Stationary waves

A transmitter produces microwaves with a frequency of 10.7 GHz. The microwaves are directed towards a metal plate. A microwave receiver R is placed between the transmitter and the metal plate, as shown.

When R is moved along the line AB, the intensity of microwaves received is observed to increase from minimum to maximum a number of times. The distance between successive minima is about 1.5 cm.

Explain these observations. Your answer should include a calculation.

A student wound a piece of string around the head of an electric toothbrush. The student attached a small mass to the other end of the string, as shown in Photograph 1.

The toothbrush was switched on and the head started to vibrate. The student rotated the toothbrush slowly to unwind the string. At a particular length, large vibrations were observed on the string. The string formed a loop, as shown in Photograph 2.

Explain how the vibrations of the toothbrush head caused the loop to form.

The student continued to unwind the string and the loop disappeared. When the length of the unwound string was twice that shown in Photograph 2 two loops were seen. Three loops were seen when the unwound length was three times that shown in Photograph 2 and so on.

Determine the frequency of vibration of the toothbrush head.

unwound length of string with 4 loops = 0.69 m mass on string = 0.010 kg mass per unit length of string = 9.1 × 10^–4kg m^–1

Rosslyn Chapel is a 15th century chapel in Scotland. Inside the chapel, small sandstone cubes protrude from a number of arches. It has been suggested that carvings on these cubes bear a resemblance to standing wave patterns that can be produced on a vibrating metal plate.

A metal plate is made to vibrate and sand is scattered evenly across its surface. At a certain frequency the sand moves to produce the standing wave pattern shown below.

(Source: https://skullsinthestars.com/2013/05/02/physics-demonstrations-chladni-patterns/#jp-carousel-7353)

Explain why the sand moves to different positions when a standing wave is formed on the plate.

The speed of sound in air can be determined by creating a standing wave in a column of air. The diagram shows a tuning fork just above the open end of a tube.

The tuning fork produces a sound wave of known frequency f. Several tuning forks are available, each with a different frequency.

A student adjusted the length L of the air column. A loud sound was heard when a standing wave was produced. A node was formed at the water surface, and an antinode was formed at the open end of the tube.

The student used values of L and f to determine a value for the speed of sound.

Describe a graphical method that the student could have used to determine a value for the speed of sound.

The photograph shows a stringed instrument called a cello being played with a bow.

(Source: © Vadim Ponomarenko/Alamy Stock Photo)

A standing wave forms on a cello string when the bow moves across the string.

Deduce whether a thicker string will produce a note of higher or lower frequency compared with a thinner string.

Assume each string is the same length and at the same tension.

The harp is a musical instrument with many strings, as shown.

All the strings are under tension.

The strings on one type of harp are made from nylon of density 1070 kg m⁻³. One string has a diameter of 1.14 mm.

Show that the mass per unit length μ of the string is about 1.1 10⁻³kg m⁻¹.

When the middle of the string is plucked, a note of frequency 440 Hz is produced.

Calculate the tension in the string.

length of string = 41.0 cm

The graph shows how the Young modulus E of the nylon varies with temperature.

When the harp is played, the temperature of the string increases.

Explain how this temperature change would affect the frequency of the note produced when the string is plucked.

A student used a sonometer to investigate the properties of a stretched wire. The sonometer is a long hollow wooden box.

A steel wire is attached to one end of the box and rests on two wooden bridges. The wire is placed under tension T by hanging a mass from the end of the wire, as shown.

The student placed the base of a vibrating tuning fork in contact with the wire, at one of the bridges. This set the wire into oscillation. He adjusted the position of the other bridge until a single-loop standing wave was produced on the wire between the bridges.

Explain how an antinode is produced at the mid-point of the wire between the bridges.

The student repeated this for a series of tuning forks with different frequencies f. For each fork he measured the distance L between the bridges.

The steel wire, of mass per unit length μ, was placed under tension T by hanging a mass of 2.10 kg from the end of the wire.

State one safety precaution that should be taken when carrying out the investigation.

The student plotted a graph of L² against 1/f². Show that the gradient of this graph is equal to

The student then found a value of μ for a brass wire, using a different method.

He measured the diameter d of the wire using a micrometer.

Explain one technique the student should use when measuring d.

The student obtained the following data.

The stated value of μ for the brass wire used by the student was

Deduce whether the student’s data supports this value for μ.

density of brass = 8700 kg m⁻³ ± 200 kg m⁻³

The diagram represents an arrangement used to generate standing waves on a string.

A standing wave pattern with two nodes is obtained as shown.

Which of the following single changes could produce a standing wave pattern with three nodes?

The photograph shows a guitar.

When a guitar string is plucked, a standing wave is created.

Explain how a standing wave is created on the string.

The diagram shows a standing wave on a guitar string.

The oscillating length of the guitar string is 66 cm.

State the wavelength for this standing wave.

Calculate the frequency of vibration for this standing wave.

tension in guitar string = 88.6 N

mass per unit length of guitar string = 4.47 10⁻³kg m⁻¹

One end of the guitar string is wrapped around a cylindrical tuning peg. Turning the peg changes the total length of the string and hence changes the tension in the string. This changes the frequency of vibration of the string.

The length of one string is 68 cm.

Calculate the extension required to produce a tension of 93.4 N in the string.

Young modulus of string material = 1.8 × 10⁹N m⁻²

cross-sectional area of string = 6.6 10⁻⁷m²

The vibrating length of string is unchanged by turning the tuning peg.

Explain the effect that tightening the string has on the frequency of the sound produced.

The photograph shows a guitar.

When a guitar string is plucked, a standing wave is created.

Explain how a standing wave is created on the string.

The diagram shows a standing wave on a guitar string.

The oscillating length of the guitar string is 66 cm.

State the wavelength for this standing wave.

Calculate the frequency of vibration for this standing wave.

tension in guitar string = 88.6 N

mass per unit length of guitar string = 4.47 10⁻³kg m⁻¹

One end of the guitar string is wrapped around a cylindrical tuning peg. Turning the peg changes the total length of the string and hence changes the tension in the string. This changes the frequency of vibration of the string.

The length of one string is 68 cm.

Calculate the extension required to produce a tension of 93.4 N in the string.

Young modulus of string material = 1.8 × 10⁹N m⁻²

cross-sectional area of string = 6.6 10⁻⁷m²

The vibrating length of string is unchanged by turning the tuning peg.

Explain the effect that tightening the string has on the frequency of the sound produced.

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

The mouse repeller produces ultrasound of frequency 26.0 kHz.

Seat belts are being tested by a car manufacturer. In the test, a car moving at a steady speed of 28 m s^–1 collides with a wall and stops.

A crash-test dummy in the driving seat is wearing a seat belt made from polyester webbing. The seat belt has a cross-sectional area of 0.85 cm² and a total length of 2.0 m. A student suggests that in the collision the seat belt absorbs all the kinetic energy of the dummy.

Show that the energy per unit volume that would have to be absorbed by the seat belt is about 2 10⁸ J m^–3.

mass of dummy = 75 kg

The graph shows how stress varies with strain for the seat belt.

Show that the area under the graph represents the energy stored per unit volume in the seat belt.

Use the graph to determine whether the seat belt absorbs all the kinetic energy of the dummy from part (a).

In this collision, the maximum strain of the seat belt is 0.075