C.3 Wave phenomena

Two coherent sources each produce waves of intensity I₀ at a point P. The waves arrive at P in phase. The intensity of a wave is proportional to the square of its amplitude.

What is the intensity at P?

The diagram shows wavefronts of a wave passing from medium 1 into medium 2. The wavefront spacing is 1.5 cm in medium 1 and 1.0 cm in medium 2.

What is the ratio n₂/n₁ of the refractive index of medium 2 to that of medium 1?

Straight water waves of wavelength 2.0 cm approach a gap of width 10 cm in a barrier, as shown. The gap is then narrowed to 2.0 cm while the frequency of the waves is kept constant.

Which row describes the waves beyond the narrower gap, compared with those beyond the 10 cm gap?

The two headlamps of a parked car illuminate the same white wall, but no interference fringes can be seen where the two beams overlap.

Which statement explains this?

An upward pulse travels along each of two long horizontal ropes towards the right-hand end of the rope. Rope 1 is tied to a rigid wall. Rope 2 is tied to a light ring that can slide without friction on a smooth vertical pole, so that this end is free to move.

Which row describes the reflected pulse on each rope?

In a double-slit experiment with monochromatic light the fringe spacing on the screen is s. The whole apparatus, including the slits and the screen, is then immersed in water of refractive index 1.33. The light source, the slit separation and the slit-to-screen distance are unchanged.

What is the new fringe spacing?

Two coherent sources S₁ and S₂ emit waves of wavelength λ in phase. At point P the path difference from the two sources is 1.5λ. At P, the wave from S₁ alone would have amplitude 2A and the wave from S₂ alone would have amplitude A. The intensity at P due to S₁ alone is I₁. The intensity of a wave is proportional to the square of its amplitude.

What is the intensity at P when both sources emit?

Two pulses of the same width and shape travel towards each other along a stretched bungee cord. Pulse P has a maximum displacement of +3.0 cm and pulse Q has a maximum displacement of −5.0 cm. What is the displacement of the cord at the point where the two maxima coincide?

In a Young's double-slit set-up, the slit separation is halved and the distance from the slits to the screen is doubled. The wavelength is unchanged. What happens to the fringe spacing?

Monochromatic light is incident normally on a diffraction grating. The grating is replaced by a wider grating that has the same line spacing, so that many more lines are illuminated. What happens to the principal maxima on a distant screen?

In an optical fibre, the speed of light is 2.00×10⁸ m s⁻¹ in the core and 2.05×10⁸ m s⁻¹ in the cladding. Which expression gives the critical angle c at the core–cladding boundary?

A beam of monochromatic light passes from air into a glass block. Which row correctly describes what happens to the frequency, speed and wavelength of the light in the glass?

The refractive indices of three materials are: diamond 2.42, crown glass 1.52 and water 1.33. For which journey of a light ray can total internal reflection occur at the boundary?

Light of wavelength 600 nm passes through two narrow slits that are 0.30 mm apart. Interference fringes are observed on a screen 2.0 m from the slits. What is the distance between adjacent bright fringes?

Infrared light of wavelength 1500 nm in a vacuum enters the core of an optical fibre, which has refractive index 1.50. What is the wavelength of the light in the core?

Monochromatic light passes through a single slit and forms a diffraction pattern on a distant screen. The slit width is doubled and the wavelength is halved.

What happens to the width of the central maximum?

Monochromatic light passes through two slits and forms an interference pattern on a distant screen. The pattern of bright fringes is modulated by the single-slit diffraction pattern of each slit.

The width of each slit is now reduced while the separation of the centres of the slits is kept the same. Which row describes the change in the separation of adjacent bright fringes and in the number of bright fringes within the central maximum of the diffraction envelope?

Monochromatic light is incident normally on a diffraction grating. The second-order maximum is observed at an angle of 30.0° to the straight-through direction.

What is the angle of the first-order maximum?

A ray of light passes from medium X into medium Y. The diagram, drawn on a square grid, shows the incident ray, the refracted ray and the normal at the point of incidence.

What is the ratio (wavelength of the light in Y)/(wavelength of the light in X)?

An equilateral glass prism PQR is completely immersed in water of refractive index 1.33. A ray of light enters face PQ along the normal and then strikes face PR, as shown.

What is the least refractive index of the glass for which the ray is totally internally reflected at PR?

Two loudspeakers S₁ and S₂, 3.0 m apart, emit sound of wavelength 1.0 m in phase. A microphone is placed 4.0 m from S₁ along a line perpendicular to S₁S₂.

What is detected at the microphone?

In a double-slit experiment the slits are 0.50 mm apart and the screen is 2.0 m from the slits. The diagram shows the bright fringes seen on the screen and the distance between the centres of the two outermost bright fringes.

What is the wavelength of the light?

Two loudspeakers are connected to the same signal generator, but the leads to one of them are reversed, so that the speakers emit sound of wavelength λ in antiphase.

At which of the following path differences from the two speakers is the sound a minimum?

Light of wavelength 500 nm passes through a single slit and forms a diffraction pattern on a screen 2.00 m from the slit. The graph shows how the intensity on the screen varies with the distance y from the centre of the pattern.

What is the width of the slit?

A layer of cooking oil of refractive index 1.47 floats on water of refractive index 1.33 in a glass dish. The oil and water surfaces are horizontal. A ray of light in air strikes the oil surface at an angle of incidence of 50°.

What is the angle between the ray in the water and the normal?

Monochromatic light is incident normally on two narrow slits a distance d apart, and the pattern is observed on a distant screen. The two slits are then replaced by a diffraction grating with many slits, also a distance d apart, illuminated by the same light.

Which statements about the principal maxima of the grating pattern, compared with the maxima of the two-slit pattern, are correct?

I. They are at the same angles.

II. They are narrower.

III. They are further apart.

Two pulses travel towards each other along a stretched rope, each at a speed of 20 cm s⁻¹. The diagram shows the rope at time t = 0.

What is the displacement of point P at t = 2.0 s?

Two loudspeakers connected to the same signal generator emit sound of frequency 680 Hz in phase. The speed of sound is 340 m s⁻¹. A student walks along a line parallel to the line joining the speakers, starting at the central loud point, and passes through three quiet points.

What is the path difference from the two speakers at the third quiet point?

White light containing all wavelengths from 400 nm to 700 nm is incident normally on a diffraction grating with 300 lines per mm. Complete spectra are formed on both sides of the central maximum.

Which statements are correct?

I. The first-order spectrum and the second-order spectrum overlap.

II. The second-order spectrum and the third-order spectrum overlap.

III. In each spectrum, violet light is diffracted through a smaller angle than red light.

Laser light passes through an adjustable single slit and forms a diffraction pattern on a distant screen. The laser is replaced with one of 1.5 times the wavelength, and the slit width is halved. All diffraction angles remain small. By what factor does the angular width of the central maximum change?

Two radio transmitters emit in phase at the same wavelength λ. A receiver moves sideways from a point on the perpendicular bisector of the line joining the transmitters. The signal falls to a minimum for the third time. What is the path difference from the transmitters to the receiver at this position?

A double slit has slit separation d and each slit has width b, where d = 4b. The slits are illuminated with monochromatic light. How many bright interference fringes can be seen within the central maximum of the single-slit diffraction envelope?

A double-slit arrangement is used to test a component for an underwater laser communication link. In air the fringe spacing is 2.66 mm. The whole apparatus, including the screen, is then submerged in water of refractive index 1.33. What is the new fringe spacing?

Monochromatic light passes through a single slit and forms a diffraction pattern on a distant screen. The angles involved are small. The wavelength is doubled and the slit width is halved. By what factor does the angular width of the central maximum change?

Monochromatic light passes through a single slit, and the diffraction pattern is viewed on a distant screen. The slit width is reduced from 0.20 mm to 0.10 mm. What happens to the angle between the centre of the pattern and the first minimum?

A short pulse of light is sent along a long step-index optical fibre. The pulse emerges spread out in time because of multimode (modal) dispersion. Which single change would reduce this dispersion?

Two loudspeakers emit sound of wavelength 0.34 m. They are connected so that they vibrate in antiphase. A microphone is 3.00 m from one loudspeaker and 3.85 m from the other. What does the microphone detect?

In an aquarium, a diver shines a torch through water (refractive index 1.33) onto a flat, parallel-sided glass wall (refractive index 1.52). The beam meets the glass at 30° to the normal and then passes into air in the visitor area. What is the angle between the beam in air and the normal?

Light of wavelength 550 nm is incident normally on a diffraction grating with 600 lines per millimetre. What is the total number of principal maxima produced?

In medical ultrasound imaging, a pulse travels through soft tissue at 1540 m s⁻¹ and strikes a boundary with bone, where its speed is about 4000 m s⁻¹. The pulse meets the boundary at an angle to the normal. Which statement is correct?

Monochromatic light is incident on a double slit. Each slit has width b and the slit separation is 4b. How many bright interference fringes are seen within the central maximum of the single-slit diffraction envelope?

In a gym, a light training rope is tied to the end of a much heavier rope. Waves travel more slowly in the heavy rope. An upward pulse is sent along the light rope towards the join. Which describes the reflected and transmitted pulses?

A bathymetric lidar system maps the sea floor using a green laser of wavelength 532 nm in air. The refractive index of sea water is 1.33. What is the wavelength of the laser light in the water?

Two radio transmitters X and Y emit waves of frequency 150 MHz in phase with each other. A receiver is 1500.0 m from X and 1503.0 m from Y. The speed of radio waves is 3.00×10⁸ m s⁻¹. Which statement describes the signal at the receiver?

A diffraction grating has 300 lines per millimetre. Light of wavelength 500 nm is incident normally on the grating. What is the total number of bright maxima that can be seen, including the central maximum?

In a double-slit experiment the slit separation is doubled and the distance from the slits to the screen is halved. The wavelength is unchanged. How does the fringe spacing on the screen change?

The core of an optical fibre has refractive index 1.48 and the cladding has refractive index 1.46. What is the critical angle at the boundary between the core and the cladding?

Sound travels at 340 m s⁻¹ in air and at 1500 m s⁻¹ in water. Total internal reflection of sound can occur at a flat air–water surface.

Which row gives the critical angle, and the medium in which the sound must be travelling towards the surface?

A discharge lamp emits two monochromatic lines of wavelength 440 nm and 660 nm. The light is incident normally on a diffraction grating with 250 lines per mm.

What is the smallest non-zero angle at which a maximum of one wavelength coincides with a maximum of the other?

A straight optical fibre has a core of refractive index 1.50 and a cladding of refractive index 1.40. A ray of light in air strikes the flat end face of the core at an angle θ to the axis of the fibre, as shown, and refracts into the core.

What is the largest value of θ for which the ray is totally internally reflected at the core–cladding boundary?

Red light of wavelength 650 nm is incident normally on a diffraction grating with 600 lines per millimetre. What is the total number of principal maxima that can be observed, including the central maximum?

Two identical pulses travel towards each other on a taut string. One pulse is displaced upwards and the other downwards. At one instant they overlap exactly and the string is momentarily straight. Which statement describes the energy of the pulses at that instant?

A laboratory spectrometer uses a diffraction grating with 500 lines per millimetre. Monochromatic light of wavelength 600 nm is incident normally on the grating. What is the total number of bright maxima produced, including the central maximum?

Two narrow parallel slits are cut in a coated sensor window. Each slit has width b, and their centres are separated by 4b. Monochromatic light passes through the slits. How many bright interference fringes appear within the central maximum of the single-slit diffraction envelope?

An astronomical spectrometer uses a diffraction grating with 500 lines per millimetre. It is illuminated normally with light of wavelength 500 nm. How many bright maxima can be observed in total?

Monochromatic light passes through two slits. Each slit has width b, and the slit separation is d = 3b. How many bright interference fringes are seen within the central maximum of the single-slit diffraction envelope?

White light containing wavelengths from 400 nm to 700 nm is incident normally on a diffraction grating. Which statement about the spectra is correct?

An optical fibre in air has a core of refractive index n₁ and a cladding of lower refractive index n₂. A manufacturer replaces the cladding with one whose refractive index is closer to n₁, though still less than n₁. What happens to the critical angle at the core–cladding boundary, and to the acceptance angle? The acceptance angle is the maximum angle of incidence at the air–core end face for which light is still guided along the fibre by total internal reflection.

A double-slit interference pattern is modulated by a single-slit diffraction envelope. The width of each slit is halved. The slit separation, the wavelength and the screen distance are unchanged. Which is correct?

Plane water waves travel from deep water, where their speed is 0.32 m s⁻¹, into shallow water, where their speed is 0.20 m s⁻¹. In the deep water the wavefronts make an angle of 50° with the boundary. What angle do the wavefronts make with the boundary in the shallow water?

White light containing wavelengths from 400 nm to 700 nm is incident normally on a diffraction grating. Which statement about the spectra seen on a distant screen is correct?

A laser beam falls on a double slit of separation 0.400 mm and an interference pattern forms on a screen. The student moves the screen to five positions and measures the distance D from the screen to the front edge of the stand that holds the slide. For each position the student measures the distance across 10 fringe spacings with a ruler; each measurement has an uncertainty of ±0.5 mm.

The graph shows the fringe spacing s against D. The line of best fit has been extended to the D axis.

Short pulses of light from a laser diode are sent into one end of a reel of optical fibre. A detector at the other end is connected to an oscilloscope, which displays the time t between the electrical trigger signal and the arrival of the light pulse. The measurement is repeated for reels of different length L. The time includes a fixed delay in the electronics and connecting cables. Each value of t is uncertain by ±2 ns.

The graph shows t against L with the line of best fit.

A ray box sends a narrow ray of light into the curved face of a semicircular glass block, along a radius, so that it reaches the centre of the flat face and leaves the block into air. The block is placed on a printed polar grid so that the angle of incidence inside the glass, θ₂, can be set precisely. The angle of refraction in air, θ₁, is read from the grid to ±1°.

Student X suggests that θ₁/θ₂ is constant. Student Y suggests that sin θ₁/sin θ₂ is constant.

Light from a laser passes through an adjustable slit and forms a diffraction pattern on a screen 1.500 m away. The slit width is set with a screw and read from a scale as b_s. The width w of the central maximum on the screen (between the first minima on either side) is measured with a ruler.

For small angles the slit width b is related to w by w = 2λD/b. The student plots b_s against 1/w. The graph shows the data and the line of best fit.

A food scientist builds a refractometer to find the sugar concentration of fruit juices. A laser beam passes along a radius of a semicircular dish holding the solution and meets the flat face at its centre. The angle of incidence is increased until the refracted ray just skims along the flat face; this is the critical angle c, measured to ±0.2°. The refractive index is then n = 1/sin c.

Solutions of known sugar concentration C (percentage by mass) are used to plot the calibration graph of n against C.

A student uses a diffraction grating on a spectrometer to study light from a hydrogen discharge tube. The light is made parallel and falls normally on the grating. A telescope is rotated to each maximum and its angular position is read from a circular scale. Positions to the right of the scale zero are positive and to the left negative.

For the red hydrogen line of wavelength 656.3 nm, the first three orders are found on both sides. The telescope reading for the straight-through beam was not recorded. A violet line is seen in first order only on this table.

A student estimates the Planck constant using five light-emitting diodes (LEDs). Each LED in turn illuminates a double slit of separation 0.250 mm, and the distance across 10 fringe spacings is measured on a screen 1.800 m away. The wavelength λ is calculated from the fringe spacing. The student then finds the threshold voltage V_th, the smallest potential difference at which the LED just begins to emit light.

The model is eV_th = hc/λ, so a graph of V_th against 1/λ should be a straight line through the origin. The graph shows the data (note the false origins).

In an electron-diffraction tube, electrons accelerated from rest through a potential difference V pass through a thin layer of graphite and form bright rings on a fluorescent screen. The radius r of the inner ring is measured with vernier calipers. For this ring the geometry of the tube gives r/L = λ/d, where L = 0.130 m and d is a spacing between planes of carbon atoms.

The graph shows r against V^−1/2 with the line of best fit.

The graph shows how the speed of long water waves depends on the depth of the water. Ocean swell of period 8.0 s travels over a sea bed where the water depth changes suddenly from 4.0 m to 1.0 m along a straight underwater ledge.

Two athletes training with a heavy battle rope each flick one end, sending a single pulse towards the other. The centres of the two pulses are initially 6.0 m apart. Each pulse travels at 4.0 m s⁻¹. Pulse A has a maximum displacement of +12 cm and pulse B has a maximum displacement of −7 cm. The pulses have the same width and shape.

Green light of wavelength 520 nm in air comes from a phone's display. It travels through air and strikes the glass cover of a smartphone at an angle of incidence of 35°. The refractive index of the glass is 1.52.

A long-distance telecommunications link uses infrared light of vacuum wavelength 1550 nm in a glass fibre. The core has refractive index 1.47. Take c = 3.00×10⁸ m s⁻¹.

A student shines a green laser through a double slit with slit separation 0.20 mm. The interference pattern is formed on a screen 2.5 m from the slits. The distance between the centre of the 1st bright fringe and the centre of the 9th bright fringe is 5.32 cm.

At a mountain base camp, two loudspeakers A and B are mounted on posts and emit the same steady tone of frequency 680 Hz, in phase. The speed of sound in the air is 340 m s⁻¹. A hiker stands at a point P that is 8.30 m from A and 6.55 m from B.

A geostationary communications satellite transmits to the Earth at a frequency of 12.0 GHz from a circular dish of diameter 2.40 m. The angle between the centre of the beam and its first minimum can be estimated from the single-slit result θ = λ/b, where b is the diameter of the dish.

Light from a laser diode passes through a double slit and falls on a screen 1.60 m away. A light sensor is moved across the screen, and the graph shows the variation of the intensity I it records with position y on the screen. The centres of the slits are 0.180 mm apart.

A diver at depth h below the calm surface of a swimming pool looks upwards. She sees the whole sky above the pool compressed into a bright circle (a "window") on the surface directly above her. Outside this circle the surface looks like a mirror. The refractive index of the water is 1.33.

The data on a DVD are stored along a spiral track. Adjacent turns of the track are a distance d apart, so the disc acts as a reflection grating. A laser beam of wavelength 650 nm is directed normally onto the disc, and the first-order reflected beams make an angle of 61.4° with the incident beam. The grating equation nλ = d sin θ applies.

A step-index optical fibre of length 2.00 km has a core of refractive index 1.480 and a cladding of refractive index 1.460. Pulses of infrared radiation of wavelength 1310 nm are sent along it. Some of the radiation in a pulse travels along the axis of the core, and some zig-zags along the core, meeting the core–cladding boundary at the critical angle.

A microwave transmitter T emits waves of wavelength 3.0 cm towards a metal sheet containing two narrow slits S₁ and S₂ whose centres are 15.0 cm apart. A receiver is moved along a line parallel to the sheet and 1.20 m from it, as shown. O is on the central axis.