Physics A Level
Chapter 14: Stationary waves 14.4 Determining the wavelength and speed of sound
Physics A Level
Chapter 14: Stationary waves 14.4 Determining the wavelength and speed of sound
Since we know that adjacent nodes (or antinodes) of a stationary wave are separated by half a wavelength, we can use this fact to determine the wavelength $\lambda $ of a progressive wave. If we also know the frequency f of the waves, we can find their speed v using the wave equation $v = f\lambda $.
One approach uses Kundt’s dust tube (Figure 14.16). A loudspeaker sends sound waves along the inside of a tube. The sound is reflected at the closed end. When a stationary wave is established, the dust (fine powder) at the antinodes vibrates violently. It tends to accumulate at the nodes, where the movement of the air is zero. Hence, the positions of the nodes and antinodes can be clearly seen.
PRACTICAL ACTIVITY 14.2
Using stationary sound waves to determine λ and v This method is shown in Figure 14.17; it is the same arrangement as used for microwaves (Practical Activity 14.1). The loudspeaker produces sound waves, and these are reflected from the vertical board.
The microphone detects the stationary sound wave in the space between the speaker and the board, and its output is displayed on the oscilloscope. It is simplest to turn off the time-base of the oscilloscope, so that the spot no longer moves across the screen. The spot moves up and down the screen, and the height of the vertical trace gives a measure of the intensity of the sound.
By moving the microphone along the line between the speaker and the board, it is easy to detect nodes and antinodes. For maximum accuracy, we do not measure the separation of adjacent nodes; it is better to measure the distance across several nodes.
Questions
5) a: For the arrangement shown in Figure 14.17, suggest why it is easier to determine accurately the position of a node rather than an antinode.
b: Explain why it is better to measure the distance across several nodes.
6) For sound waves of frequency $2500 Hz$, it is found that two nodes are separated by $20 cm$, with three antinodes between them.
a: Determine the wavelength of these sound waves.
b: Use the wave equation $v = f\lambda $ to determine the speed of sound in air.
EXAM-STYLE QUESTIONS
1) Which statement is not correct about stationary waves? [1]
A: A stationary wave always has transverse oscillations.
B: A stationary wave must have at least one node.
D: The separation between two adjacent nodes is , where $\lambda $ is the wavelength of the progressive wave.
The superposition of two progressive waves travelling in opposite directions will produce a stationary wave.
2) A string is fixed between points X and Y.
A stationary wave pattern is formed on the stretched string.
The distance between X and Y is $78.0 cm$. The string vibrates at a frequency of $120 Hz$.
What is the speed of the progressive wave on the string? [1]
A: $11.7\,m\,{s^{ - 1}}$
B: $23.4\,m\,{s^{ - 1}}$
C: $46.8\,m\,{s^{ - 1}}$
D: $93.6\,m\,{s^{ - 1}}$
3) This diagram shows a stationary wave on a string.
a: On a copy of the diagram, label one node (N) and one antinode (A). [1]
b: Mark on your diagram the wavelength of the progressive wave and label it $\lambda $. [1]
c: The frequency of the vibrator is doubled. Describe the changes in the stationary wave pattern. [1]
[Total: 3]
4) A tuning fork that produces a note of $256 Hz$ is placed above a tube that is nearly filled with water. The water level is lowered until resonance is first heard.
a: Explain what is meant by the term resonance. [1]
b: The length of the column of air above the water when resonance is first heard is $31.2 cm$.
Calculate the speed of the sound wave. [2]
[Total: 3]
5) a: State two similarities and two differences between progressive waves and stationary waves. [4]
b: This diagram shows an experiment to measure the speed of a sound in a string. The frequency of the vibrator is adjusted until the stationary wave shown is formed.
i- On a copy of the diagram, mark a node (label it N) and an antinode (label it A). [2]
ii- The frequency of the vibrator is $120 Hz$. Calculate the speed at which a progressive wave would travel along the string. [3]
c: The experiment is now repeated with the load on the string halved. In order to get a similar stationary wave the frequency has to be decreased to $30 Hz$. Explain, in terms of the speed of the wave in the string, why the frequency must be adjusted. [2]
[Total: 11]
6) This diagram shows a stationary wave, of frequency $400 Hz$, produced by a loudspeaker in a closed tube.
a: Describe the movement of the air particles at:
i- A [2]
ii- B [1]
b: The length the tube is $63.8 cm$.
Calculate the speed of the sound. [3]
[Total: 6]
7) a: Explain what is meant by:
i- a coherent source of waves. [2]
ii- phase difference. [2]
b: A student, experimenting with microwaves, sets up the arrangement shown in this diagram.
With the metal plate at position A there is a very small signal. He slowly moves the plate back, leaving the receiver in the same position. As he does so, he finds that the intensity initially rises until it becomes a maximum, then falls back to a minimum. This cycle repeats a total of five times until the plate reaches position B, where once again there is a minimum.
i- Explain why a series of maxima and minima are heard. [2]
ii- Determine the frequency of the microwaves. [5]
c: Explain why there was a minimum when the plate was at position A, next to the detector. [2]
[Total: 13]
8) This diagram shows an experiment to measure the speed of sound in air.
A small amount of dust is scattered along the tube. The loudspeaker is switched on. When the frequency is set at $512 Hz$ the dust collects in small piles as shown in the diagram.
a: Determine the wavelength of the sound wave and calculate the speed of sound in the air in the tube. [3]
b: On a copy of the diagram, show the movement of the air particles at positions P, Q, R, S and T. [3]
c: Mark two points on your diagram where the movements of the air particles are ${180^ \circ }$ out of phase with each other. Label them A and B. [1]
[Total: 7]
9) The speed v of a transverse wave on a stretched wire is given by the expression $v \propto \sqrt T $ where T is the tension in the wire.
A length of wire is stretched between two fixed point. The tension in the wire is T. The wire is gently plucked from the middle. A stationary wave, of fundamental frequency $210 Hz$, is produced.
The tension in the wire is now increased to $1.4T$. The percentage uncertainty in new tension is $8.0\% $. The length of the wire is unchanged.
Calculate the new value for the fundamental frequency when the wire is plucked in the middle. Your answer must include the absolute uncertainty written to an appropriate number of significant figures. [4]
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| explain the formation of stationary waves using graphical methods | 14.1, 14.3 | |||
| describe experiments that demonstrate stationary waves using microwaves, stretched strings and air columns | 14.3 | |||
| state what is meant by nodes and antinodes | 14.2 | |||
| recall the separation between neighbouring nodes (or antinodes) in terms of the wavelength of the progressive wave | 14.3 | |||
| determine the wavelength of sound using stationary waves. | 14.4 |