C.2 Wave model

A sound wave has a frequency of 250 Hz and a wavelength of 1.36 m.

What is its speed?

A loudspeaker produces a sound wave in air.

Which statement about the sound wave is correct?

A wave travels along a stretched horizontal rope. A small ribbon tied to the rope moves up and down as the wave passes.

Which statement about this wave is correct?

The displacement–time graph shows one point on a wave.

What is the frequency of the wave?

Which statement is correct for all electromagnetic waves?

The parking radar of a car emits electromagnetic waves of frequency 24 GHz.

Which row gives the wavelength of these waves in air and the region of the electromagnetic spectrum to which they belong?

A dental X-ray machine emits X-rays. A Wi-Fi router emits microwaves of frequency 5 GHz. Which row compares the X-rays with the microwaves when both travel through a vacuum?

A long-wave radio transmitter and the echo sounder of a ship both emit waves of frequency 200 kHz. The radio waves travel through air. The sound from the echo sounder travels through seawater, in which its speed is 1500 m s⁻¹.

What is (wavelength of the radio waves)/(wavelength of the sound waves)?

A medical ultrasound probe emits waves of frequency 5.0 MHz into soft tissue. The speed of sound in the tissue is 1540 m s⁻¹. What is the wavelength of the ultrasound in the tissue?

Which list places regions of the electromagnetic spectrum in order of increasing wavelength?

A space probe sends radio waves to Earth, and a star emits visible light and X-rays. In a vacuum, which property is the same for all of these electromagnetic waves?

During an earthquake, primary (P) seismic waves travel through rock as longitudinal waves. Which statement about the motion of the rock particles as a P-wave passes is correct?

A hospital uses X-rays for imaging, ultraviolet lamps for sterilisation, infrared cameras for thermal scans and microwaves in its kitchen ovens. Which list puts these radiations in order of increasing wavelength?

A swimmer floating just below the surface of a pool hears music played by a violinist standing at the poolside. The speed of sound is about 340 m s⁻¹ in air and about 1480 m s⁻¹ in water. How do the frequency and the wavelength of the sound in the water compare with those in the air?

An ocean buoy anchored offshore bobs up and down as swell waves pass it. It does not drift towards the beach. What do the waves transfer towards the beach?

Which property of electromagnetic waves provides evidence that they are transverse?

The graph shows the displacement against distance along a transverse wave at one instant. Each point on the wave takes 0.020 s to complete one oscillation.

What is the speed of the wave?

Light of frequency f and wavelength λ in a vacuum enters a glass block of refractive index 1.5.

Which row gives the frequency and the wavelength of the light in the glass?

A transverse wave travels with speed v along a stretched rope. A point on the rope takes a time t to move from a position of maximum displacement to its equilibrium position.

What is the wavelength of the wave?

A student sends two continuous transverse waves, P and Q, one after the other along the same long stretched spring. The tension in the spring is the same for both waves. The graph shows the displacement of the spring against distance for P (solid line) and for Q (dashed line), each at one instant.

What is (frequency of P)/(frequency of Q)?

A wave machine in a science museum is a row of horizontal rods joined at their centres by a wire. A transverse wave travels along the machine from rod X towards rod Y, which is a quarter of a wavelength further along. In each graph the solid line shows the displacement y of the tip of X against time t. Which graph shows, as a dashed line, the displacement of the tip of Y?

A submarine's sonar pulse has its amplitude reduced to one third of its original value after passing through a layer of water containing many bubbles. By what factor has the intensity of the pulse changed?

A lifeguard standing on a pier blows a whistle. Some of the sound travels from the air into the sea below. The speed of sound in water is greater than in air. What happens to the frequency and the wavelength of the sound as it enters the water?

A small Wi-Fi router acts as a point source radiating equally in all directions. At a distance of 2.0 m from the router, the amplitude of the electric field of the wave is A. Absorption can be ignored. What is the amplitude at a distance of 6.0 m?

A sensor clipped to a vibrating bridge cable records the displacement of one point on the cable against time. The shortest time between the point being at maximum positive displacement and next being at zero displacement is 2.0 ms. The wavelength of the wave on the cable is 0.60 m. What is the speed of the wave?

A lamp used to sterilise hospital equipment emits electromagnetic radiation of frequency 1.0×10¹⁵ Hz. Which row gives the wavelength in a vacuum and the region of the spectrum?

A sinusoidal transverse wave travels in the +x direction along a taut steel cable. At one instant, point P on the cable has zero displacement, and the displacement of the cable increases as x increases through P. What is the motion of P at this instant?

A buoy on an offshore wave-energy converter moves up and down as ocean waves pass. Its displacement–time graph shows exactly 3 complete oscillations in 12 s. Satellite images show that adjacent wave crests are 18 m apart. What is the speed of the waves?

A distant pulsar emits a burst of radio waves and a burst of X-rays at the same instant. Both travel through the vacuum of space to an observatory in orbit around Earth. Which statement is correct?

A loudspeaker above a swimming pool produces a note. Sound waves pass from the air into the water, where the speed of sound is greater. What happens to the frequency and the wavelength of the sound as it enters the water?

An athlete training with 'battle ropes' moves one end of a long rope up and down 2.5 times per second. This sends transverse waves along the rope, and adjacent crests are 1.6 m apart. How long does a crest take to travel 8.0 m along the rope?

A loudspeaker emits a sinusoidal sound wave into air. Consider an air molecule located at the centre of a compression at a particular instant. What are the displacement of this molecule from its equilibrium position and its speed at that instant?

A fishing boat's echo-sounder sends a short ultrasound pulse vertically down to the sea bed. The reflected pulse returns 0.80 s after it was emitted. The speed of sound in sea water is 1500 m s⁻¹. What is the depth of the sea bed below the boat?

A microphone records the steady hum of a drone. A displacement–time graph for one air molecule shows exactly 5 complete oscillations in 20 ms. A displacement–distance graph of the sound wave at one instant shows that adjacent points of maximum positive displacement are 1.36 m apart. What is the speed of the sound?

A home Wi-Fi router transmits electromagnetic waves of frequency 2.4 GHz. The speed of electromagnetic waves in air is 3.0×10⁸ m s⁻¹. What is the wavelength of these waves?

A plucked guitar string vibrates and produces a sound wave in the surrounding air. Which statement correctly compares the wave on the string with the sound wave in the air?

A railway engineer strikes a steel rail with a hammer. A colleague 1020 m away along the track hears the sound twice: once through the rail and once through the air. The speed of sound is 5100 m s⁻¹ in steel and 340 m s⁻¹ in air. What is the time interval between the two sounds?

A sound wave and a light wave each pass from air into water.

Which row describes the change in the wavelength of each wave?

A transverse wave travels to the right along a rope. The graph shows the displacement of the rope against distance at one instant. P and Q are two points on the rope.

Which row describes the velocity of P and the velocity of Q at this instant?

In a computer simulation, a sound wave travels through air in the positive x-direction. The graph shows how the displacement s of one air molecule from its undisturbed position varies with time t. A displacement in the positive x-direction is positive.

At which time is the molecule at the centre of a compression?

Water waves travel past a post at the end of a pier. A student starts a stopwatch as one crest passes the post, counting it as crest 1, and stops the stopwatch as crest n passes. The stopwatch reads t. In a photograph of the waves, the horizontal distance between a crest and the adjacent trough is d.

What is the speed of the waves?

A point source of sound has power P. At distance r from the source, the sound has amplitude A. The power of the source is doubled and the listener moves to a distance 2r. Absorption can be ignored. What is the new amplitude?

A longitudinal sound wave travels to the right along a pipe. A displacement is taken as positive when it is to the right. At one instant, a point P in the air is at the centre of a compression. Which statement describes the air near P at that instant?

A longitudinal wave travels along a long spring used to test vibration dampers. On a displacement–distance graph, positive displacement means displacement in the +x direction. At one instant the displacement is zero at x = 0.20 m. It is positive for points just below x = 0.20 m and negative for points just above x = 0.20 m. The displacement is a positive maximum at x = 0.10 m. Which statement is correct at this instant?

A drone films a transverse wave travelling along an overhead power line. In the frame taken at t = 0, a crest is at x = 0.30 m. In the frame taken at t = 0.050 s, the same crest is at x = 0.45 m. The wave has moved less than one wavelength in this time, and the wavelength is 0.80 m. What is the frequency of the wave?

A student uses a data handbook to investigate how the speed of sound v in a gas depends on the molar mass M of the gas. To make a fair comparison, the student chooses five noble gases and uses the values listed for a temperature of 0 °C. The handbook states that each speed is accurate to ±1 %.

The student's hypothesis is that v = KMⁿ, where K is a constant and n = −0.5. The graph shows log v against log M; the error bars are too small to show.

Two microphones, A and B, are placed in a straight line with a small metal plate that is struck with a hammer. Microphone A is close to the plate and microphone B is a distance d further away. A data-logger starts timing when the signal from A exceeds a set level and stops when the signal from B does, giving the time interval Δt. Each value of Δt is the mean of five strikes and has an uncertainty of ±0.15 ms. The air temperature is 20 °C.

A student measures the wavelength of ultrasound in air. An ultrasonic transmitter, driven by a signal generator set to a frequency of (40.0 ± 0.1) kHz, faces a receiver that slides along a metre rule. A dual-trace oscilloscope shows the signal sent to the transmitter and the signal from the receiver.

The student moves the receiver slowly away from the transmitter, starting at a position where the two traces are in phase (n = 0). The traces come back into phase again and again; the student records the position x of the receiver after every fifth return to phase. Each position has an uncertainty of ±0.2 cm. The air temperature is 22 °C. The error bars are too small to be shown on the graph.

A student investigates how the speed of sound in air depends on temperature. Two ultrasonic transducers are fixed (0.500 ± 0.001) m apart inside a sealed, insulated duct that contains a heater. A timer measures the time t for a short pulse to travel from one transducer to the other, with an uncertainty of ±0.5 μs. The temperature θ of the air is measured with a thermometer, with an uncertainty of ±0.5 °C.

The student's hypothesis is that the speed v is proportional to √T, where T is the absolute temperature. The error bars are too small to be shown on the graph.

A university hydraulics laboratory publishes data from a long wave flume. A paddle at one end produces long water waves of period 6.0 s. Two wave-height gauges, (4.00 ± 0.01) m apart, record the time t for a wave crest to travel from the first gauge to the second, for different still-water depths h. The published uncertainty in each time is ±0.02 s.

A student uses the published data to test the model v = √(gh) for the speed v of water waves whose wavelength is much greater than the depth of the water.

A student investigates whether the speed of ripples in a ripple tank depends on their frequency. The water has a uniform depth h = (5.0 ± 0.2) mm. A lamp above the tank casts a shadow pattern of the ripples onto a screen below it. For each frequency f of the dipper, set with a stroboscope to ±0.1 Hz, the student measures the distance D on the screen across five wavelengths, with an uncertainty of ±0.2 cm, and calculates the speed v of the pattern on the screen and its uncertainty Δv.

During a thunderstorm an observer sees a flash of lightning and hears the thunder 4.2 s later. The speed of sound in air is 340 m s⁻¹.

African elephants communicate with low-frequency rumbles of frequency 14 Hz. Some bats hunt insects by emitting pulses of ultrasound of frequency 45 kHz and detecting the echoes. The speed of sound in air is 340 m s⁻¹. Humans can hear sounds with frequencies between about 20 Hz and 20 kHz.

An ultrasound probe used in a medical scan sends pulses of ultrasound of frequency 3.50 MHz into soft tissue, in which the speed of sound is 1540 m s⁻¹. Figure 1 shows, at one instant, the displacement of particles of the tissue from their undisturbed positions against distance along the beam, for part of a pulse. A displacement in the direction in which the pulse travels is positive.

A small robotic helicopter flies in the thin carbon dioxide atmosphere of Mars. A microphone on a rover 80 m away records the sound of the helicopter's rotor blades, which has a frequency of 84 Hz. The speed of sound in the Martian atmosphere is 240 m s⁻¹. The speed of sound in air on the Earth is 340 m s⁻¹.

Spectators in a sports stadium make a crowd wave. Each spectator stands up and then sits down again as the wave reaches them, so that the wave travels along the rows of seats. In one stand the seats are in a straight row 150 m long. A video shows that a crowd wave travels the 150 m in 12.5 s and that each spectator takes 0.80 s to stand up and sit down again. Treat the motion of each spectator's head as vertical.

A small instrument buoy floats freely in the open ocean. Swell waves pass the buoy. The time between successive crests passing the buoy is 8.0 s, and the distance between successive crests is 96 m.

A 5G mobile-phone network uses electromagnetic waves of frequency 28 GHz.

A speed gun at a tennis tournament measures the speed of a serve. It emits microwaves of frequency 24 GHz towards the ball.

A railway engineer strikes a steel rail with a hammer. A sensor 1.20 km further along the track detects two sounds: one that travelled through the rail and one that travelled through the air. The table gives the speed of sound in some materials.

Material — speed of sound / m s⁻¹
Air — 340
Water — 1480
Steel — 5900

A medical ultrasound transducer emits pulses of frequency 3.5 MHz into a patient's abdomen. The speed of ultrasound in soft tissue is 1540 m s⁻¹.

At a football stadium, spectators create a 'crowd wave'. Each person stands up and sits down again as the wave reaches them, and the disturbance moves horizontally along the stand. The crest of the crowd wave travels 72 m along the stand in 6.0 s.

A loudspeaker at the left-hand end of a long tube produces a sound wave of frequency 850 Hz that travels to the right through the air in the tube. Figure 1 shows, at one instant, the displacement s of layers of air from their undisturbed positions against the undisturbed position x of each layer. A displacement to the right is positive.

A blue whale produces a low-pitched call of frequency 17 Hz. A hydrophone (an underwater microphone) on a research ship records the call.

Speed of sound in seawater = 1.50 × 10³ m s⁻¹; speed of sound in air = 340 m s⁻¹.

An infrared heat lamp hangs above a pen of young chicks to keep them warm. The tungsten filament of the lamp has a temperature of 2400 K and radiates a total power of 250 W. Treat the filament as a black body.

An earthquake produces two kinds of seismic wave that travel through the Earth's crust: P waves, which are longitudinal, and S waves, which are transverse. In the crust P waves travel at 6.0 km s⁻¹ and S waves travel at 3.5 km s⁻¹.

A self-driving car uses three types of sensor to measure the distance to nearby objects: ultrasonic parking sensors that emit pulses of sound of frequency 48 kHz, a radar that emits microwaves of frequency 77 GHz, and a lidar that emits pulses of infrared light of wavelength 905 nm. Each sensor measures the time between emitting a pulse and detecting its echo.

Speed of sound in air at 20 °C = 343 m s⁻¹

In a fitness exercise an athlete holds one end of a heavy rope. The other end is tied to a post 15.0 m away. The athlete moves the end of the rope up and down with a frequency of 2.0 Hz, sending waves along the rope towards the post. Figure 1 shows, drawn to scale, the first 6.0 m of the rope at time t = 0. Over this length the rope is clear of the floor and the decrease in amplitude is negligible.

A student uses a vibrating dipper to produce water waves of different frequencies f in a ripple tank of constant depth. The student measures the wavelength λ for each frequency.

f / Hz: 5.0, 8.0, 10.0, 12.5, 16.0
λ / cm: 4.8, 3.0, 2.4, 1.9, 1.5

A rover on Mars sends data to Earth using radio waves of frequency 8.4 GHz. At the time of transmission, Mars is 2.25×10¹¹ m from Earth.

A data buoy floats on the open ocean. Its displacement–time graph is sinusoidal. The buoy reaches its maximum upward displacement of +1.2 m at t = 0. It next reaches its maximum downward displacement of −1.2 m at t = 3.5 s. The ocean waves travel at 11 m s⁻¹.

A rover on Mars sends data to Earth using radio waves of frequency 8.4 GHz. At the time of transmission, the distance between Earth and Mars is 2.25×10¹¹ m.

A longitudinal wave of frequency 15 Hz travels in the +x direction along a long spring used to test a vehicle suspension. The table shows the displacement s of coils at positions x along the spring at one instant. Positive s means displacement in the +x direction.

x / cm: 0, 10, 20, 30, 40, 50, 60, 70, 80
s / mm: 0, +4, 0, −4, 0, +4, 0, −4, 0

A student investigates how the speed of sound in dry air depends on temperature. The results are shown below.

Temperature / °C: 0, 10, 20, 30, 40
Speed of sound / m s⁻¹: 331, 337, 343, 349, 355

A tuning fork of frequency 512 Hz is sounded in the air.

A rover on Mars sends data to Earth using radio waves of frequency 8.4 GHz. At the time of transmission, Mars is 2.25×10¹¹ m from Earth. The speed of electromagnetic waves is 3.00×10⁸ m s⁻¹.

A transverse wave travels along a long horizontal rope. P and Q are two points on the rope 0.90 m apart. Graph 1 shows how the vertical displacement y of P (solid line) and of Q (dashed line) varies with time t. Upward displacements are positive. It is known that the distance PQ is less than half a wavelength.

A sonic anemometer at a weather station measures the wind speed with ultrasound. Two transducers, P and Q, face each other 0.150 m apart on an east–west line, with P to the west. P sends a short pulse of ultrasound to Q and then Q sends a pulse to P. The wind blows from west to east along the line PQ.

The speed of a pulse relative to the ground is the speed of sound in still air plus or minus the wind speed. In one measurement the time taken from P to Q is 433.5 μs and the time taken from Q to P is 449.1 μs.