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```Standing Waves
[93 marks]
1. A student blows across the top of a cylinder that contains water. A first- [1 mark]
harmonic standing sound wave is produced in the air of the cylinder. More
water is then added to the cylinder. The student blows so that a first-harmonic
standing wave is produced with a different frequency.
What is the nature of the displacement in the air at the water surface and the
change in frequency when the water is added?
Markscheme
D
Examiners report
[N/A]
2. In an experiment to determine the speed of sound in air, a tube that is
[1 mark]
open at the top is filled with water and a vibrating tuning fork is held over
the tube as the water is released through a valve.
An increase in intensity in the sound is heard for the first time when the air
column length is x. The next increase is heard when the air column length is y.
Which expressions are approximately correct for the wavelength of the sound?
I. 4x
II. 4y
4y
III. 3
A. I and II
B. I and III
C. II and III
D. I, II and III
Markscheme
B
Examiners report
The question was well answered by students.
3. A third-harmonic standing wave of wavelength 0.80 m is set up on a
[1 mark]
string fixed at both ends. Two points on the wave are separated by a
distance of 0.60 m. What is a possible phase difference between the two points on
the wave?
Markscheme
C
Examiners report
This question had a low discrimination index with response D the most popular
and an even spread between the other 3 answers. A third-harmonic standing
wave of wavelength 0.8m must be on a string of length 1.2m giving 3 loops of
0.4m each. Depending on where the initial point is chosen, two points
separated by 0.6m will either be in adjacent loops e.g. at 0.1m and 0.7 m with
a phase difference of π or in the two end loops e.g at 0.3 m and 0.9m with a
phase difference of 0. So for a standing wave there are only two possible
answers, π (response C) or 0 (not included in these responses).
The diagram shows the direction of a sound wave travelling in a metal sheet.
4a. Particle P in the metal sheet performs simple harmonic oscillations.
[2 marks]
When the displacement of P is 3.2 μm the magnitude of its acceleration
is 7.9 m s-2. Calculate the magnitude of the acceleration of P when its
displacement is 2.3 μm.
Markscheme
Expression or statement showing acceleration is proportional to displacement
✔
so «7.9 × 2.3
»
3.2
= 5.7«m s−2 » ✔
Examiners report
This was well answered at both levels.
4b. The wave is incident at point Q on the metal–air boundary. The wave
[2 marks]
makes an angle of 54° with the normal at Q. The speed of sound in the
metal is 6010 m s–1 and the speed of sound in air is 340 m s–1. Calculate the angle
between the normal at Q and the direction of the wave in air.
Markscheme
sin θ =
340
6010
× sin 54∘ ✔
θ = 2.6° ✔
Examiners report
Many scored full marks on this question. Common errors were using the
calculator in radian mode or getting the equation upside down.
The frequency of the sound wave in the metal is 250 Hz.
4c. State the frequency of the wave in air.
Markscheme
f = 250 «Hz» OR Same OR Unchanged ✔
[1 mark]
Examiners report
Many used a ratio of the speeds to produce a new frequency of 14Hz (340 x
250/6010). It would have helped candidates if they had been aware that the
command term ‘state’ means ‘give a specific name, value or other brief
4d. Determine the wavelength of the wave in air.
[1 mark]
Markscheme
λ = « 340
= »1.36 ≈ 1.4«m» ✔
250
Examiners report
[N/A]
4e. The sound wave in air in (c) enters a pipe that is open at both ends. The
diagram shows the displacement, at a particular time T, of the standing
wave that is set up in the pipe.
[1 mark]
On the diagram, at time T, label with the letter C a point in the pipe that is at the
centre of a compression.
Markscheme
any point labelled C on the vertical line shown below ✔
eg:
Examiners report
This was answered well at both levels.
The diagram shows the direction of a sound wave travelling in a metal sheet.
5a. Particle P in the metal sheet performs simple harmonic oscillations.
[2 marks]
When the displacement of P is 3.2 μm the magnitude of its acceleration
is 7.9 m s-2. Calculate the magnitude of the acceleration of P when its
displacement is 2.3 μm.
Markscheme
Expression or statement showing acceleration is proportional to displacement
✔
so «7.9 × 2.3
» = 5.7«ms–2»
3.2
✔
Examiners report
This was well answered at both levels.
5b. The wave is incident at point Q on the metal–air boundary. The wave
[2 marks]
makes an angle of 54° with the normal at Q. The speed of sound in the
metal is 6010 m s–1 and the speed of sound in air is 340 m s–1. Calculate the angle
between the normal at Q and the direction of the wave in air.
Markscheme
sin θ =
θ = 2.6
340
6010
0
× sin 540 ✔
✔
Examiners report
Many scored full marks on this question. Common errors were using the
calculator in radian mode or getting the equation upside down.
5c. The frequency of the sound wave in the metal is 250 Hz. Determine the
wavelength of the wave in air.
[1 mark]
Markscheme
λ = « 340
= »1.36 ≈ 1.4«m» ✔
250
Examiners report
The sound wave in air in (c) enters a pipe that is open at both ends. The diagram
shows the displacement, at a particular time T, of the standing wave that is set up
in the pipe.
A particular air molecule has its equilibrium position at the point labelled M.
5d. On the diagram, at time T, draw an arrow to indicate the acceleration of
this molecule.
Markscheme
horizontal arrow «at M» pointing left ✔
[1 mark]
Examiners report
Very few candidates could interpret this situation and most arrows were
shown in a vertical plane.
5e. On the diagram, at time T, label with the letter C a point in the pipe that
is at the centre of a compression.
Markscheme
any point labelled C on the vertical line shown below ✔
eg:
Examiners report
This was answered well at both levels.
[1 mark]
Sound of frequency f = 2500 Hz is emitted from an aircraft that moves with speed
v = 280 m s–1 away from a stationary observer. The speed of sound in still air is c
= 340 m s–1.
5f. Calculate the frequency heard by the observer.
[2 marks]
Markscheme
f ′ = 2500 ×
340
340+280
✔
f ′ = 1371 ≈ 1400«Hz» ✔
Examiners report
This was answered well with the most common mistake being to swap the
speed of sound and the speed of the aircraft.
5g. Calculate the wavelength measured by the observer.
[1 mark]
Markscheme
λ′ =
340
1371
≈ 0.24/0.25«m» ✔
Examiners report
Answered well with ECF often being awarded to those who answered the
previous part incorrectly.
6. Two strings of lengths L1 and L2 are fixed at both ends. The wavespeed is [1 mark]
the same for both strings. They both vibrate at the same frequency. L1
vibrates at its first harmonic. L2 vibrates at its third harmonic.
What is
A.
L1
?
L2
1
3
B. 1
C. 2
D. 3
Markscheme
D
Examiners report
[N/A]
A pipe is open at both ends. A first-harmonic standing wave is set up in the pipe.
The diagram shows the variation of displacement of air molecules in the pipe with
distance along the pipe at time t = 0. The frequency of the first harmonic is f.
7a. Sketch, on the diagram, the variation of displacement of the air molecules[1 mark]
with distance along the pipe when t = 3 .
4f
Markscheme
horizontal line shown in centre of pipe ✔
Examiners report
[N/A]
7b. An air molecule is situated at point X in the pipe at t = 0. Describe the
motion of this air molecule during one complete cycle of the standing
wave beginning from t = 0.
Markscheme
«air molecule» moves to the right and then back to the left ✔
returns to X/original position ✔
Examiners report
[N/A]
[2 marks]
7c. The speed of sound c for longitudinal waves in air is given by
[4 marks]
c = √ Kρ
where ρ is the density of the air and K is a constant.
A student measures f to be 120 Hz when the length of the pipe is 1.4 m. The
density of the air in the pipe is 1.3 kg m–3. Determine the value of K for air. State
Markscheme
wavelength = 2 × 1.4 «= 2.8 m» ✔
c = «f λ =» 120 × 2.8 «= 340 m s−1» ✔
K = «ρc2 = 1.3 × 3402 =» 1.5 × 105 ✔
kg m –1 s–2 ✔
Examiners report
[N/A]
A transmitter of electromagnetic waves is next to a long straight vertical wall that
acts as a plane mirror to the waves. An observer on a boat detects the waves both
directly and as an image from the other side of the wall. The diagram shows one
ray from the transmitter reflected at the wall and the position of the image.
7d. Demonstrate, using a second ray, that the image appears to come from
the position indicated.
[1 mark]
Markscheme
construction showing formation of image ✔
Another straight line/ray from image through the wall with line/ray from
intersection at wall back to transmitter. Reflected ray must intersect boat.
Examiners report
[N/A]
7e. Outline why the observer detects a series of increases and decreases in [2 marks]
the intensity of the received signal as the boat moves along the line XY.
Markscheme
interference pattern is observed
OR
interference/superposition mentioned ✔
maximum when two waves occur in phase/path difference is nλ
OR
minimum when two waves occur 180° out of phase/path difference is (n + ½)λ
✔
Examiners report
[N/A]
8. A first-harmonic standing wave is formed on a vertical string of length 3.0 [1 mark]
m using a vibration generator. The boundary conditions for this string are
that it is fixed at one boundary and free at the other boundary.
The generator vibrates at a frequency of 300 Hz.
What is the speed of the wave on the string?
A.
0.90 km s–1
B.
1.2 km s–1
C.
1.8 km s–1
D.
3.6 km s–1
Markscheme
D
Examiners report
[N/A]
9. A string stretched between two fixed points sounds its second harmonic
at frequency f.
[1 mark]
Which expression, where n is an integer, gives the frequencies of harmonics that
have a node at the centre of the string?
A.
n+1
2 f
B.
nf
C.
2nf
D.
(2n + 1)f
Markscheme
B
Examiners report
[N/A]
A loudspeaker emits sound towards the open end of a pipe. The other end is
closed. A standing wave is formed in the pipe. The diagram represents the
displacement of molecules of air in the pipe at an instant of time.
10a. Outline how the standing wave is formed.
[1 mark]
Markscheme
the incident wave «from the speaker» and the reflected wave «from the closed
end»
superpose/combine/interfere
Do not allow meet/interact
[1 mark]
Examiners report
[N/A]
X and Y represent the equilibrium positions of two air molecules in the pipe. The
arrow represents the velocity of the molecule at Y.
10b. Draw an arrow on the diagram to represent the direction of motion of
the molecule at X.
[1 mark]
Markscheme
Horizontal arrow from X to the right
MP2 is dependent on MP1
Ignore length of arrow
[1 mark]
Examiners report
[N/A]
10c. Label a position N that is a node of the standing wave.
Markscheme
P at a node
[1 mark]
Examiners report
[N/A]
–1
[1 mark]
10d. The speed of sound is 340 m s–1 and the length of the pipe is 0.30
m. Calculate, in Hz, the frequency of the sound.
Markscheme
wavelength is λ = « 4×0.30
=» 0.40 «m»
3
340
f = « 0.40
» 850 «Hz»
Award  for a bald correct answer
Allow ECF from MP1
[2 marks]
Examiners report
[N/A]
[2 marks]
The loudspeaker in (a) now emits sound towards an air–water boundary. A, B and
C are parallel wavefronts emitted by the loudspeaker. The parts of wavefronts A
and B in water are not shown. Wavefront C has not yet entered the water.
10e. The speed of sound in air is 340 m s –1 and in water it is 1500 m s –1.
[2 marks]
The wavefronts make an angle θ with the surface of the water. Determine
the maximum angle, θmax, at which the sound can enter water. Give your
answer to the correct number of significant figures.
Markscheme
sin θ c
340
=
1
1500
θc = 13«°»
Award  for a bald correct answer
Award  for a bald answer of 13.1
Answer must be to 2/3 significant figures to award MP2
[2 marks]
Examiners report
[N/A]
10f. Draw lines on the diagram to complete wavefronts A and B in water for [2 marks]
θ < θmax.
Markscheme
correct orientation
greater separation
Do not penalize the lengths of A and B in the water
Do not penalize a wavefront for C if it is consistent with A and B
MP1 must be awarded for MP2 to be awarded
[2 marks]
Examiners report
[N/A]
11a. Outline how the standing wave is formed.
[1 mark]
Markscheme
the incident wave «from the speaker» and the reflected wave «from the closed
end»
superpose/combine/interfere
Do not allow meet/interact
[1 mark]
Examiners report
[N/A]
11b. Draw an arrow on the diagram to represent the direction of motion of
the molecule at X.
Markscheme
Horizontal arrow from X to the right
MP2 is dependent on MP1
Ignore length of arrow
[1 mark]
Examiners report
[N/A]
[1 mark]
11c. Label a position N that is a node of the standing wave.
[1 mark]
Markscheme
P at a node
[1 mark]
Examiners report
[N/A]
11d. The speed of sound is 340 m s–1 and the length of the pipe is 0.30
m. Calculate, in Hz, the frequency of the sound.
Markscheme
wavelength is λ = « 4×0.30
=» 0.40 «m»
3
340
f = « 0.40
» 850 «Hz»
Award  for a bald correct answer
Allow ECF from MP1
[2 marks]
[2 marks]
Examiners report
[N/A]
11e. The speed of sound in air is 340 m s –1 and in water it is 1500 m s –1.
[2 marks]
The wavefronts make an angle θ with the surface of the water. Determine
the maximum angle, θmax, at which the sound can enter water. Give your
answer to the correct number of significant figures.
Markscheme
sin θ c
340
=
1
1500
θc = 13«°»
Award  for a bald correct answer
Award  for a bald answer of 13.1
Answer must be to 2/3 significant figures to award MP2
[2 marks]
Examiners report
[N/A]
11f. Draw lines on the diagram to complete wavefronts A and B in water for [2 marks]
θ < θmax.
Markscheme
correct orientation
greater separation
Do not penalize the lengths of A and B in the water
Do not penalize a wavefront for C if it is consistent with A and B
MP1 must be awarded for MP2 to be awarded
[2 marks]
Examiners report
[N/A]
12. A pipe of fixed length is closed at one end. What is
third harmonic frequency of pipe
?
first harmonic frequency of pipe
A. 15
B. 13
C. 3
D. 5
Markscheme
C
[1 mark]
Examiners report
[N/A]
13. The diagram shows a second harmonic standing wave on a string fixed at [1 mark]
both ends.
What is the phase difference, in rad, between the particle at X and the particle at
Y?
A. 0
B. π4
C. π2
D. 34π
Markscheme
A
Examiners report
[N/A]
14. Two pulses are travelling towards each other.
[1 mark]
What is a possible pulse shape when the pulses overlap?
Markscheme
A
Examiners report
[N/A]
15. The frequency of the first harmonic standing wave in a pipe that is open [1 mark]
at both ends is 200 Hz. What is the frequency of the first harmonic in a
pipe of the same length that is open at one end and closed at the other?
A. 50 Hz
B. 75 Hz
C. 100 Hz
D. 400 Hz
Markscheme
C
Examiners report
[N/A]
16. Water is draining from a vertical tube that was initially full. A vibrating
[1 mark]
tuning fork is held near the top of the tube. For two positions of the water
surface only, the sound is at its maximum loudness.
The distance between the two positions of maximum loudness is x.
What is the wavelength of the sound emitted by the tuning fork?
A. x2
B. x
C. 32x
D. 2x
Markscheme
D
Examiners report
[N/A]
A student investigates how light can be used to measure the speed of a toy train.
Light from a laser is incident on a double slit. The light from the slits is detected
by a light sensor attached to the train.
The graph shows the variation with time of the output voltage from the light
sensor as the train moves parallel to the slits. The output voltage is proportional
to the intensity of light incident on the sensor.
17a. Explain, with reference to the light passing through the slits, why a
series of voltage peaks occurs.
[3 marks]
Markscheme
«light» superposes/interferes
pattern consists of «intensity» maxima and minima
OR
consisting of constructive and destructive «interference»
voltage peaks correspond to interference maxima
Examiners report
[N/A]
17b. The slits are separated by 1.5 mm and the laser light has a
[1 mark]
wavelength of 6.3 x 10–7 m. The slits are 5.0 m from the train track.
Calculate the separation between two adjacent positions of the train when the
output voltage is at a maximum.
Markscheme
«s = λD
d
=
6.3×10−7×5.0
1.5×10−3
=» 2.1 x 10–3 «m»
If no unit assume m.
Examiners report
[N/A]
17c. Estimate the speed of the train.
Markscheme
correct read-off from graph of 25 m s
2.1×10−3
v = « xt =
=» 8.4 x 10–2 «m s–1»
25×10−3
Allow ECF from (b)(i)
Examiners report
[N/A]
[2 marks]
17d. In another experiment the student replaces the light sensor with a
[2 marks]
sound sensor. The train travels away from a loudspeaker that is
emitting sound waves of constant amplitude and frequency towards a reflecting
barrier.
The sound sensor gives a graph of the variation of output voltage with time along
the track that is similar in shape to the graph shown in the resource. Explain how
this effect arises.
Markscheme
ALTERNATIVE 1
«reflection at barrier» leads to two waves travelling in opposite directions
mention of formation of standing wave
maximum corresponds to antinode/maximum displacement «of air molecules»
OR
complete cancellation at node position
Examiners report
[N/A]
18a. Explain, with reference to the light passing through the slits, why a
series of voltage peaks occurs.
[3 marks]
Markscheme
«light» superposes/interferes
pattern consists of «intensity» maxima and minima
OR
consisting of constructive and destructive «interference»
voltage peaks correspond to interference maxima
Examiners report
[N/A]
18b. The slits are separated by 1.5 mm and the laser light has a
[1 mark]
wavelength of 6.3 x 10–7 m. The slits are 5.0 m from the train track.
Calculate the separation between two adjacent positions of the train when the
output voltage is at a maximum.
Markscheme
«s = λD
d
=
6.3×10−7×5.0
1.5×10−3
=» 2.1 x 10–3 «m»
If no unit assume m.
Examiners report
[N/A]
18c. Estimate the speed of the train.
Markscheme
correct read-off from graph of 25 m s
2.1×10−3
v = « xt =
=» 8.4 x 10–2 «m s–1»
25×10−3
Allow ECF from (b)(i)
Examiners report
[N/A]
[2 marks]
A student investigates how light can be used to measure the speed of a toy train.
Light from a laser is incident on a double slit. The light from the slits is detected
by a light sensor attached to the train.
The graph shows the variation with time of the output voltage from the light
sensor as the train moves parallel to the slits. The output voltage is proportional
to the intensity of light incident on the sensor.
As the train continues to move, the first diffraction minimum is observed when the
light sensor is at a distance of 0.13 m from the centre of the fringe pattern.
18d. Determine the width of one of the slits.
Markscheme
angular width of diffraction minimum = 0.13
5.0
slit width = « λ
d
=
6.3×10−7
0.026
=» 2.4 x 10–5 «m»
Award [1 max] for solution using 1.22 factor.
Examiners report
[N/A]
[2 marks]
18e. Suggest the variation in the output voltage from the light sensor that
will be observed as the train moves beyond the first diffraction
minimum.
Markscheme
«beyond the first diffraction minimum» average voltage is smaller
«voltage minimum» spacing is «approximately» same
OR
rate of variation of voltage is unchanged
OWTTE
Examiners report
[N/A]
[2 marks]
18f. In another experiment the student replaces the light sensor with a
[2 marks]
sound sensor. The train travels away from a loudspeaker that is emitting
sound waves of constant amplitude and frequency towards a reflecting barrier.
The graph shows the variation with time of the output voltage from the sounds
sensor.
Explain how this effect arises.
Markscheme
«reflection at barrier» leads to two waves travelling in opposite directions
mention of formation of standing wave
maximum corresponds to antinode/maximum displacement «of air molecules»
OR
complete cancellation at node position
Examiners report
[N/A]
19. A pipe of length L has two open ends. Another pipe of length L′ has one
open end and one closed end.
[1 mark]
′
The frequency of the first harmonic of both pipes is the same. What is L ?
L
A. 2
B. 32
C. 1
D. 12
Markscheme
D
Examiners report
[N/A]
A longitudinal wave is travelling in a medium from left to right. The graph shows
the variation with distance x of the displacement y of the particles in the medium.
The solid line and the dotted line show the displacement at t=0 and t=0.882 ms,
respectively.
The period of the wave is greater than 0.882 ms. A displacement to the right of
the equilibrium position is positive.
20a. (i) Calculate the speed of this wave.
[4 marks]
(ii) Show that the angular frequency of oscillations of a particle in the medium
Markscheme
(i)
ALTERNATIVE 1
«distance travelled by wave =» 0.30 m
v =≪
distance
time
=≫ 340ms−1
ALTERNATIVE 2
0.882×10−3×1.6
evaluates T =
«=4.7ms» to give f = 210 or 212 Hz
0.3
uses λ=1.6 m with v=fλ to give 340ms–1
(ii)
ALTERNATIVE 1
λ=1.60m
ω =≪ 2πf =≫ 2π ×
340
1.60
= 1.3 × 103 or 1.34×103rads–1
ALTERNATIVE 2
2π×3
«0.882 ms is 0.3
of cycle so whole cycle is»
1.6
16×0.882×10−3
Allow ECF from (b)(i).
Examiners report
[N/A]
20b. One particle in the medium has its equilibrium position at x=1.00 m.
(i) State and explain the direction of motion for this particle at t=0.
(ii) Show that the speed of this particle at t=0.882 ms is 4.9ms−1.
[4 marks]
Markscheme
(i)
the displacement of the particle decreases OR «on the graph» displacement is
going in a negative direction OR on the graph the particle goes down OR on
the graph displacement moves towards equilibrium/0
to the left
Do not allow “moving downwards”.
(ii)
y=–1.5mm
v = 2π × 212 × √(4.0 × 10−3 ) − (1.5 × 10−3 )
2
«v=4.939≈4.9ms-1»
Allow ECF from (b)(ii).
4.3mm
Do not allow 0.882ms
= 4.87ms−1 .
Examiners report
[N/A]
2
20c. The travelling wave in (b) is directed at the open end of a tube of length [3 marks]
1.20 m. The other end of the tube is closed.
(i) Describe how a standing wave is formed.
(ii) Demonstrate, using a calculation, that a standing wave will be established in
this tube.
Markscheme
(i)
the superposition/interference of two oppositely moving/reflected «identical
travelling» waves
(ii)
the allowed wavelengths in the tube are
λ=
4L
n
=
480
, n = 1, 3, 5,…
n
OR
diagram showing 34 of a standing wavelength in the tube
1.6 =
4.80
n
⇒n=3
OR
justification that 34
× 1.6 = 1.2m
Allow diagram showing 34 of a wavelength for MP1.
Examiners report
[N/A]
21a. Outline what is meant by a black hole.
[2 marks]
Markscheme
region of space with extreme/very large curvature of spacetime
such that light cannot escape the region OR escape speed within region is > c
Do not allow “large” or omission of degree of curvature.
Examiners report
[N/A]
21b. An observer views a distant spacecraft that is 23.0 km from the centre [3 marks]
of a black hole. The spacecraft contains a clock that ticks once every
second and the ticks can be detected by the distant observer. In 2.00 minutes the
observer counts 112 ticks of the clock.
Determine the mass of the black hole.
Markscheme
time for 1 second spacecraft tick in observer frame =1.07s
1.07 =
M =≪
1.00
√1−
RS
OR RS=2.96×103m
2.3×104
c2×2.96×103
2×6.67×10−11
=≫ 2.0 × 1030 kg
Examiners report
[N/A]