Properties of waves

State what is meant by a transverse wave.

Figure 2 shows apparatus that is used to investigate stationary waves on a stretched wire.

A block of weight W is used to keep the wire under tension.

The frequency of the ac supply is varied until a stationary wave is produced on the wire.

Explain how a stationary wave is produced on the wire.

Figure 3 shows a small section of the wire at one instant. Five points on the wire are labelled P₁ to P₅. The dashed line represents the position of the wire when the ac supply is turned off.

Describe how the phase of the oscillating particles varies along the wire

A student investigates stationary waves on a wire using the apparatus in Figure 2. The investigation requires the student to produce the first five harmonics on the wire.

The student needs to choose one of two wires, A or B, for the investigation.

The mass of a 2.00 m length of wire A is 1.32 g. The mass of a 2.00 m length of wire B is 2.94 g.

The ac supply can produce signals in the range 1 Hz to 50 Hz. The length of the wire that vibrates between the vibration generator and the pulley is 1.50 m.

The student needs to choose one value for weight W for the investigation. W can be either 1.0 N or 5.0 N.

Determine, in kg m⁻¹, the mass per unit length of each wire. Go on to suggest which wire and which value of W the student should use to produce the first five harmonics.

An experiment is done to investigate stationary waves on a string. A string of length 1.05 m is attached between a clamp stand and a vibration generator. A stationary wave is formed on the string when the vibration generator frequency is 625 Hz.

Figure 6 shows the variation of displacement with distance from one end of the string at time t = 0 At this time all points on the string have their maximum displacement. S is one point on the string.

The stationary wave is produced by two progressive waves travelling in opposite directions on the string.

Deduce the amplitude of one of the progressive waves.

Determine, in m s⁻¹, the speed of one of the progressive waves.

State the phase relationship between the two waves when t = 0

Sketch, on Figure 7, a graph to show how the displacement of S varies with t.

Figure 7

A laser emits light of wavelength 600 nm for 10 ns.

What is the number of complete waves emitted by the laser?

A pencil is weighted with a thin coil of wire. The volume of the wire is negligible. Figure 14 shows the pencil and wire floating in equilibrium in water.

Figure 14 Figure 15

In Figure 14 the combined weight of the pencil and wire is equal to an upwards force called the buoyancy force. The length of the pencil that is submerged is l. A student pushes the pencil down through a displacement y as shown in Figure 15. The buoyancy force is now greater than the weight. There is a resultant upward force F acting on the pencil when the student releases it. The magnitude of F for any value of y is given by

F = Aρgy

where A is the cross-sectional area of the pencil ρ is the density of water g is the acceleration due to gravity.

The pencil is pushed down and released. The pencil then oscillates vertically about the equilibrium position.

Show that the pencil moves with simple harmonic motion.

The time period T of the vertical oscillations is given by

The measured value of l in Figure 15 is 85 mm. The pencil is pushed down 5.0 mm and released.

Calculate the maximum acceleration of the pencil.

A ship floating in the sea can be modelled by the pencil floating in water. The ship can oscillate vertically. These oscillations are called heave oscillations.

Wave motion causes forced oscillations of the ship. Under certain conditions, heave resonance may then occur.

Explain what is meant by resonance.

Figure 16 shows a ship moving through continuous waves of wavelength 118 m

The ship is moving steadily at 8.0 m s⁻¹ relative to the seabed in the same direction as the waves.

The natural frequency of heave oscillations of the ship is 0.13 Hz.

A crew member needs an emergency operation. The ship’s doctor is confident that she can do the operation if the ship remains fairly steady.

There are two options:

• stop the ship’s motors and loosely anchor the ship to the seabed

• continue to sail the ship at 8.0 m s⁻¹ in the same direction.

Deduce which is the better option. Support your answer with a calculation.

Section B

Each of Questions 07 to 31 is followed by four responses, A, B, C and D.

For each question select the best response.

A string with a length of 1.2 m vibrates at its second harmonic. The diagram shows the displacement – time graph for a point on the string.

What are the wavelength and frequency of the wave on the string?

Figure 1 shows apparatus used to measure the speed of sound in a steel rod.

The steel rod is suspended from a beam using rubber bands. When the hammer is in contact with the end L of the steel rod, a circuit is completed and the signal generator is connected to the oscilloscope.

Figure 2 shows the waveform then displayed on the oscilloscope.

Which control on the oscilloscope should be used to centre the trace vertically on the screen? Tick (✓) one box.

When the hammer hits end L, a sound wave travels along the steel rod and is reflected at end R. When the wave returns to L the rod bounces away from the hammer and the circuit is broken.

Figure 3 shows the waveform produced by the brief contact between the hammer and L. Note that the waveform has now been centred vertically.

Figure 4 shows the time-base setting of the oscilloscope.

The distance between L and R in Figure 1 is 0.870 m.

Deduce the speed of sound in the steel rod.

A student repeats the experiment using a steel rod of twice the length.

Explain:

• how using the longer rod affects the waveform displayed

• any changes needed to get an accurate result for the speed.

You should include numerical detail.

The diagrams show the displacement–distance graph for a wave and the displacement–time graph for a point in the wave.

Which is correct for this wave?

The diagram shows a stationary wave on a string at one instant in time. P, Q and R are three points on the string.

Which row is correct?

A student investigates the interference of sound waves using two loudspeakers, P and Q, connected to a signal generator (oscillator). Each loudspeaker acts as a point source of sound.

Figure 3 shows the arrangement.

Point O is the midpoint between P and Q.

Explain why the two loudspeakers are coherent sources of sound waves.

The student faces the two loudspeakers at point A. Point A is at equal distances from P and Q. He then moves to point B, at right angles to the line OA, still facing the two loudspeakers. As his head moves from A to B the amplitude of the sound wave he hears decreases and then increases. The amplitude starts to decrease again as he moves beyond B.

Explain why the variation in amplitude occurs as he moves from A to B.

The student records the following data:

separation of the two loudspeakers = 0.30 m distance OA = 2.25 m distance from A to B = 0.95 m

Show that the path difference for the sound waves from the two loudspeakers to point B is about 0.1 m.

The frequency of the sound wave is 2960 Hz.

Calculate the speed of sound from the student’s data.

The student moves his head to point C as shown in Figure 4. The emitted frequency of the sound from the loudspeakers is then gradually decreased.

Discuss the effect that this decrease in frequency has on the amplitude of the sound wave heard by the student.

A wave travels across the surface of water. The diagram shows how the displacement of water particles at the surface varies with distance.

Which row correctly describes both w and z ?

The diagram shows the cross-section of a progressive transverse wave travelling at 24 cm s⁻¹ on water. The amplitude of the wave is 2.0 cm and the frequency is 4.0 Hz.

Which statement is correct?

Two pendulums A and B oscillate with simple harmonic motion. The time period of A is 2.00 s and the time period of B is 1.98 s.

A and B are released in phase.

What is the number of oscillations of A before A and B are next in phase?

Figure 4 shows the structure of a violin and Figure 5 shows a close-up image of the tuning pegs.

The strings are fixed at end A. The strings pass over a bridge and the other ends of the strings are wound around tuning pegs that have a circular cross-section. The tension in the strings can be increased or decreased by rotating the tuning pegs.

Explain how a stationary wave is produced when a stretched string is plucked.

The vibrating length of one of the strings of a violin is 0.33 m When the tension in the string is 25 N, the string vibrates with a first-harmonic frequency of 370 Hz

Show that the mass of a 1.0 m length of the string is about 4 × 10^–4 kg

Determine the speed at which waves travel along the string in question 03.2 when it vibrates with a first-harmonic frequency of 370 Hz

Figure 6 shows how the tension in the string in question 03.2 varies with the extension of the string.

The string with its initial tension of 25 N is vibrating at a frequency of 370 Hz The diameter of the circular peg is 7.02 mm

Determine the higher frequency that is produced when the string is stretched by rotating the tuning peg through an angle of 75°

Assume that there is no change in the diameter of the string.

In the diagram, P is the source of a wave of frequency 50 Hz

The wave travels to R by two routes, P → Q → R and P → R. The speed of the wave is

What is the path difference between the two waves at R in terms of the wavelength λ of the waves?

What is the period of the wave?

A microwave transmitter T and a receiver R are arranged on a line marked on the bench.

A metal sheet M is placed on the marked line perpendicular to the bench surface.

Figure 1 shows side and plan views of the arrangement. The circuit connected to T and the ammeter connected to R are only shown in the plan view.

The distance y between T and R is recorded.

T is switched on and the output from T is adjusted so a reading is produced on the ammeter as shown in Figure 2.

M is kept parallel to the marked line and moved slowly away as shown in Figure 3

The reading decreases to a minimum reading which is not zero. The perpendicular distance x between the marked line and M is recorded.

The ammeter reading depends on the superposition of waves travelling directly to R and other waves that reach R after reflection from M.

State the phase difference between the sets of waves superposing at R when the ammeter reading is a minimum. Give a suitable unit with your answer.

Explain why the minimum reading is not zero when the distance x is measured.

When M is moved further away the reading increases to a maximum then decreases to a minimum.

At the first minimum position, a student labels the minimum n = 1 and records the value of x. The next minimum position is labelled n = 2 and the new value of x is recorded. Several positions of maxima and minima are produced.

Describe a procedure that the student could use to make sure that M is parallel to the marked line before measuring each value of x. You may wish to include a sketch with your answer.

It can be shown that

The student plots the graph shown in Figure 4.

and adds error bars to the graph.

Determine

the maximum gradient G_max of a line that passes through all the error bars

the minimum gradient G_min of a line that passes through all the error bars.

Explain how the graph in Figure 4 can be used to obtain the value of y. You are not required to determine y.

Suppose that the data for n = 13 had not been plotted on Figure 4.

Add a tick () in each row of Table 1 to identify the effect, if any, on the results