Wave Properties and Superposition

A wave carries energy, not matter

A ripple crosses a pond. A leaf on the water does not travel across the pond. It only moves up and down. The wave moves. The water mostly stays where it is.

So a wave carries energy from one place to another, and it does not carry the material along with it.

• In a transverse wave, the material moves at right angles to the direction of the wave. A wave on a string is an example.
• In a longitudinal wave, the material moves back and forth along the direction of the wave. Sound is an example.

Describing a wave

These are the words that we use.

• Wavelength $\lambda$ is the distance between two neighbouring crests.
• Frequency $f$ is the number of crests that pass a point in one second. Its unit is the hertz (Hz).
• Period $T$ is the time for one full oscillation. It is $T = \dfrac{1}{f}$.
• Amplitude is the biggest displacement from the rest position. It tells how much energy the wave carries. For sound, a bigger amplitude is a louder sound.

The speed of a wave

In one period, the wave moves forward by one wavelength. So the speed is the distance divided by the time.

An example. A note of $440\ \text{Hz}$ travels in air at $340\ \text{m/s}$. Its wavelength is $\lambda = \dfrac{v}{f} = \dfrac{340\ \text{m/s}}{440\ \text{Hz}} \approx 0.77\ \text{m}$. Here we wrote the law as $\lambda = v/f$, because we wanted $\lambda$.

The medium sets the speed. The speed does not depend on the source. It depends on the medium that the wave travels in.

• On a string, $v = \sqrt{\dfrac{T}{\mu}}$. Here $T$ is the tension and $\mu$ is the mass of the string for every metre of length. A tighter string gives a faster wave.
• Sound in air at room temperature goes at about $340\ \text{m/s}$.

So what happens if the source vibrates faster? The frequency goes up. The speed stays the same. Then $v = f\lambda$ tells us that the wavelength must become shorter.

Superposition

Two waves can be in the same place at the same time. At every point, the displacements simply add.

After they pass through each other, each wave goes on as if nothing had happened. This one rule explains interference, beats and standing waves.

Interference

Two sources send out identical waves, with the same frequency, and both start together. At the place where you stand, the two waves arrive after travelling two different distances. The difference is the path difference.

• If the path difference is $0, \lambda, 2\lambda, \ldots$, a whole number of wavelengths, crest meets crest. This is constructive interference. The sound is loud.
• If the path difference is $\tfrac{\lambda}{2}, \tfrac{3\lambda}{2}, \ldots$, an odd number of half wavelengths, crest meets trough. This is destructive interference. The sound is quiet.

Simulation: PhET Interactive Simulations, University of Colorado Boulder (CC BY 4.0).

Beats

Now two sources have frequencies $f_1$ and $f_2$ that are very close to each other. Sometimes the two waves rise together, and sometimes one is up while the other is down. You hear one note, at the average frequency. But its loudness rises and falls. The number of rises every second is

Musicians use this to tune an instrument. When two strings come closer in pitch, the beats become slower. When the strings match, the beats are gone.

You should find that the swells come closer together as the two frequencies move apart, and disappear when they are equal.

Practise with new numbers

Try it in Python

This program finds the wavelength of four notes in air.

Now we add two waves. This program draws wave 1, wave 2 and their sum. The frequencies are low, so that you can see each wave.

Remember these

• $v = f\lambda$. The medium sets the speed. The source sets the frequency.
• At every point, the displacements of two waves add.
• Path difference of whole wavelengths: loud. Odd half wavelengths: quiet.
• Beats: $f_{\text{beat}} = |f_1 - f_2|$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A note has a frequency of $850\ \text{Hz}$, and sound travels at $340\ \text{m/s}$. What is its wavelength?

2. Tuning forks of $480\ \text{Hz}$ and $486\ \text{Hz}$ sound together. How many beats do you hear each second, and what pitch?

3. Two speakers send out the same note, starting together, with a wavelength of $0.8\ \text{m}$. At your place the path difference is $2.4\ \text{m}$. Loud or quiet?

4. A source in a room starts to vibrate faster. What happens to the speed, the frequency and the wavelength of its sound?