C.3 Wave phenomena

Light meets the boundary between two transparent media.

Which row gives the conditions needed for total internal reflection to occur?

Light travels at 2.25 × 10⁸ m s⁻¹ in a transparent liquid. In a vacuum it travels at 3.00 × 10⁸ m s⁻¹.

What is the refractive index of the liquid?

Plane water waves in a ripple tank pass through a gap in a barrier. The width of the gap is then reduced until it is about equal to one wavelength.

What happens to the waves beyond the gap?

Two identical lamps placed side by side illuminate a screen. No interference fringes are seen on the screen.

What is the reason?

A mobile-phone network transmits signals at 700 MHz and at 28 GHz. In the region just behind tall buildings, the 700 MHz signal is received much better than the 28 GHz signal.

Which statement explains this in terms of diffraction?

Plane water waves in a ripple tank travel towards a straight barrier. The rays make an angle of 30° with the normal to the barrier, as shown.

Which diagram shows the reflected wavefronts and their direction of travel?

Plane water waves of wavelength 2.0 cm in a ripple tank meet two obstacles in turn: a thin rod of diameter 3 mm, and a solid block 25 cm wide.

Which statement describes the water just behind each obstacle?

A small dipper vibrates up and down at one point on the surface of the water in a ripple tank.

Which row describes the wavefronts and the rays of the waves that spread out from the dipper?

At a football stadium, two loudspeakers are driven in phase by the same amplifier. They emit sound of wavelength 0.28 m. A spectator sits 3.4 m from one loudspeaker and 4.1 m from the other. What does the spectator hear, and why?

The two headlights of a car emit light of the same colour. No interference fringes are seen where their beams overlap on a wall. What is the main reason?

In a medical endoscope, light passes from air into the glass core of an optical fibre. What happens to the frequency, the wavelength and the speed of the light?

A ray of light in air strikes the flat surface of a new transparent material for smartphone screens. The angle of incidence is 60°, and the angle of refraction is 30°. What is the refractive index of the material?

Ocean swell moves from deep water into the shallower water near a surfing beach. In the shallower water the waves travel more slowly. What happens to the frequency of the waves and to the separation of adjacent wavefronts?

A medical endoscope carries light along glass fibres. Each fibre has a core surrounded by a cladding of lower refractive index. Which pair of conditions is necessary for total internal reflection at the core–cladding boundary?

A medical ultrasound transducer has a fixed diameter. The frequency of the ultrasound is increased, while its speed in tissue stays the same. What happens to the spreading of the beam, and why?

A layered optical sensor contains two transparent materials. Material X has refractive index 1.2 and material Y has refractive index 1.6. What is the ratio \(\frac{\text{speed of light in X}}{\text{speed of light in Y}}\)?

During a sound check in a sports hall, two loudspeakers emit coherent sound in phase with a wavelength of 0.60 m. A microphone is 4.20 m from one speaker and 5.10 m from the other. What is detected at the microphone, and why?

Two pulses travel towards each other along a stretched guitar string. One pulse has a maximum displacement of 3.0 cm upwards and the other has a maximum displacement of 5.0 cm downwards. What is the displacement of the string at the instant the points of maximum displacement of the two pulses coincide?

Ocean swell heading towards a harbour passes from deep water into the shallower water at the harbour entrance. How do the frequency, speed and wavelength of the waves in the shallow water compare with their values in the deep water?

The cover glass of a smartphone screen has refractive index 1.50. The speed of light in a vacuum is 3.0×10⁸ m s⁻¹. What is the speed of light in the cover glass?

A climber ties a light cord to the end of a heavy rope. Waves travel more slowly in the rope than in the cord. A single upward pulse is sent along the cord towards the junction. What are the orientations of the reflected pulse and the transmitted pulse?

A dental curing lamp emits blue light of wavelength 480 nm in a vacuum. Inside the polymer light guide of the lamp, the wavelength of this light is 300 nm. What is the refractive index of the polymer?

A wave passes from medium 1 into medium 2. The angle between the ray and the normal is 50° in medium 1 and 30° in medium 2.

What is (speed of the wave in medium 2)/(speed of the wave in medium 1)?

Monochromatic light of wavelength λ passes through two narrow slits and forms bright and dark fringes on a screen. Point P is at the centre of the second bright fringe beyond the central bright fringe. Point Q is at the centre of the second dark fringe from the central bright fringe, on the same side.

Which row gives the path difference at P and at Q?

In a double-slit experiment with monochromatic light, the distance between the slits and the screen is doubled and the slit separation is halved.

What is (new fringe separation)/(original fringe separation)?

Two pulses, X and Y, travel towards each other along a string. The diagram shows the string at one instant, drawn on a square grid.

Which diagram shows the string at the instant when the two pulses exactly overlap?

A plane wavefront travelling in air meets the flat surface of a glass block at an angle to the normal, and passes into the glass.

What happens to the spacing of the wavefronts and to the direction of the rays?

Plane waves of wavelength λ₁ in medium 1 meet a straight boundary with medium 2 at an angle of incidence θ₁. In medium 2 the wavelength is λ₂ and the angle of refraction is θ₂.

Which equation is correct?

Two dippers in a ripple tank vibrate in phase with the same frequency. At point X the two waves arrive in phase and the water oscillates with an amplitude of 7.0 mm. At a nearby point Y the two waves arrive in antiphase and the amplitude is 1.0 mm. Each wave has the same amplitude at X as at Y.

What are the amplitudes of the two waves?

A ray of light travelling in glass of refractive index 1.50 meets a flat glass–air boundary at an angle of incidence of 30°.

Which row gives the angle of reflection and the angle of refraction?

In a double-slit experiment, the laser is replaced by a white-light filament lamp with a narrow single slit in front of it, so that the single slit illuminates the double slit.

Which statement describes the pattern seen on the screen?

A radar reflector on a lifeboat contains two flat metal plates joined along one edge at 90°. A microwave ray, travelling in a plane perpendicular to the join, meets plate 1 at an angle of incidence of 25°. It is reflected on to plate 2 and is reflected again.

Which row gives the angle of incidence at plate 2 and the total angle through which the ray has been turned by the two reflections?

An optical fibre used in a broadband connection has a core of refractive index 1.50 and a cladding of refractive index 1.40. What is the critical angle at the core–cladding boundary?

A double-slit experiment with a laser gives a fringe spacing s on a screen in air. The whole apparatus, including the slits and the screen, is then submerged in a tank of water of refractive index 1.33. The slit separation and the slit-to-screen distance stay the same. What is the new fringe spacing?

A spear-fisher standing on a rock looks down at an angle at a fish in calm water. Which statement correctly describes what happens?

A dental curing lamp emits blue light of wavelength 450 nm in air. In the resin filling, the wavelength of this light is 300 nm. The speed of light in air is 3.0 × 10⁸ m s⁻¹. What is the speed of the light in the resin?

A cut diamond (refractive index 2.4) sparkles because light is totally internally reflected inside it. The diamond is now placed in water (refractive index 1.3) instead of air. What happens to the critical angle at the diamond surface, and what is the effect?

A person stands in a corridor, out of line of sight of an open doorway 0.9 m wide. They can clearly hear a band playing in the room but cannot see the band. Which statement best explains this?

A car dashboard uses an acrylic light pipe of refractive index n₁, coated with a transparent polymer of refractive index n₂, where n₂ < n₁. Light travels along the pipe by total internal reflection at the acrylic–polymer boundary. What is the sine of the critical angle at this boundary?

A diver looking straight up from below a calm water surface sees the sky through a bright circle. Outside this circle the surface looks like a mirror. The water is replaced by a transparent liquid of greater refractive index, and the diver stays at the same depth. What happens to the bright circle?

A Young's double-slit apparatus produces fringes with spacing s in air. The entire apparatus, including the source region, slits and screen, is then submerged in water of refractive index 1.33. The slit separation and slit-to-screen distance are unchanged. What is the new fringe spacing?

A jeweller is told that light travels through a gemstone at 1.25 × 10⁸ m s⁻¹. The speed of light in a vacuum is 3.00 × 10⁸ m s⁻¹. What is the refractive index of the gemstone?

Laser light has wavelength 600 nm and frequency 5.0 × 10¹⁴ Hz in air. It enters a plastic spectacle lens of refractive index 1.5. Which row gives the wavelength and frequency of the light inside the lens?

A double-slit arrangement is illuminated with white light instead of monochromatic light. Which statement correctly describes the pattern seen on the screen?

In a double-slit experiment, each slit on its own produces light of intensity I at the centre of the screen. Both slits are now opened, and the light from the slits is coherent. What is the intensity at the centre of the central bright fringe?

A medical endoscope uses optical fibres with a core of refractive index 1.52 and a cladding of refractive index 1.48. What is the critical angle at the core–cladding boundary?

Ultrasound travels at 1540 m s⁻¹ in soft tissue and at 3080 m s⁻¹ in a bone. A beam in the soft tissue strikes the tissue–bone boundary. Which statement is correct?

A diver is 2.0 m below the flat, calm surface of a pool (refractive index of water 1.33). Looking upward, she sees the whole sky compressed into a bright circle on the surface, directly above her. The critical angle for water is 48.8° and tan 48.8° = 1.14. What is the diameter of the bright circle?

In a test of materials for a laser projector, a ray of light passes from transparent plastic A into transparent plastic B. The angle of incidence in A is 30° and the angle of refraction in B is 45°. Which statement is correct?

In a double-slit experiment the slit separation is d and the screen is a distance D from the slits. The distance between the centres of the second dark fringes on either side of the central bright fringe is y.

What is the wavelength of the light?

An optical fibre has a core of refractive index 1.50 surrounded by cladding of refractive index 1.45.

What is the critical angle at the boundary between the core and the cladding?

Two loudspeakers emit sound of the same frequency and amplitude, but exactly in antiphase. Point X is the same distance from both loudspeakers. At point Y the path difference is half a wavelength.

Which row describes the sound detected at X and at Y?

A double slit and a screen are fixed inside a glass tank. Monochromatic laser light passes through the slits and forms fringes of separation s on the screen. The tank is then filled with water of refractive index 1.33, without moving the slits or the screen.

What is (new fringe separation)/s?

Two small dippers in a ripple tank are 6.0 cm apart and vibrate in phase, producing waves of wavelength 2.5 cm.

How many points of maximum disturbance are there on the straight line between the two dippers?

Light passes from air into glass of refractive index 1.50. The angle of incidence θ₁ is increased from 0° to 90°, and the angle of refraction θ₂ is measured.

Which graph shows how θ₂ varies with θ₁?

The speed of sound is 340 m s⁻¹ in air and 1480 m s⁻¹ in water. Total internal reflection of sound can occur at the flat surface of a calm lake.

Which row gives the direction in which the sound must travel to the surface for this to happen, and the critical angle?

The critical angle for light travelling in medium X towards a boundary with air is 35°. For light travelling in medium Y towards a boundary with air it is 50°.

What is the critical angle for light travelling in X towards a boundary with Y?

An engineer testing an audio system accidentally reverses the wires to one of two identical loudspeakers, so that they emit in antiphase at the same frequency. Point X is the same distance from both loudspeakers. At point Y, the path difference is half a wavelength. What is heard at X and at Y?

Two small loudspeakers A and B emit sound in phase at the same frequency. They are 3.0 wavelengths apart. A microphone is moved slowly along the straight line from A to B. How many minima does it detect between A and B?

An LED strip along the edge of an aquarium wall sends light inside the glass (refractive index 1.50). A ray in the glass strikes the glass–water boundary (refractive index of water 1.33) at an angle of incidence of 60°. What happens at this boundary?

Light travelling in water (refractive index 1.33) passes into a horizontal glass plate (refractive index 1.50) floating on the water. Above the plate is air. As the angle of incidence in the water is gradually increased, light eventually stops emerging into the air. At what angle of incidence in the water does this first happen?

A laser beam in air is directed at the flat surface of water in a transparent tank. The student marks the path of the beam in the water on a card held vertically in the tank and measures the angle of incidence i and the angle of refraction r with a protractor. Each angle has an uncertainty of ±1°.

A student has five lasers of known wavelength λ and one double-slit slide. The slide is fixed at a distance D = (2.500 ± 0.005) m from a screen. For each laser the student measures the distance across five fringe separations, 5s, with a ruler. Each measurement of 5s has an uncertainty of ±0.5 mm.

The student's hypothesis is that the fringe separation s is proportional to λ.

Light from a red laser passes through a double slit. The slit separation, measured with a travelling microscope, is d = (0.250 ± 0.005) mm. Fringes are seen on a screen at a distance D from the slits. For each value of D the student measures the distance across 10 fringe separations with a ruler whose uncertainty is ±1 mm, and calculates the fringe separation s.

Short pulses of infrared light from a laser are sent into reels of optical fibre of different lengths L. A fast photodiode at the far end of each fibre is connected to an oscilloscope, which is triggered by the laser. The time t between the trigger and the arrival of the pulse is read from the oscilloscope screen. The uncertainty in each time is ±10 ns, which is too small to show on the graph. The fibre lengths are those printed on the reels.

A student builds a simple refractometer to measure the sugar content of drinks. A semicircular dish with thin, transparent walls is filled with a sugar solution. A laser beam is directed through the curved wall towards the centre of the flat wall, and the dish is rotated on a turntable until the beam refracted into the air at the flat wall just disappears. The angle of incidence at this moment is the critical angle c, read from the turntable scale with an uncertainty of ±0.3°. The thin flat wall does not affect the critical angle, so the refractive index of the solution is n = 1/sin c.

The student uses six solutions of known concentration (grams of sugar per 100 g of solution, %). The graph shows n against concentration with error bars.

A student tests Snell's law for water waves. A glass plate lies on the floor of a ripple tank so that the water is 12 mm deep on one side of its straight edge and 3 mm deep above it. A straight dipper vibrating at 5.0 Hz sends plane waves across the edge (Figure 1).

For six orientations of the dipper the student photographs the waves with a ruler in the photograph and measures, with a protractor, the angle θ₁ between the ray and the normal to the edge in the deep water and the angle θ₂ in the shallow water. The uncertainty in each angle is ±2°.

On one photograph the student also measures the distance across four wavelengths: (233 ± 2) mm in the deep water and (145 ± 2) mm in the shallow water.

A narrow beam of light enters a semicircular glass block through its curved surface, along a radius, so that it meets the centre of the flat surface from inside the glass at an angle of incidence θ. The block stands on a turntable whose scale is read to ±0.5°.

A light sensor measures the power of the beam reflected back into the glass at the flat surface. The student calculates the reflected fraction R = (reflected power)/(power arriving at the flat surface), with an uncertainty of ±0.02.

Acrylic rods are used as light guides. A student shines light along a straight rod of diameter D and bends it round cylinders of decreasing radius R (Figure 1). R is the radius of the inner surface of the bend. The student records the largest radius at which light is first seen escaping from the outer surface of the bend. D is measured with callipers (±0.02 mm); the cylinders are available in steps of 0.5 mm and the uncertainty in each recorded R is ±1.0 mm.

A simple model, considering the ray that grazes the inner surface, predicts that light escapes when R < D/(n − 1), where n is the refractive index of the acrylic.

A student uses a green laser and six double-slit slides with different slit separations d. The screen is fixed at a distance D = 2.400 m from the slits. For each slide the student measures the distance across 10 fringe separations, 10s, and calculates log d and log s.

The student's hypothesis is that the fringe separation s is inversely proportional to d.

Two small loudspeakers, 0.80 m apart, are connected to the same signal generator so that they emit sound of the same frequency f in phase. The experiment is done on a playing field to avoid reflections. A microphone, about 2.5 m in front of the loudspeakers, is moved sideways from the position of maximum loudness on the centre line until it detects the first minimum. The distances r₁ and r₂ from the microphone to the two loudspeakers are then measured with a tape measure, each with an uncertainty of ±0.5 cm, and the path difference Δ = r₂ − r₁ is calculated.

The air temperature is 20 °C. The graph shows Δ against 1/f with error bars.

A student investigates road mirages: on a hot day the road far ahead seems to be covered with water. The student takes values of the refractive index n of dry air at atmospheric pressure, for yellow light, from an online physical-properties database. For a gas, (n − 1) is proportional to the density of the gas.

The student's hypothesis is that (n − 1) is inversely proportional to the absolute temperature T, as the ideal gas model predicts. The database values are given to ±0.1 in the second column.

A small lamp L is fixed to the floor of a swimming pool, 1.80 m below the flat surface of the water. The refractive index of the water is 1.33 and that of air is 1.00. The figure, drawn to scale, shows two rays, A and B, from L reaching the surface, and a point S on the surface 2.5 m horizontally from the point directly above L.

Ocean swell of period 10.0 s approaches a coral reef. Over the deeper water the waves travel at 15.0 m s⁻¹; over the shallow reef they travel at 9.0 m s⁻¹. The edge of the reef is straight. Figure 1 is a plan view, drawn to scale, of the wavefronts in the deeper water, which meet the edge of the reef at an angle of incidence of 40°.

When a guitar string is plucked, the sound contains several harmonics. A microphone records the note from one string. In a simple model, the sound at the microphone consists of only two components: the first harmonic and the second harmonic.

Figure 1 shows how the displacement of the air at the microphone that each component would produce on its own varies with time t. The speed of sound in air is 340 m s⁻¹.

Glaciologists measure the thickness of a glacier with radar carried by a helicopter (Figure 1). The antenna sends short pulses of radio waves of frequency 60 MHz vertically downwards and detects the echoes. The refractive index of ice for these waves is 1.78.

For one pulse, the echo from the surface of the ice arrives 0.400 μs after the pulse is sent, and the echo from the rock bed arrives 3.10 μs after the pulse is sent.