Stationary Waves (AQA A Level Physics)

Exam Questions

3 hours44 questions
1a2 marks

State two necessary conditions for a stationary wave to be set up on a string.

1b2 marks

State the principle of superposition.

1c2 marks

A stationary wave is made up of nodes and anti–nodes. 

Define what is meant by a: 

            (i) node 

            (ii) anti–node

1d2 marks

Figure 1 shows a stationary wave on a string fixed at both ends.

Figure 1

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Label on Figure 1 the nodes (N) and anti–nodes (A).

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2a2 marks

Explain the differences in amplitudes in undamped progressive transverse waves and stationary waves.

2b2 marks

Complete the sentences by using an answer from the box. You may use a word more than once.

 

Constructive

Interference

Destructive

Phase

 

   (i) Nodes are areas of _____________ _____________ 

   (ii) Anti–nodes are areas of _____________ _____________

2c3 marks

A particular type of guitar 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 four loops.

Figure 1

3-2-s-q--q2c-easy-aqa-a-level-physics

Compare the amplitude and whether the points are in phase or anti–phase between: 

            (i) A and C 

            (ii) B and D 

            (iii) A and D

2d2 marks

The frequency of the first harmonic of the stationary wave in Figure 1 is 60 Hz. 

Calculate the frequency of the stationary wave in Figure 1.

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3a2 marks

In Figure 1a, AB is a stretched string of length L.

Figure 1a

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On Figure 1b, sketch 

            (i) The fundamental mode of vibration for the string. 

            (ii) The third harmonic vibration for the string.

Figure 1b

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3b2 marks

L is equal to 0.72 m. 

Calculate the wavelength of the standing wave produced when the string is plucked.

3c3 marks

The speed of the wave in the first harmonic is 24 m s–1 

Calculate the frequency of the first harmonic.

3d2 marks

The mass per unit length of the string is 4.1 × 10–3 kg m–1. 

Show that a tension of 2.4 N is needed for the string to have the same frequency as that calculated in part (c).

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4a2 marks

Explain the difference between energy transfers in progressive waves and stationary waves.

4b3 marks

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. This produces a stationary wave.

Figure 1

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Explain why a stationary wave is formed on the string.

4c4 marks

A clarinet produces different notes from different harmonics of stationary waves. 

Draw a line from each note in the clarinet to its matching harmonic. One has been done for you.

3-2-s-q--q4c-easy-aqa-a-level-physics

4d2 marks

State and explain which harmonic would produce the highest note. 

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5a2 marks

Describe: 

(i)
The phase difference between two adjacent points on a progressive transverse wave. 
(ii)
The phase difference between two adjacent points on a standing wave. 
5b3 marks

Figure 1 shows four types of progressive transverse waves.

Figure 1

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State which pairs of waves superpose to create the following standing waves

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You may choose a wave more than once.

5c2 marks

Figure 2 shows a stationary wave on a stretched guitar string of length 0.58 m when it is plucked.

Figure 2

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State the harmonic shown on the guitar string and the wavelength of the standing wave.

5d2 marks

The frequency of the note played on the guitar string is 276 Hz. 

Hence or otherwise, calculate the speed of the transverse waves along the string. 

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1a3 marks

Figure 1 shows a standing wave set up on a wire of length 59 cm. The wire is vibrated at a frequency of 150 Hz. The dashed line represents the equilibrium position.

Figure 1

3-2-s-q--q1a-medium-aqa-a-level-physics

Calculate the speed of transverse waves along the wire.

1b2 marks

Calculate the first harmonic frequency of the wire in Figure 1.

1c3 marks

Figure 2 shows points X, Y and Z on the wire.

Figure 2

3-2-s-q--q1c-medium-aqa-a-level-physics

State the phase relationship between points: 

  • X and Y
  • Y and Z
  • X and Z
1d3 marks

The wire is then vibrated at a frequency of 300 Hz. 

Sketch the new shape of the stationary wave on Figure 3 and show on your diagram three points, A, B and C that oscillate in phase.

Figure 3

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2a2 marks

Figure 1 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 is at maximum displacement.

Figure 1

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Mark clearly on the diagram the direction in which points Q, R, S and T are about to move.

2b2 marks

In Figure 1, the frequency of vibration 180 Hz. 

Calculate the frequency of the second harmonic for this string.

2c3 marks

The speed of the transverse waves along the string is 60 m s–1. 

Calculate the length of the string.

2d2 marks

Compare the amplitude and phase of points R and S on the string.

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3a3 marks

Figure 1 shows a vibrating piano string fixed at both ends.

Figure 1

3-2-s-q--q3a-medium-aqa-a-level-physics

Explain how a stationary wave is produced on the string.

3b2 marks

The piano string in Figure 1 vibrates with a frequency of 328 Hz. 

Calculate the first harmonic of the piano string.

3c3 marks

The piano string in Figure 1 has a tension of 735 N and a mass per unit length of 1.98 × 10–2 kg m–1.  

Calculate the length of the piano string.

3d3 marks

In order to achieve the second harmonic frequency, calculate how you would change the length and tension of the piano string independently to achieve this.            

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4a3 marks

Figure 1 represents a stationary wave formed on a violin string fixed at P and Q when it is plucked at its centre. X is a point on the string at maximum displacement.

Figure 1

3-2-s-q--q4a-medium-aqa-a-level-physics

Explain why a stationary wave is formed on the string.

4b3 marks

The stationary wave in Figure 1 is the D string of the violin which has a frequency of 294 Hz. 

Calculate the time taken for the string at point X to move from maximum displacement to its next maximum displacement.

4c3 marks

The progressive waves on the string travel at a speed of 190 m s–1 

Calculate length of the D string.

4d3 marks

Figure 2 shows the string between P and Q. 

A violinist presses on the string at C to shorten it and create the higher note ‘E’. The distance between C and Q is 0.29 m.

Figure 2

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The speed of the progressive wave remains at 190 m s–1 and the tension remains constant. 

Calculate the frequency of the note E.

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5a3 marks

Explain clearly how the following vary in a stationary wave:                       

  • Amplitude
  • Phase
  • Energy transfer
5b6 marks

A stationary wave in the third harmonic is formed on a stretched string. 

Discuss the formation of this wave and its properties. Your answer must include: 

  • An explanation of how the stationary wave is formed
  • A description of the features of this particular harmonic of the stationary wave
  • A description of the processes that produce these features 

The quality of written communication will be assessed in your answer.

5c2 marks

On Figure 1, draw the stationary wave that would be formed on the string in part (b) with two more nodes and two more antinodes. State the harmonic of this new stationary wave.

Figure 1

3-2-s-q--q1d-medium-aqa-a-level-physics

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1a3 marks

Figure 1a shows the structure of a violin and Figure 1b shows a close-up of the tuning pegs

Figure 1a

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Figure 1b

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The strings are fixed at end A. The strings pass over a bridge and the other ends of the strings are wound around cylindrical spools, fixed to tuning pegs. 

The mass of a particular string is 1.4 × 10–4 kg and it has a vibrating length of 0.35 m. When the tension in the string is 25 N, it vibrates with a first-harmonic frequency of 357 Hz. 

Show that the first harmonic frequency doubles when the tension in the string quadruples.

1b4 marks

Determine the speed at which waves travel along the string in question (a) when the tension in the string is 50 N.  

1c5 marks

Figure 2 shows how the tension in the string varies with the extension of the string.

Figure 2

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The string, under its original tension of 25 N is vibrating at a frequency of 357 Hz. The diameter of the cylindrical spool is 6.50 × 10–3 m. 

Determine the higher frequency that is produced when the tuning peg is rotated through an angle of 60º.

1d2 marks

State and explain the assumption that must be made in order to carry out the calculation in part (c). 

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2a2 marks

Sound waves, like waves on string, can produce stationary waves inside air columns. 

A physics technician demonstrates how sound can set up stationary waves using a tall tube of cross-sectional area 3.0 × 10–3 m2 and a loudspeaker connected to a signal generator, as shown in Figure 1.

Figure 1

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The signal generator is switched on so that sound waves enter the tube. He slowly pours water into the tube and the sound gradually increases in volume, reaching a first maximum at a particular instant. Immediately after the volume begins to decrease. 

Water continues to be added until the volume rises again, reaching a second and final maximum after a further 2.5 × 10–3 m3 of water is poured in. 

Determine the wavelength of the sound waves.

2b3 marks

To help explain the demo, the technician sketches how particles of air move around in a tube when a stationary sound wave is set up. This sketch is shown in Figure 2.

Figure 2

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On Figure 2, label all positions where there are displacement nodes.

2c3 marks

One of the technician’s students asks, “Are positions of displacement nodes the same as

positions where the air pressure is maximum?” 

With reference to the meaning of a displacement node, and using Figure 2, suggest how the technician might respond to the student.

2d4 marks

In the space provided in Figure 3 below, sketch the shape of the stationary sound wave set up in Figure 2

Figure 3

3-2-s-q--q2d-hard-aqa-a-level-physics

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3a5 marks

Figure 1 shows an experiment designed by a student to investigate vibrations in a stretched nylon string of fixed length l. 

The student measures how the frequency f of the first harmonic varies with the mass m suspended from the string.

Figure 1

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Table 1 shows the results they record from the experiment:

                     Table 1

m / g

f / Hz

500

110

800

140

1200

170

 

With reference to an appropriate proportionality, discuss the validity of the results in Table 1.

3b4 marks

The student notes that the string has a uniform diameter of 4.0 × 10–4 m. 

The fixed length l of the vibrating string was also measured to be 0.80 m. 

Determine the density of the nylon string.

3c4 marks

The student realises that a graph of frequency f against square root of T  would be a straight line, where T is the tension in the string, if she used her readings in Table 1. 

Discuss how the graph would look for tensions that are much larger than those used by the student in Table 1.

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4a4 marks

A horizontal sonometer wire of length 0.50 m and mass per unit length 8.0 × 10–3 kg m–1 is maintained taut by hanging masses from a mass hanger at one end, while an alternating current is passed through it.

 A pair of magnets is placed on either side of the wire, with the poles arranged as shown in Figure 1.

Figure 1

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The frequency of the signal generator is adjusted until a stationary wave is set up, with maximal amplitude, as shown in Figure 1. 

With reference to progressive waves, explain how a stationary wave is set up in the sonometer wire.

4b3 marks

The signal generator can be thought of as a vibration generator – sending transverse waves along the stretched wire. The speed c of these transverse waves is given by the equation: 

               c = square root of T over mu end root

where T is the tension in the wire and μ is the mass per unit length of the wire. 

Show that c has units of speed.

4c4 marks

The frequency of the signal generator is gradually adjusted to 50 Hz until a stationary wave is formed as shown in Figure 2.

Figure 2

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Calculate the mass attached to the end of wire.

4d3 marks

It can be shown that the general expression for the nth order of harmonic frequencies is: 

               fn = nf1 

where f1 is the frequency of the first harmonic and fn are higher harmonics. 

By referring to the frequency of the first harmonic, discuss what is observed when the alternating current is set to 25 Hz.

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