Exercises: Types of wave motion

Classify the following waves as mechanical or electromagnetic and explain the criterion you have used:

1. a) The sound emitted by a loudspeaker.
2. b) The light from a lighthouse.
3. c) The microwaves in an oven.
4. d) Ocean waves.
5. e) Seismic waves from an earthquake.

Explain the difference between a transverse wave and a longitudinal wave, indicating how the medium oscillates in each case relative to the direction in which the disturbance travels, and give a real-life example of each type.

The end of a string is made to oscillate vertically with simple harmonic motion.

1. a) Explain why the wave generated in the string is called a harmonic wave.
2. b) Describe the relationship between the motion of each point on the string and the motion of the end being shaken.

In many science fiction films, thunderous ship explosions are heard in outer space.

1. a) Reason, based on the nature of sound, why no explosion could be heard in the vacuum of space.
2. b) Explain how astronauts communicate by voice with mission control and what type of wave is used for this.

Seismic P-waves are longitudinal and seismic S-waves are transverse.

1. a) Explain what each of these statements means, describing how the ground moves as each type of wave passes.
2. b) Reason which of the two types cannot propagate through the fluid layers of the Earth's interior and why.

When a guitar string fixed at both ends is plucked, a standing wave is formed as shown in the figure.

1. a) Explain what nodes and antinodes of a standing wave are.
2. b) State, in terms of the wavelength $\lambda$, the distance separating two consecutive nodes.

The speed of transverse waves in a taut string is $v = \sqrt{\frac{T}{\mu}}$, where $T$ is the tension and $\mu$ is the linear density of the string.

1. a) Reason how the speed changes if the tension is increased without changing the string.
2. b) Reason how it changes if it is replaced by a thicker string (greater $\mu$) under the same tension.
3. c) Relate these results to what a guitarist does when turning the tuning pegs to tune the instrument.

A cork floats on the surface of a pond when a wave train arrives: the cork moves up and down, but does not advance with the waves, as shown in the figure.

1. a) Reason, from this observation, what a wave transports and what it does not transport.
2. b) Explain how the disturbance then reaches large distances without any net horizontal displacement of the water along with it.

A piano string of length $1\ \text{m}$ has a mass of $10\ \text{g}$ and is subjected to a tension of $400\ \text{N}$.

1. a) Calculate its linear density $\mu$, in $\text{kg/m}$.
2. b) Calculate the propagation speed of transverse waves in the string using $v = \sqrt{\frac{T}{\mu}}$.
3. c) Calculate the time taken for a pulse to travel along the string from one end to the other.

A sharp blow is struck at the end of a steel pipe of length $1000\ \text{m}$, filled with water and surrounded by air, so that sound travels through all three media simultaneously. The speeds of sound are $5100\ \text{m/s}$ in steel, $1480\ \text{m/s}$ in water and $340\ \text{m/s}$ in air.

1. a) Calculate the time taken for the sound to reach the other end through each of the three media.
2. b) Explain why three separate signals are perceived in time at the other end.
3. c) Reason, relating it to the internal structure of each medium, why sound travels faster in a solid than in a gas.

A standing wave with four antinodes is formed on a string of $1,2\ \text{m}$ fixed at both ends.

1. a) Draw the shape of the string in your notebook and mark the nodes and antinodes.
2. b) Calculate the wavelength, taking into account that four half-wavelengths fit on the string.
3. c) If waves travel along the string at $240\ \text{m/s}$, calculate the frequency of the vibration.

You can do it on paper and send a photo with your answer.

A seismic monitoring station is located at $300\ \text{km}$ from the epicentre of an earthquake. P-waves travel through that region at $7,5\ \text{km/s}$ and S-waves at $4\ \text{km/s}$.

1. a) Calculate the time taken for each type of wave to reach the station.
2. b) Calculate the time difference between both arrivals.
3. c) Explain why this time difference allows the distance to the epicentre to be estimated.

In $2015$, the LIGO observatory detected gravitational waves for the first time, coming from the merger of two black holes located $1,3\cdot10^{9}$ light years away. Gravitational waves travel through a vacuum at $c = 3\cdot10^{8}\ \text{m/s}$. Data: $1$ light year $= 9,46\cdot10^{15}\ \text{m}$; $1$ year $= 3,156\cdot10^{7}\ \text{s}$.

1. a) Express the distance to the source in metres, in standard form.
2. b) Calculate the time the signal travelled before reaching Earth, in seconds.
3. c) Reason whether gravitational waves are electromagnetic or mechanical waves and what distinguishes them from both.

The echo sounder of a fishing vessel emits an ultrasound pulse that takes $0,12\ \text{s}$ to go down to a shoal of fish and return. Sound propagates in seawater at $1500\ \text{m/s}$. The diagram in the figure shows the path of the pulse.

1. a) Calculate the depth at which the shoal is located, taking into account that the measured time corresponds to the two-way trip.
2. b) If the pulse has a frequency of $50\ \text{kHz}$, calculate its wavelength in water.
3. c) Explain, from the frequency value, why these waves are called ultrasound.

A guitar string of length $0,65\ \text{m}$ vibrates in its fundamental mode, that is, with a single antinode between the two fixed ends. Waves travel along it at $260\ \text{m/s}$.

1. a) Reason, from the fixed-end condition, that the wavelength of the fundamental mode is $\lambda_1 = 2 \cdot L$.
2. b) Calculate the fundamental frequency using $f_1 = \frac{v}{2 \cdot L}$.

A string of length $0,8\ \text{m}$ fixed at both ends allows modes of vibration with frequencies $f_n = \frac{n \cdot v}{2 \cdot L}$, with $n = 1, 2, 3, \dots$ Waves travel along it at $160\ \text{m/s}$. The figure shows the first three modes of vibration.

1. a) Calculate the frequencies of the first three modes of vibration.
2. b) Explain what mathematical relationship these frequencies have with the fundamental frequency and why they are called harmonics.

In the open sea, a tsunami propagates as a gravity wave whose speed depends on the depth $h$ according to $v = \sqrt{g \cdot h}$. Take $g = 9,8\ \text{m/s}^2$.

1. a) Calculate the speed of the tsunami in an area of the ocean with a depth of $4000\ \text{m}$, in $\text{m/s}$, and express it in $\text{km/h}$.
2. b) Calculate the time it would take to cover $2000\ \text{km}$ of open ocean at that speed, in hours.
3. c) As it approaches the coast, the depth decreases: reason what happens to the tsunami's speed and explain, using the conservation of energy transported by the wave train, why its height increases on reaching the coast.

The P-waves ($7\ \text{km/s}$) and S-waves ($4\ \text{km/s}$) from an earthquake arrive at a seismic station with a time difference of $54\ \text{s}$.

1. a) Write down the expression for the travel time of each wave as a function of the distance $d$ to the epicentre.
2. b) Set up the equation for the time difference and calculate the distance $d$.
3. c) Explain why data from a single station are not enough to locate the epicentre and what additional information having three stations provides (triangulation).

Two identical guitar strings should emit the same note, but their fundamental frequencies turn out to be $440\ \text{Hz}$ and $430\ \text{Hz}$. The fundamental frequency of a string is $f_1 = \frac{1}{2 \cdot L} \cdot \sqrt{\frac{T}{\mu}}$.

1. a) Reason whether the tension of the lower-pitched string must be increased or decreased to match the other one.
2. b) Calculate by what percentage the tension of that string must change, taking into account that the frequency is proportional to $\sqrt{T}$.
3. c) Explain why such a small change in tension is already noticeable to the ear.

After a major earthquake, stations on the opposite side of the planet, more than about 103° from the epicentre, do not record S-waves. P-waves do not arrive directly in an intermediate band either, the so-called "shadow zone" (roughly between 103° and 143°), although they reappear farther away after crossing the core along curved paths. The diagram in the figure summarizes these observations.

1. a) Explain what conclusion about the internal structure of the Earth can be deduced from the absence of S waves in the shadow zone.
2. b) Explain, relating it to their transverse nature, why S waves do not pass through the fluid outer core.
3. c) P waves change speed upon entering the core and their paths curve: state the name of this phenomenon and explain why it occurs when a wave changes medium.

An oceanographic research sonar must distinguish objects of $5\ \text{cm}$ on the seabed; to resolve a detail, a wavelength of the order of the size of the object or smaller is needed. Sound travels in water at $1500\ \text{m/s}$.

1. a) Calculate the minimum frequency that the sonar must use.
2. b) If the seabed is at $3000\ \text{m}$, calculate the time that elapses between the emission of the pulse and the reception of its echo.
3. c) A conventional fishing sonar operates at $20\ \text{kHz}$: calculate its wavelength and reason whether it would serve to distinguish those objects.

An organ pipe open at both ends measures $0,85\ \text{m}$ and sound travels through the air at $340\ \text{m/s}$. In an open pipe, the fundamental wavelength is $\lambda_1 = 2 \cdot L$. The figure represents the pipe vibrating in its fundamental mode.

1. a) Calculate the fundamental frequency of the pipe.
2. b) Calculate the frequency of its second harmonic.
3. c) If the same pipe were closed at one end, its fundamental wavelength would become $\lambda_1 = 4 \cdot L$: calculate the new fundamental frequency and explain why the pipe then sounds lower-pitched.

Two loudspeakers separated by a few metres emit the same pure tone of $f = 170\ \text{Hz}$ in phase, which travels through the air at $340\ \text{m/s}$. The diagram in the figure shows the set-up with a listening point P.

1. a) Calculate the wavelength of the sound.
2. b) A listener situated at a point equidistant from both loudspeakers hears the tone at high intensity: explain, based on the path difference travelled by the two waves, why they reinforce each other there.
3. c) At another point, the distances to the loudspeakers are $5\ \text{m}$ and $4\ \text{m}$: calculate the path difference, express it in terms of $\lambda$ and explain what the listener hears there.
4. d) State the general condition, in terms of $\lambda$, that the path difference must satisfy for the sound to be reinforced and for it to be cancelled.

On a string fixed at both ends, two identical harmonic waves travelling in opposite directions superimpose, forming a standing wave.

1. a) Explain, by superposition of the two waves, how points that never oscillate (nodes) and points that oscillate with maximum amplitude (antinodes) appear.
2. b) The string measures $1\ \text{m}$ and vibrates forming three antinodes at a frequency of $150\ \text{Hz}$: calculate the wavelength and the propagation speed of the waves travelling along the string.
3. c) Explain how a standing wave differs from a travelling wave in terms of the net transport of energy along the string.

Unit challenge. As a future researcher, experimentally model the propagation of a transverse mechanical wave on a taut string or cable at home.

1. Classify the type of wave according to the medium, dimensionality, and direction of propagation. Analytically justify how the SHM of an individual point gives rise to the general equation $y(x,t) = A \sin(\omega t - kx + \varphi_0)$.

2. Measure the usable length ($L \pm 0,001\text{ m}$) with a tape measure. Generate sharp transverse pulses and time the round-trip ($t \pm 0,01\text{ s}$) by recording a slow-motion video with your smartphone and counting frames for 4 different lengths, carrying out 3 repetitions per trial.

3. Construct a table with mean values and absolute uncertainties. Plot the graph of distance travelled versus mean time and, from the slope of the line, obtain the propagation speed $v$.

4. Prepare a short report that includes the data, the graph, and the derivation of the parameters $\omega$, $k$, and $\lambda$ to write the full wave equation resulting from disturbing the end with $A = 0,04\text{ m}$, $f = 2,5\text{ Hz}$ and zero initial phase ($\varphi_0 = 0$).

Assessment criteria:

• Rigour in wave classification and formal justification of the kinematic link with SHM.

• Metrological treatment: correctness of the data table, uncertainty calculation, and graphical fitting to deduce $v$.

• Exact analytical determination of the equation $y(x,t)$ expressed in SI units.