C.4 Standing waves and resonance

A lightly damped mass–spring system is driven by a periodic force. After some time the system oscillates with a constant amplitude.

Which statement about the steady oscillation is correct?

A string of length 1.2 m, fixed at both ends, vibrates at a frequency of 150 Hz in the mode shown.

What is the lowest frequency at which a standing wave can form on this string?

Two waves of the same amplitude, frequency and wavelength λ travel in opposite directions along a string and superpose to form a standing wave.

Which statements about the standing wave are correct?

I. Its points oscillate at the same frequency as the travelling waves.

II. Adjacent nodes are a distance λ apart.

III. It transfers energy along the string from one end to the other.

A model building on a shaking table has a natural frequency of sway of 2.5 Hz and is lightly damped. The table now moves the base sinusoidally at 2.0 Hz until a steady oscillation is reached.

Which row gives the frequency of the steady oscillation of the model and its amplitude compared with the amplitude when the table is driven at 2.5 Hz?

A narrow glass tube of length L is sealed at both ends and contains air. What is the wavelength of the sound wave in the fundamental (first harmonic) standing wave of the air column?

A section of gas pipeline is sealed at both ends. The gas inside forms a standing sound wave in its fourth harmonic. How many displacement nodes are there along the section, including the ends?

A microwave oven with its turntable removed sets up a standing wave inside the cooking chamber. A flat bar of chocolate placed in the oven melts first at a row of spots. These spots mark adjacent antinodes and are 6.1 cm apart. What is the wavelength of the microwaves?

Which property belongs to a standing wave but not to a travelling wave?

In a hospital acoustics test, an air-filled duct of length 0.85 m is sealed at both ends. The speed of sound in the air is 340 m s⁻¹. What is the lowest frequency at which a standing wave can form in the duct?

A beekeeper stretches a light cord of length 1.2 m between two vertical poles. A small ring at each end of the cord slides freely on its pole, so each end of the cord is free to move. The cord vibrates in its fundamental mode. What is the wavelength of the standing wave?

Fine powder in a horizontal glass tube settles into small heaps at points where the air is not moving. A loudspeaker sends sound of frequency 2000 Hz along the tube and the heaps form with a separation of 0.085 m between neighbouring heaps. What is the speed of sound in the tube?

A wing of a Formula 1 car vibrates strongly when the engine runs at a particular speed. A tuned damper is fitted to the wing and the engine speed is swept through the same range. Compared with the undamped wing, the amplitude-frequency response of the damped wing has a peak that is

In a sports-science laboratory, a taut elastic cord of length 0.40 m is fixed at both ends. It vibrates in its fundamental mode at a frequency of 300 Hz. What is the speed of transverse waves on the cord?

A railway signal whistle is a pipe that is closed at one end and open at the other. Which row gives the ratio of the frequencies of its three lowest resonances?

A rail-tunnel ventilation shaft acts as an air column open at both ends. It has length 4.25 m. The speed of sound in the air is 340 m s^-1. What is the frequency of the third harmonic?

A pipe closed at one end has a first-harmonic frequency of 100 Hz. A pipe of the same length open at both ends is sounded.

What is the frequency of its second harmonic?

A string of length L is fixed at both ends. Two consecutive resonant frequencies of the string are f_A and f_B, where f_B > f_A. The speed of transverse waves on the string is v.

What is L?

The diagram shows the displacement nodes and antinodes of a standing wave in the air in a pipe that is open at both ends. The frequency of this standing wave is 850 Hz.

What is the first-harmonic frequency of the pipe?

A mass on a spring is driven by a periodic force of variable frequency. The amplitude of the resulting oscillation is measured for a range of driving frequencies, first with light damping and then with the damping increased.

Which row gives the effect of increasing the damping on the resonance curve?

A string of length 1.20 m fixed at both ends vibrates in its third harmonic, so that nodes are found at 0, 0.40 m, 0.80 m and 1.20 m from one end. P, Q and R are points on the string at 0.20 m, 0.30 m and 0.60 m from that end.

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

A light elastic cord of length 0.90 m is clamped at one end. The other end is tied to a light ring that can slide without friction along a smooth vertical rod, so that this end of the cord is a free end. The speed of transverse waves on the cord is 18 m s⁻¹.

What is the second-lowest frequency at which a standing wave can form on the cord?

A pipe open at both ends has a first-harmonic frequency f. One end of the pipe is closed with a cap, and then the other end is also closed with a cap. The speed of sound in the air in the pipe does not change.

What is the first-harmonic frequency of the pipe with both ends closed?

A stretched wire vibrates in its second harmonic. A stroboscope photograph is taken at an instant when every point of the wire is at its equilibrium position, so the wire is straight.

Which statement describes the motion of the points of the wire at this instant?

Two elastic cords P and Q have the same length L. Both ends of P are tied to light rings that slide without friction on smooth vertical rods, so that both ends of P are free. One end of Q is clamped and its other end is free. The speed of transverse waves on Q is twice the speed of transverse waves on P.

What is (first-harmonic frequency of P)/(first-harmonic frequency of Q)?

Two loudspeakers face each other along a straight line and emit sound of frequency 680 Hz in phase. The speed of sound is 340 m s⁻¹. A microphone moves at a constant speed of 0.85 m s⁻¹ along the line between the loudspeakers, and the loudness that it detects rises and falls regularly.

What is the time between successive maxima of loudness?

A sealed air duct in an aircraft cabin is 0.85 m long and is closed at both ends. The speed of sound in the air is 340 m s⁻¹. Which frequency is a possible standing-wave frequency for the air in the duct?

In a model of a tethered buoy, a taut cable of length L is fixed at its lower end. Its upper end is attached to a light ring that slides without friction on a vertical pole. The upper end is therefore free to move transversely. Which wavelength is a possible standing wave on the cable?

A vibration-testing rig for satellite components drives a mass–spring system over a range of frequencies. The amplitude of the mass is recorded at each frequency. The damping in the system is then increased and the test is repeated. Which describes the change to the amplitude–frequency curve?

As a washing machine's drum speeds up, the machine's frame vibrates violently at one particular spin rate. At higher spin rates the vibration becomes much smaller. What is the best explanation for the violent vibration?

A string of length L is fixed at both ends and vibrates in its third harmonic. Point X is 0.10L from one end, point Y is 0.25L from the same end, and point Z is 0.40L from the same end. What are the phase differences between X and Y, and between X and Z?

The human ear canal can be modelled as a tube of length 2.5 cm, open at the outer end and closed by the eardrum. The speed of sound in the canal is 340 m s⁻¹. The lowest resonant frequency is 3400 Hz. What is the next higher resonant frequency?

A car's suspension is driven by bumps in the road and shows a resonance peak. Its shock absorbers wear out, so the damping is reduced. Compared with new shock absorbers, how does the graph of oscillation amplitude against driving frequency change?

A 490 Hz tuning fork is held above a vertical tube whose lower end is in water. The water level is lowered. Resonance is first heard when the air column is 0.17 m long, and next heard when it is 0.52 m long. What is the speed of sound in the air?

A tube closed at one end is held vertically and the water level is lowered. Sound from a loudspeaker of fixed frequency produces resonance when the air column is 0.160 m long and again when it is 0.500 m long. The speed of sound is 340 m s⁻¹. What is the frequency of the sound?

The beekeeper's cord (free ends, fundamental frequency 12 Hz) is now driven so that it vibrates in its fourth harmonic. Counting the end points, how many nodes and antinodes does the standing wave have, and what is its frequency?

A standing wave is set up on a stretched string. Which statement correctly compares the motion of points on the string between two adjacent nodes?

A taut string of length L has one end fixed. The other end is attached to a light ring that slides without friction on a vertical rod, so that the end is free to move. The lowest resonant frequency of the string is f. What is the next-lowest resonant frequency of the string?

A tuning fork of frequency 680 Hz is held over a vertical tube, closed at the lower end by water, whose level is lowered. Resonance is first heard when the air column is 0.115 m long and again when it is 0.365 m long. What is the speed of sound in the air?

A railway footbridge behaves as a driven oscillator with light damping, driven by regular footsteps of varying frequency. Extra damping is then added to the bridge. Which change to the graph of amplitude against driving frequency is correct?

A tennis racket string is stretched between two fixed points. A stringing machine increases the tension in the string by a factor of 4. The vibrating length of the string is also halved. The mass per unit length is unchanged. By what factor does the fundamental frequency change?

A string of length L is fixed at both ends and vibrates in its third harmonic. The maximum amplitude on the string is A. Point P is a distance L/12 from one end. Point Q is a distance L/2 from the same end. Which statement about the oscillations of P and Q is correct?

A violin string has a fundamental frequency of 440 Hz. A player presses the string so that its vibrating length becomes 2/3 of the original length. The tuning peg also slips, increasing the tension by 44%. The speed of waves on the string is proportional to the square root of the tension. What is the new fundamental frequency?

An air duct in a building resonates at 450 Hz and at 750 Hz. There are no resonant frequencies between these two values. Which row correctly describes the duct and its fundamental frequency?

A standing wave on a string has an amplitude of 6.0 mm at an antinode. What is the amplitude of vibration of a point on the string that is one twelfth of a wavelength from the nearest node?

A pipe of length L is closed at one end and open at the other, and its fundamental frequency is f. A second pipe of length 2L is closed at both ends and contains the same gas. Which statement describes the resonant frequencies of the second pipe?

A string of length 1.20 m and mass per unit length 2.5 × 10^-3 kg m^-1 is fixed at both ends. It vibrates in its third harmonic at a frequency of 150 Hz. What is the tension in the string?

An elastic cord of length 1.20 m lies along a horizontal bench. The end at x = 0 is attached to a vibration generator of very small amplitude, so that it behaves almost as a fixed end. The end at x = 1.20 m is tied to a light ring that slides freely on a smooth vertical rod. The generator frequency is 15.0 Hz and a standing wave forms.

The amplitude A of the cord is measured at different distances x from the generator with an uncertainty of ±1 mm. The graph shows the data with a smooth curve.

A string passes over a pulley at one end of a bench and carries a hanging load of mass M. A vibration generator near the other end drives the string, and the vibrating length between the generator and the pulley is 0.800 m. For each load the student finds the frequency f₁ of the first harmonic.

The speed of waves on a string is v = √(F/μ), where F is the tension and μ the mass per unit length. The student plots f₁² against M.

A metal strip is clamped at one end so that it can oscillate horizontally; an aluminium vane is fixed to its free end. The strip is plucked and its free oscillation is recorded with a motion sensor (Figure 1). The clamp is then attached to a vibration generator, and for each driving frequency f the student records the steady amplitude A of the vane, taking readings every 0.5 Hz (curve X, Figure 2).

Two strong magnets are then placed on either side of the vane, without touching it, and the measurements are repeated (curve Y).

A long horizontal tube is surrounded by a water jacket whose temperature θ is read on a thermometer. A loudspeaker at the open end emits sound of frequency 1500 Hz. A piston inside the tube is moved slowly away from the loudspeaker and a microphone detects successive resonances. The distance Δx between the piston positions of two adjacent resonances is measured with a metre rule.

Kinetic theory predicts that the square of the speed of sound v is proportional to the absolute temperature of the air. The graph shows v² against θ with the line of best fit.

A cello string has a vibrating length of 0.690 m between its two fixed ends and a mass per unit length μ of 1.50 × 10⁻³ kg m⁻¹. The speed of transverse waves on the string is v = √(T/μ), where T is the tension. The string is wound around a peg of radius 4.0 mm, which is turned to tune it.

A thin wine glass rings at a frequency of 880 Hz when it is tapped. A small loudspeaker connected to a signal generator is placed 0.30 m from the glass. The speed of sound in air is 340 m s⁻¹.

A steel stay cable on a pedestrian footbridge is 24 m long. It is fixed at both ends. In strong wind, engineers observe a standing wave on the cable with three loops between the ends. The cable oscillates at a frequency of 1.8 Hz.

An acoustic levitator suspends small droplets of a medical sample in air without touching them. An ultrasonic transducer emits sound towards a flat reflector placed directly opposite it. The droplets settle at a line of positions, each a node of the standing wave, spaced 4.3 mm apart. The speed of sound in the air is 344 m s⁻¹.

A hospital sound laboratory uses an air-filled pipe of length 0.60 m that is sealed at both ends. Take the speed of sound in air as 340 m s⁻¹.

A tuning fork of frequency 680 Hz is held above a tube closed at one end. The water level in the tube is lowered until the first resonance is heard. Take the speed of sound as 340 m s⁻¹ and neglect end corrections.

A sports-science team studies a battle rope of length 9.0 m. Both ends of the rope are effectively fixed and act as nodes. Transverse waves travel along the rope at 13.5 m s^-1.

A train horn is a pipe of length 0.60 m, closed at one end and open at the other. Take the speed of sound in air as 340 m s^-1 and ignore end corrections.

A microwave oven produces electromagnetic waves of frequency 2.45 GHz. The waves are reflected by the metal walls of the oven. A student removes the rotating turntable and heats a flat plate of cheese for a short time. The cheese melts only in small spots arranged in lines.

A gas sensor uses a rigid tube of length 0.180 m sealed at both ends, as shown. A small loudspeaker in one cap drives the gas; a microphone in the other cap detects the amplitude of the sound. The frequency of the loudspeaker is swept and the frequencies at which the microphone signal is a maximum are recorded.

P and Q are points on the axis of the tube, 0.045 m and 0.135 m from the left cap.

A test rig for car shock absorbers holds a mass of 300 kg on a spring. The mass is pulled down 50 mm and released, and the graph shows its displacement x against time t for three different shock absorbers P, Q and R.

A cryogenic transfer line contains a straight cylindrical section of length 1.40 m. The section is sealed with rigid metal caps at both ends and is filled with helium gas at 300 K. A small speaker and microphone inside the section detect acoustic resonances at 365 Hz, 730 Hz and 1095 Hz. There is no resonance below 365 Hz.

An unsealed desert road has developed regular ridges, called corrugations, spaced 0.75 m apart. Each wheel assembly of a vehicle, together with its tyre and suspension, has a natural vertical oscillation frequency of 12 Hz.

Vibrating-wire strain gauges are embedded in a concrete dam to monitor stress. Each gauge contains a steel wire fixed at both ends. The wire is plucked electromagnetically and vibrates at its fundamental frequency. In a calibration test, the tension is kept constant and the length L of the wire is varied. The fundamental frequency f is measured for each length:

L = 0.100 m, f = 1495 Hz
L = 0.125 m, f = 1204 Hz
L = 0.150 m, f = 998 Hz
L = 0.200 m, f = 752 Hz

In pharmaceutical research, an ultrasonic levitator holds small droplets in mid-air, so that no container can contaminate them. An ultrasound transducer faces a flat reflector, and a standing sound wave forms in the air between them. Six tiny polystyrene beads become trapped at six consecutive pressure nodes. The distance between the first and sixth bead is 21.4 mm. The speed of sound in air is 343 m s⁻¹.

An overhead power cable is fixed at two towers 60.0 m apart. Wind makes the cable vibrate. Sensors record the frequencies of several standing-wave modes, where n is the harmonic number.

n: 2, 3, 5, 7
f / Hz: 1.90, 2.85, 4.75, 6.65

A sound source of unknown frequency is held above a vertical tube. The lower end of the tube is in water, and the water level is slowly lowered. The first resonance occurs when the air column is 0.142 m long, and the second when it is 0.462 m long. The speed of sound on that day is 346 m s⁻¹.

A simple flute is a pipe 0.330 m long, open at both ends. On Earth the speed of sound is 343 m s⁻¹.

A light elastic cord of length 1.8 m has a smooth ring at each end that slides on a vertical rod, so both ends of the cord are free. Transverse waves travel along the cord at 8.4 m s⁻¹.

A student measures the speed of sound using a tube closed at one end and a tuning fork of frequency 425 Hz. The first resonance occurs when the air column is 0.185 m long and the second when it is 0.585 m long.

A pipe of length 0.85 m is sealed at both ends. A researcher sets up the third harmonic standing wave. Take the speed of sound as 340 m s⁻¹.

Two identical loudspeakers face each other 2.40 m apart in a long corridor and emit sound in phase at 425 Hz. A microphone is moved along the line joining them. Take the speed of sound as 340 m s⁻¹.

In a simple model of a pulsating star, the gas pulsates radially as a standing sound wave. The centre of the star is a displacement node and the surface is a displacement antinode. The star has radius R = 3.6 × 10¹⁰ m and the average speed of sound inside it is v = 3.0 × 10⁵ m s^-1.

A cable in a climbing-wall test rig has length 0.75 m. One end of the cable is fixed. The other end is attached to a light ring that can slide freely on a vertical rod. The cable has a tension of 28.8 N and a mass per unit length of 2.0 × 10^-3 kg m^-1.

A rail technician measures the speed of sound using a tuning fork of frequency 500 Hz held above a vertical tube partly filled with water. The water level is lowered. The first resonance is heard when the air column has length 0.158 m and the second when the length is 0.498 m.

A sports-arena air horn contains an open pipe of length 0.42 m, open at both ends. The speed of sound in air at temperature θ (in °C) is v = 331 + 0.6θ m s^-1. Ignore end corrections and any change in length of the pipe with temperature.

A simple model of the human vocal tract is a pipe of length 17.0 cm, closed at the vocal cords and open at the lips. Sound from the vibrating vocal cords is strongest at frequencies close to the resonant frequencies of the air in the tract.

The speed of sound in a gas at kelvin temperature T is v = √(γRT/M), where M is the molar mass and γ is a constant for the gas. For air, γ = 1.40 and M = 29.0 g mol⁻¹. At 20 °C the speed of sound in air is 343 m s⁻¹.

A wind turbine has three blades, each 40 m long, mounted on a tall tower. As each blade passes in front of the tower, the tower receives a small sideways push. A slight imbalance of the rotor also pushes the tower sideways once per revolution.

The graph shows the amplitude A of the sideways vibration of the top of the tower as the rotation rate of the rotor is slowly increased.

A tall office tower sways in its fundamental mode with a natural period of 6.0 s. After a single strong gust, the amplitude of sway at the top decreases from 0.40 m to 0.10 m over 10 complete oscillations. The damping is light, so the period can be taken as constant.

A concert harp string has a vibrating length of 0.65 m between its fixed ends. Its fundamental frequency is 196 Hz. The speed of sound in air is 343 m s⁻¹.

A scale-model bridge stay cable is tested in a wind tunnel. The cable vibrates in a standing wave at frequency 50 Hz. Adjacent nodes are 0.40 m apart, and the amplitude at an antinode is 6.0 mm. The amplitude A of a point a distance x from a node is given by A = A_max sin(2πx/λ), where A_max is the antinode amplitude and the angle is in radians.

In speech science, the vocal tract is modelled as a tube 17.0 cm long. It is closed at the vocal folds and open at the lips. The resonances of this tube, called formants, strengthen certain frequencies in the sound produced by the vocal folds. The speed of sound in warm, humid exhaled air is 350 m s⁻¹.

A string 0.90 m long is fixed at both ends and vibrates in its third harmonic at 240 Hz. The amplitude at each antinode is 4.0 mm. Points P, Q and R on the string are 0.10 m, 0.25 m and 0.50 m from the same end. The amplitude at a point a distance x from a node varies as A_max|sin(2πx/λ)|.

Bridge engineers check the tension in a stay cable by measuring how it vibrates. The cable is 50.0 m long between fixed anchorages and has a mass per unit length of 60 kg m⁻¹. Its fundamental frequency is measured as 2.20 Hz. The speed v of transverse waves on the cable is given by v = √(T/μ), where T is the tension and μ is the mass per unit length.