The Nature of Waves: Energy Transfer

A cork floats in the sea and a series of waves passes beneath it on their way to the shore.

1. a) Does the cork travel with the waves to the beach, or does it simply move up and down, remaining in almost the same spot?
2. b) If the water does not move forward with the wave, what is it that actually reaches the shore?

Classify each of these waves as mechanical (requires a material medium to propagate) or electromagnetic (can travel through a vacuum), and write out all six complete classifications:

1. a) Sound in air.
2. b) Sunlight.
3. c) Sea waves.
4. d) Radio waves.
5. e) The wave travelling along a guitar string.
6. f) X-rays.

Which of them could cross the space between the Earth and the Moon?

Andrea shakes the end of a rope up and down and a wave moves along it. Afterwards, she pushes a coil of a stretched spring and a compression travels through the spring. The figure represents both waves.

1. a) Which of the two waves is transverse and which is longitudinal?
2. b) Explain the difference by looking at how each point of the medium moves compared to the direction in which the wave travels.
3. c) Sound causes air to oscillate in the same direction in which it travels: to which type of wave does it belong?

The waves that a duckling generates while swimming in a pond travel at $1.2\ \text{m/s}$, and the duckling produces $0.6$ waves per second.

1. a) What is the frequency of the waves?
2. b) Calculate their wavelength using $v = \lambda \times f$.

At the beach, Aitana observes that the wave crests arrive separated by $4\ \text{m}$ from one another and that each crest rises $0.6\ \text{m}$ above the calm water level. The figure represents the waves she observes.

1. a) What is the wavelength $\lambda$?
2. b) What is the amplitude $A$ of the waves?

Sitting on a buoy, a swimmer counts that $15$ crests pass by in $30\ \text{s}$.

1. a) Calculate the frequency $f$ of the waves in hertz.
2. b) Explain in your own words what this frequency value means.

Match each wave quantity with its unit in the International System of Units (SI) and write out the four complete pairs. Also state what each quantity tells us about the wave:

1. a) Wavelength.
2. b) Period.
3. c) Frequency.
4. d) Wave speed.

Units: hertz ($\text{Hz}$), meter ($\text{m}$), second ($\text{s}$) and meter per second ($\text{m/s}$).

A wave crossing a pond has a frequency of $5\ \text{Hz}$.

1. a) Calculate its period $T$.
2. b) Explain what the period of a wave means.

An anchored boat rocks $30$ times per minute due to the waves.

1. a) Express its frequency in hertz.
2. b) Calculate the period of the rocking.

The alternating current supplying domestic sockets changes direction with a period of $0.02\ \text{s}$.

1. a) Calculate its frequency in hertz.
2. b) How many complete cycles occur in one minute?

When shaking the end of a hose, waves are formed in it whose crests are $2\ \text{m}$ apart, while the hand moves up and down $3$ times per second. Calculate the wave speed using the expression $v = \lambda \times f$.

$10$ crests pass a buoy in $4\ \text{s}$, and a photograph shows that two consecutive crests are separated by $0.6\ \text{m}$.

1. a) Calculate the frequency of the waves.
2. b) Calculate their wave speed.

The home Wi-Fi router emits electromagnetic waves with a frequency of $2.4\ \text{GHz}$.

1. a) Express this frequency in hertz, using standard form.
2. b) Calculate the wavelength of the signal, knowing that electromagnetic waves travel at $3\times10^{8}\ \text{m/s}$.

A wave travels $25\ \text{m}$ in $10\ \text{s}$ as it approaches the shore.

1. a) Calculate its speed of propagation.
2. b) If its frequency is $0.5\ \text{Hz}$, calculate its wavelength.

In a water park, the wave machine generates waves with a wavelength of $40\ \text{cm}$ at a frequency of $6\ \text{Hz}$.

1. a) Express the wavelength in meters.
2. b) Calculate the propagation speed of the waves.

Two waves travel along the same taut string, so that both have the same speed. Wave A has twice the frequency of Wave B.

1. a) Which of the two has the greater wavelength?
2. b) How many times greater? Justify your answer using $v = \lambda \times f$.

A tsunami in the open ocean travels at about $720\ \text{km/h}$ and its period is about $20\ \text{min}$. The figure shows a diagram of the tsunami wave in the open ocean.

1. a) Express its speed in $\text{m/s}$.
2. b) Express its period in seconds.
3. c) Calculate its wavelength and express it in kilometers.
4. d) In the open ocean, the height of the wave is less than $1\ \text{m}$, spread over that enormous wavelength: explain why ships barely notice the tsunami in the open sea even though it carries a gigantic amount of energy.

A wave travels along a thin string at $8\ \text{m/s}$ and passes into a second, thicker string joined to the first, along which it travels at $4\ \text{m/s}$. The frequency, $2\ \text{Hz}$, is set by the hand shaking the first string and is the same in both. The figure represents the wave passing from one string to the other.

1. a) Calculate the wavelength in each string.
2. b) State which quantity does not change when passing from one string to the other and which ones do.
3. c) Explain why the frequency cannot change at the joint: think about how the point where the two strings meet vibrates.

Along a taut string, $6$ crests pass in $3\ \text{s}$, and in a photograph it is measured that two consecutive crests are $0.5\ \text{m}$ apart.

1. a) Calculate the frequency of the wave.
2. b) Calculate its speed of propagation.
3. c) Calculate its period.
4. d) If we tighten the string further and the speed increases to $2\ \text{m/s}$ while maintaining the same frequency, what is the new wavelength?

A radio station broadcasts on FM at $100\ \text{MHz}$ and on AM at $1000\ \text{kHz}$.

1. a) Express both frequencies in hertz using standard form.
2. b) Calculate the wavelength of each broadcast (radio waves travel at $3\times10^{8}\ \text{m/s}$).
3. c) Long waves navigate obstacles better than short waves: reason which broadcast will better reach a village hidden behind a mountain.

Earthquakes emit P-waves (longitudinal) travelling at $6\ \text{km/s}$ and S-waves (transverse) travelling at $3\ \text{km/s}$. At a seismic station, the S-wave arrives $50\ \text{s}$ after the P-wave. The diagram shows the focus of the earthquake and the station.

1. a) If the focus of the earthquake is at a distance $d$ from the station, write down the time taken for each wave to arrive.
2. b) Set up the equation $t_S - t_P = 50$ and calculate the distance $d$ to the focus.
3. c) Check the result by calculating the travel time of each wave.

In a wave tank in the laboratory, a wave is photographed: the crests appear at positions $0.5\ \text{m}$, $3\ \text{m}$, $5.5\ \text{m}$, and $8\ \text{m}$, and its maximum height above the calm level is $0.3\ \text{m}$. Furthermore, it is observed that $4$ crests pass a point in the tank every second. The figure shows the photograph of the wave with the positions of the crests.

1. a) Determine the wavelength from the positions of the crests.
2. b) State the amplitude of the wave.
3. c) Calculate the frequency and the speed of propagation.

On two consecutive days, series of waves with the same wavelength and frequency arrive at the same beach, but on Monday the amplitude is $0.2\ \text{m}$ and on Tuesday it is $1\ \text{m}$. The figure compares the waves on both days.

1. a) A cork floats in the same area on both days: describe its movement each day and explain whether it reaches the shore pushed by the waves.
2. b) On which day do the waves transport more energy? What wave quantity determines this?
3. c) Relate your answer to the fact that on Tuesday swimmers are knocked about by the waves, whereas on Monday they barely notice them.

You want to measure the speed of waves in a water tank in the laboratory and you have a ruler, a stopwatch and a vibrator that generates waves.

1. a) Explain which two quantities you would measure and how you would do it using this equipment.
2. b) With the vibrator running, you count $20$ oscillations in $8\ \text{s}$ and measure $0.4\ \text{m}$ between two crests: calculate the frequency and the speed of propagation.
3. c) State a source of error in your measurement and a way to reduce it.