Exercises: Phenomena and applications

A hiker shouts in front of the vertical wall of a cliff and hears the echo $1.6\ \text{s}$ later (speed of sound in air: $340\ \text{m/s}$). The diagram represents the path of the sound.

1. a) Explain why the echo occurs, naming the wave phenomenon responsible.
2. b) Taking into account that the sound travels twice the distance to the wall, calculate how far away the cliff is.

A ray of light is incident on a plane mirror, making an angle of $35^\circ$ with the normal. The figure shows the incident ray.

1. a) State the two laws of reflection.
2. b) Determine the angle of reflection and the angle that the reflected ray makes with the surface of the mirror.

A ray of light passes from air into water, and it is observed that it bends towards the normal.

1. a) What wave phenomenon has the ray experienced? Define it.
2. b) State which wave quantity does not change when passing from one medium to another and which ones do change (propagation speed and wavelength). Justify why the frequency is conserved when changing medium.

A ray of light strikes the surface of water ($n_2 = 1.33$) from the air ($n_1 = 1$) with an angle of incidence of $40^\circ$ (data: $\sen 40^\circ \approx 0.64$). The figure shows the change in direction of the ray.

1. a) Write Snell's law of refraction and substitute the data from the statement into it, leaving $\sen r$ as the unknown.
2. b) Calculate the angle of refraction.

From the hallway, the living room television can be heard even though the screen cannot be seen. The floor plan of the house outlines the situation.

1. a) What wave phenomenon allows sound to "bend round the corner"?
2. b) Explain, by comparing the width of the doorway with the typical wavelengths of sound (of the order of $1\ \text{m}$) and light (of the order of $5 \times 10^{-7}\ \text{m}$), why sound diffracts appreciably at the doorway while light does not.

Two loudspeakers emit the same note in phase. A microphone placed on the perpendicular bisector of the line segment joining them is at the same distance from both. The diagram shows the arrangement of the loudspeakers and the microphone.

1. a) With what path difference do the two waves arrive at the microphone?
2. b) Reason whether the interference at that point is constructive or destructive and what will be heard there.

Two in-phase sources emit waves of wavelength $\lambda = 2\ \text{m}$. At a point P, the distances to the sources are $d_1 = 10\ \text{m}$ and $d_2 = 13\ \text{m}$. The figure shows the two sources and point P.

1. a) Calculate the path difference $\Delta r = d_2 - d_1$.
2. b) Determine whether $\Delta r$ is an integer number of wavelengths or an odd number of half-wavelengths, and conclude whether the interference at P is constructive or destructive.

When tilting a CD under white light, rainbow-like colours appear.

1. a) Explain the role of the microscopic grooves on the disc, which act as a reflection diffraction grating.
2. b) Explain why each colour is observed in a different direction, relating it to the constructive interference of each wavelength and to $\lambda_{\text{rojo}} > \lambda_{\text{azul}}$.

The sonar of an oceanographic vessel emits an ultrasound pulse of $50\ \text{kHz}$ towards the seabed and the echo returns $0.8\ \text{s}$ later (speed of sound in water: $1500\ \text{m/s}$). The diagram shows the path of the pulse.

1. a) Calculate the depth of the seabed, remembering that the pulse makes a round trip.
2. b) Calculate the wavelength of the pulse in water and express it in centimetres.

In the laboratory, a laser beam is directed from air ($n_1 = 1$) onto a glass plate at an angle of $50^\circ$, and an angle of refraction of $30^\circ$ is measured (given: $\sen 50^\circ \approx 0.77$; $\sen 30^\circ = 0.5$). The figure represents the measurement.

1. a) Write down Snell's law and rearrange for the refractive index of the glass.
2. b) Calculate its value.

Yellow sodium light has a frequency of $5.1 \times 10^{14}\ \text{Hz}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$; refractive index of water: $n = 1.33$).

1. a) Calculate the speed of propagation of this light in water.
2. b) Calculate its wavelength in water and express it in nanometres.

Two loudspeakers emit a tone of $340\ \text{Hz}$ in phase (speed of sound in air: $340\ \text{m/s}$).

1. a) Calculate the wavelength of the tone.
2. b) At a point P, the distances to the loudspeakers are $6\ \text{m}$ and $8.5\ \text{m}$. Determine whether at P the tone is heard loudly or barely heard, justifying it with the path difference.

A glass optical fibre ($n_1 = 1.48$) is surrounded by air ($n_2 = 1$). The figure shows a ray incident on the wall of the fibre at the critical angle.

1. a) Write down Snell's law for a ray incident on the glass-air interface at the critical angle, for which the angle of refraction is $90^\circ$.
2. b) Calculate the critical angle of the fibre.

A home router emits its Wi-Fi signal in the $2.4\ \text{GHz}$ band (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Calculate the wavelength of the signal and express it in centimetres.
2. b) Compare it with the width of a doorway ($0.8\ \text{m}$) and explain, naming the wave phenomenon responsible, why there is coverage in the adjacent room even though there is no direct line of sight to the router.

A diver points their torch towards the surface of the water ($n_{\text{agua}} = 1.33$; $n_{\text{aire}} = 1$) at an angle of $50^\circ$ to the vertical. The diagram represents the situation.

1. a) Calculate the critical angle for the transition from water to air.
2. b) Explain whether the beam from the torch will emerge into the air or undergo total internal reflection, by comparing the angle of incidence with the critical angle.

A swell with a wavelength of $8\ \text{m}$ and speed $4\ \text{m/s}$ reaches the mouth of a harbour, an opening of $10\ \text{m}$ between two breakwaters. The figure shows the harbour mouth viewed from above, with the incoming wave fronts.

1. a) Calculate the period and frequency of the waves.
2. b) Explain, by comparing the opening with the wavelength, whether the waves will diffract upon entering and how it will affect the boats anchored behind the breakwaters.
3. c) Would the situation change if the opening were $100\ \text{m}$? Justify your answer.

The Voyager 1 probe is the most distant human-made object: a radio signal sent from Earth takes $22.5\ \text{h}$ to reach it (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Express the time in seconds and calculate the distance to the probe, giving the result in standard form.
2. b) Explain why communication with space probes is carried out using electromagnetic waves and not sound waves, referring to the nature of each type of wave.
3. c) Assuming, as an idealisation, that the radio signal is reflected by the probe and returns to Earth, the echo of a sonar and this returning signal are manifestations of the same wave phenomenon: identify it and describe what it consists of.

A ray of light is incident from air ($n_{\text{aire}} = 1$) at an angle of $45^\circ$ onto a glass sheet ($n = 1.5$) with flat, parallel faces (given: $\sen 45^\circ \approx 0.71$). The figure shows the ray entering the sheet.

1. a) Calculate the angle of refraction at the first face.
2. b) Justify geometrically that the ray is incident on the second face at that same angle, and calculate the angle of refraction upon emerging into the air.
3. c) Conclude the relationship between the emerging ray and the incident ray, and state the physical principle that explains it (reversibility of optical paths).

An optical fibre has a core of index $n_1 = 1.50$ and a cladding of index $n_2 = 1.47$. The diagram shows a section of fibre with a guided ray.

1. a) Calculate the critical angle at the core-cladding interface.
2. b) Explain why light propagates through the core without escaping when incident on its walls at angles greater than this critical angle.
3. c) If the core were in direct contact with air ($n = 1$), calculate the new critical angle and comment on the practical advantage provided by the cladding.

Two loudspeakers emit a tone of $170\ \text{Hz}$ in phase (speed of sound in air: $340\ \text{m/s}$).

1. a) Calculate the wavelength of the tone.
2. b) At point P, the distances to the loudspeakers are $d_1 = 8\ \text{m}$ and $d_2 = 5\ \text{m}$: classify the interference at P.
3. c) Keeping $d_1 = 8\ \text{m}$ fixed, determine two possible values of $d_2$ for which there is a maximum at P and another two for which there is a minimum, justifying it with the interference condition.
4. d) Predict what will be perceived when moving slowly across the classroom along a line parallel to the one joining the loudspeakers.

A soap bubble is a thin film of water ($n = 1.33$) of thickness $e$. Under near-normal incidence light, the light reflected from the two faces of the film interferes constructively when $2 \times n \times e \approx \lambda / 2$ (given condition for the minimum thickness, which already includes the phase shift of $\pi\ \text{rad}$ experienced by one of the reflections). The figure shows the two interfering reflections.

1. a) Calculate the thickness of the film that reinforces the green colour of $\lambda = 532\ \text{nm} = 5.32 \times 10^{-7}\ \text{m}$, and express it in nanometres.
2. b) Explain why, when illuminating the bubble with white light, colours are seen that change from one area of the surface to another.

A router emits in two frequency bands: $2.4\ \text{GHz}$ and $5\ \text{GHz}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Calculate the wavelength of each band, in centimetres.
2. b) Explain, based on the phenomenon of diffraction, which band bends better around household obstacles (doorways, furniture) and, also taking into account absorption by materials, which passes better through walls.
3. c) Manufacturers recommend using the $5\ \text{GHz}$ band only near the router, even though it allows higher data speeds: relate this to what was calculated in the previous sections.

A spotlight submerged at a depth of $2\ \text{m}$ in a swimming pool ($n_{\text{agua}} = 1.33$; $n_{\text{aire}} = 1$) emits light in all directions (given: the tangent of the water-air critical angle is approximately $1.14$). The diagram shows the geometry of the situation.

1. a) Calculate the critical angle for the transition from water to air.
2. b) Rays arriving at the surface at angles smaller than the critical angle emerge into the air, and those arriving at larger angles undergo total internal reflection: explain why the light emerges forming a luminous circle directly above the spotlight.
3. c) Calculate the radius of that circle.
4. d) A person in a boat observes the surface far from the vertical above the spotlight: explain whether they will be able to see the spotlight directly and what the surface outside the luminous circle will look like.