Stationary Waves (AQA AS Physics)

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

Calculate the speed of transverse waves along the wire.

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

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

Figure 2

State the phase relationship between points: 

• X and Y
• Y and Z
• X and Z

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

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

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

In Figure 1, the frequency of vibration 180 Hz. 

Calculate the frequency of the second harmonic for this string.

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

Calculate the length of the string.

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

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

Figure 1

Explain how a stationary wave is produced on the string.

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

Calculate the first harmonic of the piano string.

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.

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.            

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

Explain why a stationary wave is formed on the string.

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.

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

Calculate length of the D string.

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

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

Calculate the frequency of the note E.

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
• A description of the processes that produce these features 

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

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

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

Figure 1a

Figure 1b

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.

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

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

Figure 2

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^º.

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

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

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.

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

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

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.

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

Figure 3

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

Table 1 shows the results they record from the experiment:

                     Table 1

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

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.

The student realises that a graph of frequency f against   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.

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

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.

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 = 

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.

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

Calculate the mass attached to the end of wire.

It can be shown that the general expression for the n^th order of harmonic frequencies is: 

               f_n = nf₁ 

where f₁ is the frequency of the first harmonic and f_n 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.