4.5 Standing Waves

The diagrams show the structure of a violin and a close-up of the tuning pegs.

The strings are attached at end X then pass over a bridge which acts as a fixed point. The strings are also fixed at the other end, where they are wound around cylindrical spools, fixed to tuning pegs.

Strings for musical instruments create notes according to their tension and a property of the string called mass per unit length, μ. 

The properties of the string and the frequency of the first harmonic are related by the equation:

Where f = frequency of first harmonic (Hz), L = length (m), T = tension (N) and μ = mass per unit length (kg m⁻¹).

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.

 When the tension in the string is 50 N

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

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

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 60 °.

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

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 stationary waves in the tube. Include in your answer a condition for formation of the wave and describe the wave which is formed.

For the third harmonic of the wave formed construct a three-part diagram clearly linking the wave shape, node formation and pressure differences within the tube. Start with the template provided below.

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  

A speaker is set-up directly above the top of a vertical pipe which is partially filled with water. 

Initially, there is a strong sound heard from the pipe when the distance between the loudspeaker and the water is 83 cm. This is the longest length for which a strong sound is heard. 

As the pipe is filled with more water, a second strong sound is heard from the pipe when the distance between the loudspeaker and the water is 67 cm. 

Outline how a standing wave is created between the speaker and the surface of the water.    

Predict the distance between the speaker and the water at which the next strong sound will be produced as the pipe is filled water.

The air within the pipe and the water at the bottom of the pipe are both heated to 70 °C. The speed of sound in this warmer air is 371 m s⁻¹ and the speaker now plays a sound at a constant frequency of 600 Hz. 

The speaker is brought down to the surface of the water and slowly raised until a strong sound is produced. The distance between the surface of the water and the speaker is 15.5 cm when this occurs. State what is causing the strong sound and estimate the wavelength of this sound.

If the water volume is kept constant, predict the distance that the speaker must be raised for the next strong sound to be produced and outline what causes this strong sound.

Explain clearly how the following vary in a stationary wave: 

• Amplitude
• Phase
• Energy transfer

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

On the diagram shown, 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.

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

The diagram 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.

Explain why a stationary wave is formed on the string.  

The stationary wave formed represents the "A" string of a violin which has a frequency of 440 Hz. 

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

The progressive waves on the "A" string travel at a speed of 280 m s⁻¹. 

Calculate the length of the "A" string.    

This diagram shows the string between P and Q. 

A violinist presses on the string at C to shorten it and create the higher "B" note. The distance between C and Q is 0.252 m. 

The speed of the progressive wave remains at 280 m s⁻¹ and the tension remains constant.

Calculate the frequency of the note "B".  

The diagram 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 a maximum displacement. 

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

In the diagram from part (a), the frequency of vibration is 240 Hz. 

Calculate the frequency of the second harmonic for this string.

The speed of the transverse waves along the string is 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).