
a substantial number of unit errors in this question.
Few of the candidates were comfortable with the calculation of intensity ratio from the graph.
Examiners saw many examples of 3:1 (the amplitude ratio), usually with no explanation for the
origin of the ratio. In other cases the ratio was calculated the wrong way around (1:3 or 1:9) or
candidates worked on the basis of an inverse-square law calculation.
12
(a)
(i)
Almost half the candidates revealed that they could not describe the motion of a
stretched string exhibiting a stationary (standing) wave.
(ii)
However, the written description of the necessary conditions for the string to
demonstrate the motion was better with many candidates giving a comprehensive list
(two correct conditions were required as a minimum for full marks).
Page 52 of 56
(b)
14
16
Correct calculations of the frequency of the fundamental were surprisingly rare. There is
widespread confusion about the wavelength conditions for the stationary wave. Common
incorrect answers included 80Hz (where the candidate thinks the string is 3λ long) and
360Hz (candidate multiplies rather than divides by 3). Examiners were surprised to find that
a significant minority understood the physics and implied that they intended to divide 120
by 3, but commonly obtained an answer of 60 Hz for the result.
It was evident that the depth of knowledge necessary to answer this question was not available
to the majority of candidates. Even the energy transfer section in part (c) was the source of
wrong or vague or inadequate answers.
In part (a) there was much confusion with progressive waves and only the better candidates
showed B correctly.
In part (b)(i) few candidates said that the particle at V performed simple harmonic motion. Credit
was given if it was stated that the particle was instantaneously at rest. In part (b)(ii) most
candidates knew the meaning of amplitude but the question about phase often defeated weaker
candidates. with more confusion with progressive waves
21
A large number of candidates struggled with part (a). This was mainly due to a lack of
understanding of the fact that two waves must be travelling in opposite directions in order for a
standing wave to form. They seemed to be describing one wave reflecting back and forth. Those
who understood how the stationary wave formed and added further detail went on to score two or
three marks fairly easily.
Some candidates in part (b) (i) did not multiply by two and only scored one mark out of the two
available.
A majority gained two marks in part (b) (ii). A few candidates knew what to do but their sketch
lacked acceptable accuracy, for example, the ‘loops’ were not of similar length. Only a quarter of
candidates got the wavelength wrong.
25
Most candidates did the calculation well. Those that did not usually did not correctly rearrange
the equation. Once again, those who set out calculations well tend to be more successful than
who are untidy or who miss out steps. Candidate’s drawings of the oscillation tended to be
correct. A few got it wrong because they could not identify the correct oscillation but more lost the
mark because their loops were obviously far from equal in size. Candidates should be advised to
take care with their answers. Measuring the sixes of loops would not be inappropriate. Few
candidates answered part (a) (iii) well. Stopping was mentioned and allowed even when the
candidates did not indicate that this should be done lightly.
Page 53 of 56
Part (b) was extremely badly done. There were some diagrams that showed more or less
appropriate experimental apparatus but more that showed unfamiliarity with any sensible
experiment. Candidates tended not to know the names of apparatus such as oscillators or signal
generators. Very few mentioned that the length of the vibrating string should be measured using
a metre rule or suitable alternative. Those that mentioned that a graph should be drawn almost
always failed to suggest an appropriate graph that would yield a straight line. When describing
experiments, candidates should be aware of the need to state the measurements that should be
made; the measuring equipment used; which variables to control and how; how to display the
results in a straight line graph.
26
In part (a) (i), about 60% of candidates drew one ‘loop’ and picked up the mark. However, we
were fairly lenient on the shape of the ‘loop’ and students need to practice drawing these shapes.
Part (a) (ii) was expected to be a little easier than it was. 42% scored no marks on this despite
the benefit of an error carried forward from an incorrect part (a) (i). Many did not realise the
wavelength was found from the length of the string and knowledge of the shape of the
fundamental. Some candidates used λ = v/f with v = speed of light. In contrast, most candidates
found part (a) (iii) a very easy calculation.
The majority of candidates got four antinodes in part (b) (i), but then nearly half of those lost the
second mark by either not sketching the curve carefully enough or, more commonly, forgetting to
label the antinodes.
In part (c), the vast majority correctly suggested tightening or shortening the string. A few thought
that plucking harder would increase the pitch and some suggested increasing the length, using a
thinner string, increasing the wave speed, or even ‘play faster’.
27
In part (a), many students could not make a distinction between progressive and stationary
waves, instead making generalised, unspecific and often inaccurate descriptions of aspects of
waves. For example, some thought that all stationary waves were transverse. References to
phase and amplitude were rare and confused.
Part (b) was done more effectively by many. Those that failed to gain credit tended to invert the
ratios. Students could get the correct numerical answer using the general wave equation but this
approach was not credited.
Page 54 of 56
28
Part (a)(i) was almost universally misinterpreted due to a similar question appearing on a
previous paper. Many students interpreted the question as ‘describe the motion over the next
cycle’. Those who did this often failed to point out that there was a continuing oscillation taking
place. Part (a)(ii) was very poorly answered which was a surprise. A common answer was ‘out of
phase’ for X and Y which is not equivalent to ‘antiphase’. Phase was often given in terms of
number of wavelengths, e.g. ½λ. There was little understanding of the difference between phase
difference along a progressive wave and a stationary wave. Many had measured the fraction of a
wavelength between the points and converted this into an angle as you would for a progressive
wave. It is suggested that phase difference along a stationary wave be demonstrated by referring
to the many simulations available.
Part (b)(i) presented few problems for students. In part (b)(ii) many students did 1/780 and
obtained the time for one complete cycle but did not recognise that they needed to divide by 4 to
get the time for ¼ of a cycle. A significant number thought that the time between maximum
displacement and reaching the equilibrium position was half a cycle. Some divided 780 by 4
which makes the answer 8 times greater than it should be.
For part (c)(i) most students got ‘antinode’ but a significant number put ‘node’/ ‘amplitude’/ ‘max
displacement’ / ‘stationary wave’ / ‘equilibrium’ / ‘maxima’. Part (c)(ii) presented few problems for
students. In part (c)(iii) quite a few students left this blank because they were unable to answer
the previous question. However, many of those who scored the mark did so by using an incorrect
answer to (c)(ii). Students should be encouraged not to give up; the final part of a question is not
necessarily the hardest.
29
Many students had learned the correct definition in part (a) but some gave a description, for
example ‘the greatest height of the wave from the middle’. This did not gain marks.
Surprisingly in answer to part (b), many students referred to the equilibrium position as the ‘node’
and maximum amplitude as the ‘antinode’ on a progressive wave. Many use fractions of a cycle
to describe the position of the knot but some use angles or fractions of a wavelength which are
not appropriate. The biggest loss of marks occurred in the first mark where a large number
thought that the knot would be travelling upwards initially.
Part (c) was a fairly easy question with students only needing to state that the ‘knot is at a node
on a stationary wave which is caused by superposition’ to get three marks. Most students
managed to get two of the marking points. Many did not understand how a node is formed,
believing it is the sum of a peak and a trough only, or that the whole rope is stationary, or that the
rope is only stationary at a node when cancellation occurs between waves that are 180° out of
phase. The two waves that form a stationary wave are not always 180° out of phase in order to
cancel at the nodes. Nodes are where the wave always cancels but the phase difference
between the waves repeatedly varies from zero to 2π. Cancellation everywhere on the stationary
wave only occurs when the waves are in antiphase but cancellation always happens at the nodes
because the displacements of the waves are always equal and opposite at those points (or
displacements are both zero when in phase and in antiphase). This is a complex situation but
there are many simulations available on the internet that help to get these ideas across.
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(a)
A large majority of the candidates completed this successfully.
(b)
Many candidate having completed part (a) successfully then failed to appreciate that to
produce four times the frequency, a quarter of the length is needed. Some suggested using
a shorter length without making reference to the how much shorter it needed to be. Others
incorrectly suggested changing tension / thickness or the mass per unit length.
(c)
Those who appreciated that beats would be heard often failed to go on to give value of the
beat frequency.
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