Standing Waves & Resonance (DP IB Physics: HL): Structured Questions

A standing wave is produced in a pipe that is open at both ends and placed within a chamber filled with an unknown gas. The length of the pipe is 45 cm. The frequency of the first harmonic is 381 Hz.

Calculate the speed of this standing wave.

Calculate the wavelength of the fourth harmonic for this pipe.

Calculate the frequency of the sixth harmonic.

The pipe is now submerged and filled with water.

The speed of sound in water is 1500 m s⁻¹.

Determine the period of the first harmonic in the water-filled pipe.

A loudspeaker is positioned directly above the top of a vertical pipe and emits a sound wave of constant frequency. The pipe is gradually filled with water. As the amount of water increases, the loudness of the sound heard reaches a maximum because a standing wave has formed in the tube.

Outline how the standing wave is formed.

The loudness of the sound heard in the pipe reaches a maximum for the first time when the distance between the loudspeaker and the water surface is 55 cm.

The pipe is then filled with more water. The next time a maximum is heard is when the distance between the loudspeaker and the water surface is 33 cm. A final maximum is heard before the water completely fills the pipe.

(i) Deduce the number of nodes formed when the loudness of the sound heard reaches a maximum for the first time.

[1]

(ii) Predict the distance between the loudspeaker and the water surface for which the final maximum is heard.

[2]

(iii) The speed of sound in air is 330 m s^-1. Show that the frequency of the sound emitted by the loudspeaker is 750 Hz.

[2]

The air and water within the pipe are both heated. The loudspeaker continues to emit sound at a constant frequency as the water is drained through a valve.

State and explain the effect this has, if any, on the positions of the maximum loudness.

Describe how a standing wave differs from a travelling wave in terms of

• energy transfer

• amplitude

• phase

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

(i) On the diagram shown, draw the standing wave that would be formed on the string in part (b) with two more nodes and two more antinodes.

[1]

(ii) State the harmonic of this standing wave.

[1]

Calculate the length of the string in part (c) if it oscillates at 500 cycles per second and the speed of waves travelling within it is 140 m s^–1.

On a violin, the strings vibrate between two fixed points. The diagram shows the standing wave formed on a violin string fixed at points P and Q. The string is plucked at point X, the centre of the string.

Explain how a standing wave is formed on the string.  

Point X on the string vibrates in its first harmonic to produce a frequency of 440 Hz. The length PQ of the string is 32 cm.

Show that the speed of the wave on the string is about 280 m s⁻¹.

The violinist changes the length of the string by pressing on the string at point C. The string continues to vibrate in its first harmonic.

State and explain the effect this has on the frequency of the note produced.

The diagram shows the appearance of a standing wave on a stretched string at one instant in time. In the position shown each part of the string is at a maximum displacement.

Draw arrows on the diagram to show the direction of motion of points Q, R, S and T.

The string vibrates with a frequency of 240 Hz. 

Calculate the frequency of the second harmonic for this string.

The waves move along the string with a speed of 55 m s⁻¹. 

Calculate the length of the string.

Compare the amplitude and phase of points R and S on the string in the diagram used in part (a).

A physics class investigates stationary waves in air using a tall tube of cross-sectional area 3.0 × 10^–3 m² and a loudspeaker connected to a signal generator. Initially the tube is empty of water. The signal generator is switched on so that sound waves enter the tube. Water is slowly poured into the tube.    

The class notice that 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.

One method of illustrating sound waves is shown.   

Sketch the diagram labelling all the positions of the nodes formed by the standing wave in part (a).

The teacher asks whether the positions of the nodes and antinodes are related to regions of pressure along the standing wave.

By analysing the diagram from part (b) discuss the correct response.

Using the diagram shown, sketch the shape of the stationary sound wave the students discussed in the previous part. 

When a violin string is plucked, a standing wave is produced between two fixed points, the bridge and the nut of the violin.

The speed of a wave on a string is given by

$v=\sqrt{\frac{T}{\mu }}$

where $T$ is the tension in the string and $\mu$ is the mass per unit length of the string.

The length of the string between the bridge and the nut is $L$.

Show that the frequency of the first harmonic on the violin string is $f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$.

A violin string has a vibrating length of 32.8 cm. When under a tension of 25 N, the string vibrates in its first harmonic with a frequency of 370 Hz.

Calculate

(i) the mass per unit length of the string

[1]

(ii) the tension required to produce a frequency of 294 Hz.

[1]

The tension in the strings can be increased or decreased by rotating the tuning pegs.

The graph shows how the tension in the string varies with the extension of the string.    

The string, under its initial tension of 25 N, is vibrating at a frequency of 370 Hz. The diameter of the circular spool of a peg is 7.0 mm.

Determine the angle the tuning peg must be rotated through to produce a frequency of 294 Hz.

The diagram shows a common piece of teaching laboratory equipment which can be used to demonstrate wave phenomena.   

Explain how waves from the loudspeaker form standing waves in the tube. Include in your answer a condition for formation of the wave and describe the wave which is formed.

Draw, on the diagram, the third harmonic of the standing wave formed in the tube. Label the positions of the nodes with the letter N and the positions of the antinodes with the letter A.

The speed of sound in the tube is 340 m s⁻¹ and the frequency of the sound emitted by the loudspeaker is 880 Hz.

For this equipment, calculate  

(i) the length of the tube

[2]

(ii) the wavelength of the fifth harmonic.

[2]

A student is investigating forced vertical oscillations in springs. 

Two springs, A and B, are suspended from a horizontal metal rod that is attached to a vibration generator. The stiffness of A is 3k, and the stiffness of B is k. 

Two equal masses are suspended from the springs. 

The vibration generator is connected to a signal generator. The signal generator is used to vary the frequency of vibration of the metal rod. When the signal generator is set at 6.5  Hz, the mass attached to spring A oscillates with a maximum amplitude of 4.0 cm.

Calculate the frequency at which the mass attached to spring B oscillates with maximum amplitude.

The investigation is repeated with the mass attached to spring A immersed in a beaker of oil. 

A graph of the variation of the amplitude with frequency for spring A is different for spring B.  

(i) Sketch the variation of this graph for spring A.

[3]

(ii) Explain two differences between the graphs.

[2]

(i) Sketch the graph of displacement against time for spring A after it has been immersed in a beaker of oil starting at its lowest point. 

[3]

(ii) Explain in terms of energy the reasons behind the graph you have drawn. 

[3]

When immersed in the beaker of oil, spring A is released with the same amplitude as when it was connected to the vibration generator.

Calculate the fraction of the energy lost in the oil when the amplitude of oscillations is 0.9 cm.