Stationary 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.

In an investigation of stationary waves on different wires, the frequency f of the first harmonic is measured.

Which quantity produces a straight-line graph through the origin when plotted against f ? In each case all other variables are kept constant.

This question is based on an investigation into how the frequency of the first-harmonic stationary wave on a string varies with tension (required practical activity 1).

Figure 1 shows the arrangement for this investigation.

The tension in the string is equal to the weight mg of the total mass m tied to the string.

The frequency is changed until the first-harmonic stationary wave appears on the string. This frequency f is determined using the oscilloscope.

The process is repeated for different values of m.

Identify two control variables for the arrangement shown in Figure 1.

You may annotate Figure 1 as part of your answer.

The total mass m is made from a mass hanger and up to three slotted masses.

One 20 g, one 50 g and one 100 g slotted mass are provided. The hanger has a mass of 100 g.

The hanger must not be removed from the string.

Which statement is correct?

Tick (✓) one box.

The frequency f is calculated using the waveform displayed on the oscilloscope. Figure 2 shows the waveforms displayed on the oscilloscope when m = 100 g and when m = 200 g.

m = 100 g m = 200 g

The y-voltage gain setting is 5 V div⁻¹. The time-base setting is 20 ms div⁻¹.

Determine V_rms, the root mean square voltage of the ac supply.

Determine the period of the waveform on Figure 2 when m = 100 g.

Deduce whether Figure 2 shows the expected result of doubling m from 100 g to 200 g.

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 loudspeaker producing a single-frequency sound is mounted above a tube filled with water. A tap at the bottom of the tube is opened to allow the water to run out.

A student observes the change in loudness of the sound emitted by the tube as the water runs out. When the length of the column of air in the tube reaches L₁, the loudness is at its first maximum. The next maximum is reached when the length of the column of air is L₂.

What is the wavelength of the sound emitted by the loudspeaker?

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?

A standing wave is created on a string.

Which statement about the two waves that create the standing wave is not correct?

The frequency of the first harmonic of a standing wave on a string is f. The tension in the string is T. The tension is increased to 4 T without changing the length or mass of the string.

Which harmonic has a frequency 2 f after this change?

Figure 1 shows apparatus used to investigate the properties of microwaves. The microwaves from the transmitter T are vertically polarised and have a wavelength of about 3 cm. The microwaves are detected at the receiver by a vertical metal rod R.

Explain how the apparatus can be used to demonstrate that the waves from T are vertically polarised.

Figures 2a and 2b show T and R and two different positions of a metal plate M that reflects microwaves. M is vertical and parallel to the direct transmission from T to R.

In an experiment, T and R are about two metres apart. M is moved slowly towards X Figure 2a shows the initial position of M. Figure 2b shows M when it has been moved a few centimetres.

The arrowed lines show the path of waves that reach R directly and the path of waves that reach R by reflection from M.

Explain what happens to the signal detected by R as M is moved slowly towards X.

Figure 3 shows an arrangement used in a different experiment to try to determine the wavelength of the microwaves.

A double-slit arrangement is placed between T and R.

The initial position of R is the same distance from each slit and is 0.45 m from the midpoint of the two slits. AB is a line perpendicular to the line between T and the initial position of R. R can be moved 0.25 m towards A and 0.25 m towards B along AB.

The two slits act as two coherent sources with a separation of 0.12 m.

Suggest why Young’s double-slit equation should not be used to determine the wavelength.

The wavelength is known to be about 3 cm.

Deduce whether this practical arrangement is suitable for a determination of a value for the wavelength.

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?

Stationary waves are set up on a rope of length 1.0 m fixed at both ends.

Which statement is not correct?

A stationary wave is formed on a stretched wire.

Figure 14 shows the wire, fixed at one end, supported by two bridges and passing over a pulley.

A 0.500 kg mass is attached to the free end of the wire. A uniform horizontal magnetic field is applied perpendicular to the wire between the bridges. A signal generator is connected to each end of the wire.

The oscilloscope shown is used to determine the frequency of the output of the signal generator. The wire oscillates because the alternating current in the wire interacts with the magnetic field.

Figure 15 shows the first harmonic stationary wave produced when the distance x between the bridges is adjusted.

The output potential difference (pd) of the signal generator is displayed on the oscilloscope, as shown in Figure 16.

The time-base setting of the oscilloscope is 10 ms cm¹.

Determine f, the frequency of the alternating pd.

A metre ruler is placed next to the bridges supporting the wire, as shown in Figure 17

Determine c.

Determine, in kg m¹, the mass per unit length of the wire.

A student uses digital vernier callipers to measure the diameter of a cylindrical metal rod. The student places the rod between the jaws of the callipers and records the reading indicated. Without pressing the zero button, the student removes the rod and closes the jaws.

Figure 18 shows the calliper readings in millimetres, before and after the jaws are closed.

Calculate the diameter d of the rod.

Describe relevant procedures to limit the effect of random error in the result for d.

Determine the density of the rod. The mass per unit length of the rod is 3.54 10³ kg m¹.

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.

Table 1 shows the results of the experiment.

Show that the data in Table 1 are consistent with the relationship

where T is the tension in the nylon string .

The nylon string used has a density of 1150 kg m^–3 and a uniform diameter of

Determine the length l of the string used.

The student uses the relationship in question 02.1 to predict frequencies for tensions that are much larger than those used in the original experiment.

Explain how the actual frequencies produced would be different from those that the student predicts.

and with the same tension as the first.

What length and diameter are required for the second string so that both strings have the same first-harmonic frequency?

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