C.4 Standing waves and resonance

Three oscillators with different degrees of damping are each displaced by the same amount and released at time t = 0. The graph shows how the displacement of each oscillator varies with time.

Which row identifies the critically damped oscillator and the heavily damped oscillator?

Two waves that are identical except for their direction of travel move in opposite directions along a string and form a standing wave.

What is (amplitude of the standing wave at an antinode)/(amplitude of each travelling wave)?

Which is an example of a useful application of resonance?

A smart speaker emits a steady note towards a flat, hard wall. A standing wave forms in the air between the speaker and the wall.

Which statement explains how the standing wave forms?

In a vibration test, a circuit board on a flexible mount is shaken by a vibration table. The table exerts a periodic force of constant amplitude on the mount. The natural frequency of the board on its mount is 18 Hz, and the damping is light.

The table is set to 30 Hz. Once the motion of the board is steady, which row describes the frequency of oscillation of the board, and its amplitude compared with that when the table is set to 18 Hz?

A diver jumps from the end of a springboard. The board then oscillates up and down with light damping until it comes to rest.

As the oscillations die away, which row describes the amplitude of the oscillations and the time taken for each oscillation?

A longitudinal standing wave is set up in the air inside a tube. Which statement correctly describes a point that is a displacement node?

An acoustic levitator traps tiny polystyrene beads at the nodes of a standing sound wave in air. Adjacent beads are 4.3 mm apart. The speed of sound in the air is 343 m s⁻¹. What is the frequency of the sound?

An organ pipe is open at both ends and sounds its second harmonic. How many displacement nodes are there inside the pipe?

A straight ventilation duct is open at both ends. Air in the duct resonates at its fundamental frequency. How many displacement nodes and displacement antinodes are there along the duct?

The human ear canal can be modelled as a tube closed at one end by the eardrum. A particular ear canal has a fundamental resonant frequency of 3.0 kHz. Which of the following is also a resonant frequency of this ear canal?

A clarinet player warms up the instrument by blowing through it. The air inside becomes warmer, so the speed of sound in it increases. The length of the air column does not change. What happens to the wavelength and the frequency of the fundamental?

A standing sound wave is set up in a pipe. Which row correctly describes the open end of the pipe?

A mass–spring system is driven by an oscillator of variable frequency and a graph of amplitude against driving frequency is obtained. The experiment is repeated with a vane attached to the mass so that the damping is heavier. Compared with the first curve, the second curve has

A guitar string is tuned by turning a peg. The length of the vibrating string stays the same, but the tension is increased to four times its original value. What happens to the frequency of the fundamental?

Which statement about a lightly damped system at resonance is correct?

A standing wave is set up on a string. Point X lies in one loop and point Y lies in the adjacent loop, on the other side of a node. What is the phase difference between the oscillations of X and Y?

Which row correctly compares a travelling wave with a standing wave on a string?

Travelling wave / Standing wave

A sitar string is fixed at both ends and vibrates in its fourth harmonic. How many nodes are there on the string, including the two ends?

A string of length 0.65 m is fixed at both ends. Waves travel along the string at 260 m s⁻¹.

What is the frequency of the third harmonic?

A pipe open at both ends has a first harmonic of frequency f_open. One end of the pipe is then closed, and the new first harmonic has frequency f_closed. The speed of sound is unchanged.

What is f_closed/f_open?

The tension in a string fixed at both ends is increased to four times its original value. The length of the string does not change. The speed of waves on the string is proportional to the square root of the tension.

What is (new first-harmonic frequency)/(original first-harmonic frequency)?

A system is made to oscillate by a periodic driving force whose frequency can be varied. The damping of the system is then increased.

Which statement describes the change to the graph of amplitude against driving frequency?

A standing wave and a travelling wave of the same frequency exist on two identical strings.

Which statement is true of the standing wave but not of the travelling wave?

As the drum of a washing machine speeds up, the whole machine vibrates violently at one particular drum speed. At higher drum speeds the vibration becomes much smaller.

Which statement explains this?

The air in a pipe that is open at both ends can form standing waves of two successive harmonics with frequencies 540 Hz and 720 Hz. The speed of sound in the air is v.

What is the length of the pipe?

The air in a pipe that is closed at one end and open at the other vibrates in its fifth harmonic.

Which row gives the number of displacement nodes and the number of displacement antinodes in the air column? Include any at the ends of the pipe.

A string is fixed at both ends and vibrates in its second harmonic.

Which graph shows how the amplitude A of oscillation of the points on the string varies with the distance x from one end of the string?

The air in a tube of length L that is closed at both ends vibrates in its first harmonic with frequency f.

One end of the tube is then opened. Later the other end is opened as well. The speed of sound in the air does not change.

Which row gives the frequency of the first harmonic of the air when one end is open and when both ends are open?

In an atomic force microscope, a tiny cantilever is driven at its natural frequency of 300 kHz and oscillates with a constant amplitude. Air resistance removes energy from the cantilever at a rate of 9.0 × 10⁻¹² W.

How much energy does the driving force transfer to the cantilever during each oscillation?

The shock absorbers of a car normally give damping that is close to critical. A garage replaces them with shock absorbers that give much heavier damping.

The car then goes over a single bump. Which row describes the motion of the car body afterwards, and the time it takes to settle at its equilibrium position, compared with the original shock absorbers?

Two air particles, P and Q, lie on opposite sides of the same displacement node in a longitudinal standing wave. Each particle is a distance λ/8 from the node. What is the phase difference between the oscillations of P and Q?

The human ear canal can be modelled as a tube that is open at the outer end and closed by the eardrum. Its lowest resonant frequency is 3.4 kHz. Which of the following is not a resonant frequency of this model?

A narrow ventilation duct in a building resonates in its fundamental mode. The air temperature rises, which increases the speed of sound. The duct length does not change. What happens to the fundamental frequency and to the wavelength of the fundamental?

A rocket combustion chamber of length 0.50 m behaves acoustically as a tube closed at both ends. A longitudinal instability is observed in its lowest mode at a frequency of 1.2 kHz. What is the speed of sound in the hot gas?

A longitudinal standing wave is set up in a long spring used in a laboratory demonstration. Consider these statements.
I. The amplitude of oscillation varies with position along the spring.
II. There is a net transfer of energy along the spring.
III. All coils between two adjacent nodes oscillate in phase.
Which statements are correct?

Noise-cancelling headphones detect an ambient sound wave of pressure amplitude 4.0 mPa. They emit a wave of the same frequency with amplitude 3.0 mPa, exactly in antiphase with it. What is the intensity of the resulting sound as a fraction of the intensity of the ambient sound?

A sound engineer needs two pipes with the same fundamental frequency. Pipe P is open at both ends. Pipe Q is closed at one end. What is the ratio of their lengths, L_P/L_Q?

A smartphone speaker emits a steady 850 Hz tone above a tall measuring cylinder. The cylinder is slowly filled with water, which shortens the air column. Loud resonances are heard at successive air-column lengths that differ by 0.20 m. What is the speed of sound in the air?

A tube of length L is closed at one end. The air inside vibrates at the second-lowest frequency at which resonance is possible. At which distances from the closed end are the displacement nodes?

A building's air-conditioning duct hums loudly because a fan drives one of its resonant modes. Engineers line the inside of the duct with sound-absorbing foam. Compared with the unlined duct, how does the resonance curve (amplitude against driving frequency) for this mode change?

The suspension of a car uses springs together with shock absorbers. The shock absorbers are designed to give approximately critical damping. What is the main purpose of this?

A string on a stringed instrument has fundamental frequency f. It is replaced by a string of the same material, length and tension, but with twice the diameter. What is the new fundamental frequency?

A tall building has a natural frequency of sway of 0.8 Hz. During an earthquake the ground oscillates steadily at 0.5 Hz. Once steady oscillations are established, the building sways with a frequency of

A stay cable on a cable-stayed bridge is fixed at both ends. Gusting wind makes it vibrate strongly in its third harmonic. How many antinodes and nodes, including the end points, does the standing wave on the cable have?

A piano wire sounds a fundamental of 440 Hz. Over a hot day the technician finds the wire tension has risen by 10% while the length and mass per unit length are unchanged. The new fundamental frequency is approximately

A diver jumps off the end of a springboard. The board then oscillates with light damping caused by air resistance and internal friction. Which describes the subsequent motion of the board?

A tennis racket string has a vibrating length of 0.30 m between its fixed ends. It vibrates in its third harmonic. What is the wavelength of the standing wave?

A cellist presses a string so that its vibrating length becomes three-quarters of its original length. The wave speed on the string is unchanged. What is the ratio (new fundamental frequency) / (original fundamental frequency)?

Which statement correctly describes the motion of the points on a string that lie between two adjacent nodes of a standing wave?

An engineer drives a guy wire on a radio mast with a variable-frequency shaker. Both ends of the wire are fixed. Two successive resonant frequencies are 240 Hz and 300 Hz. What is the fundamental frequency of the wire?

A long tether connects two parts of a spacecraft and is held taut, with both ends fixed. A small pulse sent along the tether takes time t to travel from one end to the other and back again. What is the fundamental frequency of standing waves on the tether?

A pipe of length L is closed at one end and open at the other. The speed of sound in the air in the pipe is v.

What is the frequency of the standing wave with the lowest frequency above that of the first harmonic?

A standing wave is set up on a string fixed at both ends. The diagram shows the string at one instant of maximum displacement (solid line) and half a period later (dashed line). P, Q and R are points on the string. The distance PQ is λ/4 and the distance PR is 3λ/4, where λ is the wavelength.

Which row gives the phase difference between the oscillations of P and Q, and between those of P and R?

A string is clamped at end X. Its other end, Y, is tied to a light ring that slides without friction on a vertical rod, so that Y is a free end.

Which diagram shows the third harmonic of the string? Each diagram shows the two extreme positions of the string.

One end of a long, heavy rope is tied to a wall in a gym. An athlete moves the other end up and down so that waves of frequency 2.5 Hz and wavelength 1.8 m travel along the rope. The waves are reflected at the wall and a standing wave forms.

Which row gives the time between successive instants at which the rope is straight and the distance between adjacent nodes?

A pipe of length L is closed at one end and open at the other. It resonates at its third harmonic. Point X lies on the axis of the pipe, a distance L/3 from the closed end. Which row describes X?

Three consecutive resonant frequencies of a pipe open at both ends are 240 Hz, 360 Hz and 480 Hz. One end of the pipe is then sealed. The length of the pipe is unchanged. What is the lowest resonant frequency of the sealed pipe?

Two wires X and Y are made of the same metal and are under the same tension. Each is fixed at both ends. Wire X has length L and diameter d. Wire Y has length 2L and diameter d/2. What is the ratio \(\frac{f_Y}{f_X}\) of their fundamental frequencies?

A string vibrates in a standing wave. At one instant every point on the string has zero displacement, so the string is straight. Which statement about this instant is correct?

A string fixed at both ends vibrates in its second harmonic at frequency f. A guitarist then lightly touches the string at a point one-third of its length from one end. This forces a node at that point, and the string vibrates at the lowest frequency now possible. What is this frequency?

A narrow plastic tube is closed at one end by a piston that can slide along the tube, so that the length L of the air column can be changed. A small loudspeaker connected to a signal generator is placed at the open end. For each length the student increases the frequency from a low value and records the frequency f at which the sound first becomes loudest. This is the first harmonic of the air column. The uncertainty in f is ±3 Hz. The air temperature is 22 °C.

One end of a string is attached to a vibration generator and the other passes over a pulley to a hanging mass. The vibrating length between the generator and the pulley is (1.500 ± 0.005) m. The frequency of the generator is slowly increased, and the frequency f is recorded each time a large-amplitude standing wave with n loops forms. Each frequency has an uncertainty of ±0.5 Hz.

A microwave transmitter faces a flat metal plate. A small probe receiver is moved along the line between them, and the student records the distance x from the plate of each position where the received signal is a minimum. The minima are numbered N = 1, 2, 3 … starting with the one nearest the plate. Each position has an uncertainty of ±0.2 cm.

A model of a car suspension consists of a mass hanging from a spring. A light vane fixed below the mass dips into a beaker of oil, which damps the motion. The mass is pulled down and released, and a motion sensor records its displacement. From the record, the student reads the amplitude A of every second oscillation, with an uncertainty of ±0.5 mm. The oscillation number n counts the oscillations completed since the first amplitude was measured.

The student's hypothesis is that the amplitude falls by the same fraction in every oscillation. If so, a graph of ln( A / mm ) against n is a straight line. The graph shows ln( A / mm ) against n with error bars.

A thin steel strip is clamped horizontally to a vibration generator so that a length L of the strip overhangs the clamp. The student increases the frequency of the generator slowly and records the frequency f at which the free end of the strip vibrates with the largest amplitude. This is repeated for different lengths L.

The student's hypothesis is that f is proportional to 1/L². The graph shows f against 1/L².

A trolley of mass (0.500 ± 0.001) kg is held between two springs on a horizontal track. The combined spring constant, measured in a separate experiment, is (40.0 ± 1.0) N m⁻¹. The far end of one spring is moved back and forth by a motor at a driving frequency f. When the motion is steady, a motion sensor records the amplitude A of the trolley. Each amplitude has an uncertainty of ±0.2 cm.

A student investigates the air in a plastic pipe of length (0.850 ± 0.002) m. One end of the pipe is open; the student cannot see whether the far end has been sealed. A small loudspeaker at the open end plays a short burst of noise, and a microphone connected to a computer records the sound. The frequency spectrum shows peaks at the resonant frequencies of the air column. The frequencies are listed in order of increasing frequency as peak number n. Each frequency has an uncertainty of ±2 Hz.

A student investigates how the damping of a driven oscillator affects its amplitude at resonance. A magnet of mass 0.250 kg hangs from a spring whose upper end is attached to a vibration generator. The generator moves up and down with an amplitude of 3.0 mm at a frequency that can be varied.

A stack of n identical thin aluminium sheets stands vertically beside the path of the magnet, close to it but not touching it. The sheets exert a damping force on the moving magnet, and the damping increases with n. For each value of n the student varies the driving frequency, finds the resonant frequency f_r, and measures the maximum amplitude A_max of the magnet from a video. The student's hypothesis is that A_max is inversely proportional to n.

The graph shows 1/A_max against n for six of the values of n, with error bars for the uncertainty in A_max.

A student investigates the positions of the frets on a guitar. When a string is pressed against fret n, the vibrating length L of the string becomes the distance from that fret to the bridge. For the open string (n = 0), L = 648.0 mm. The student measures each length with a steel tape, with an uncertainty of ±0.5 mm. Assume that pressing the string does not change its tension.

The student's hypothesis is that each fret raises the frequency of the first harmonic by the same factor r. If so, ln(L / mm) = ln(L₀ / mm) − n ln r, where L₀ is the length of the open string.

The graph shows ln(L / mm) against n for the data in the table, except n = 6.

A steel wire passes over two bridges (0.600 ± 0.001) m apart and then over a pulley to a hanger carrying masses m. The tension T in the wire is taken to be mg. The wire is plucked at its centre and a phone app measures the frequency f of the first harmonic.

The speed of transverse waves on the wire is v = √(T/μ), where μ is the mass per unit length. The student's hypothesis is that f is proportional to √T.

A student investigates how the amplitude varies along a standing wave. An elastic cord is stretched horizontally between a vibration generator and a fixed clamp. The length of the cord between them is (1.800 ± 0.005) m. The generator moves its end of the cord with an amplitude that is negligible, so both ends behave as nodes. At a frequency of (24.0 ± 0.1) Hz the cord vibrates in its third harmonic.

A vertical millimetre scale is fixed just behind the cord. In a long-exposure photograph the vibrating cord appears as a blurred band. At different distances x from the generator the student reads the full width w of the band from the scale in the photograph and calculates A = w/2.

The student's hypothesis is that A = A_N |sin(2πx/λ)|, where λ is the wavelength, A_N is the amplitude at an antinode and the angle 2πx/λ is in radians. The graph shows A against |sin(2πx/λ)| for seven of the positions, with error bars for the uncertainty in A.

The ear canal of an adult is a tube about 2.5 cm long. It is open at the outer end and closed at the inner end by the eardrum. The air in the canal can be modelled as the air in a pipe with one open end and one closed end. The speed of sound in the air in the canal is 340 m s⁻¹.

Four simple pendulums P, Q, R and S hang from the same horizontal rod. Their lengths are 0.25 m, 0.39 m, 0.60 m and 0.95 m respectively. Each bob is a light paper cone, so air resistance damps the motion. A motor makes the rod oscillate horizontally with a small amplitude, at a frequency that can be varied.

A science-centre exhibit has a horizontal cord of length 1.20 m with both ends fixed. A vibrator near one end sends waves of frequency 5.0 Hz and wavelength 0.80 m along the cord, and the waves are reflected at the far end. Figure 1 shows, at one instant, the displacement of the cord that each of the two travelling waves would produce on its own. Both waves have an amplitude of 2.0 cm.

A long spring is stretched along a smooth bench between two clamps 3.00 m apart. A vibration generator near one clamp moves a coil of the spring back and forth along the axis of the spring. At a frequency of 4.0 Hz a standing wave forms, and the coils at 0.75 m, 1.50 m and 2.25 m from the left-hand clamp stay at rest.

Figure 1 shows, at one instant of maximum displacement, the displacement s of the coils along the axis of the spring (positive to the right) against their equilibrium distance x from the left-hand clamp. P and Q are coils at x = 0.25 m and x = 1.00 m.

A medical ultrasound probe contains a thin piezoelectric disc. The disc is driven so that a longitudinal standing wave forms across its thickness. Both flat faces of the disc are free to move. The disc vibrates in its fundamental mode at 2.5 MHz. The speed of longitudinal waves in the disc material is 4000 m s⁻¹.

A didgeridoo is a long wind instrument. It can be modelled as a tube of length 1.30 m that is closed at the player's mouth and open at the far end. The speed of sound in the air inside is 340 m s⁻¹.

A safety whistle on an industrial pressure vessel contains an air column 0.35 m long. The column is closed at one end and open at the other. The speed of sound in the air is 343 m s⁻¹.

A badminton racquet string is stretched between two points on the frame 0.20 m apart. The tension in the string is 100 N and its mass per unit length is 6.0 ×10⁻⁴ kg m⁻¹. The speed of transverse waves on a string is v = √(T/μ).

At an adventure park, a steel cable on a short training zip-wire is under a tension of 2000 N. A maintenance worker strikes one end of the cable. A transverse pulse travels 50 m along the cable in 0.40 s. Assume the tension and the mass per unit length are the same everywhere along the cable.

A slackline used for balance training is a flat webbing strap anchored firmly between two trees 12 m apart. When an athlete bounces lightly at the centre, the line vibrates in its fundamental mode with frequency 1.5 Hz.

In a sports science laboratory, one end of a heavy training rope is tied to a wall bracket. A motor oscillates the other end with a small amplitude and a steady frequency, and a standing wave forms on the rope.

One end of a rubber cord is attached to a vibration generator G. The generator moves the cord with a very small amplitude, so this end behaves as a fixed end. The other end of the cord is tied to a light ring that can slide without friction along a vertical rod, so this end is free to move up and down.

The length of the cord is 1.50 m and waves travel along it at 9.0 m s⁻¹.

A simple wind instrument is modelled as a straight pipe of length 0.660 m that is open at both ends. The speed of sound in the air in the pipe is 343 m s⁻¹.