Multiple waves can be in the same position at the same time. Unlike particles, waves pass through each other without interacting.

wave speed =

When two waves are in the same position we say they are in superposition. The observed amplitude for waves in superposition is the sum of each wave's amplitude.

Ears are able to detect separate sound frequencies, even though they enter the ear in a superposition. This spectrogram lets you visualize the frequencies that make up various sounds.

The Wedge is a famous surf spot that has some dangerously fun waves because of interference from waves reflected off a jetty.

Question: A wave is created on each side of a rope with an amplitude of 0.3 m. The waves interfere as they pass through each other.

What is the amplitude of the rope when a crest lines up with a crest?
What is the amplitude of the rope when a crest lines up with a trough?

Two crests add constructively to make a larger amplitude.

$$ 0.3\, \mathrm{m} + 0.3\, \mathrm{m} = 0.6\, \mathrm{m}$$

A crest and trough add destructively to cancel out.

$$ 0.3\, \mathrm{m} - 0.3\, \mathrm{m} = 0\, \mathrm{m}$$

Waves add constructively when both wave amplitudes are positive or both are negative. This produces a higher observed amplitude. Constructive interference is how two speakers playing the same song are louder than just one.

Example: Draw the superposition of these waves. They have the same wavelength, so they only add constructively.

Waves add destructively when one wave amplitude is positive and the other is negative. This produces a lower observed amplitude. Two examples of destructive interference are noise cancelling headphones and the bulbous bow of boats.

Example: Draw the superposition of these waves. They have the same wavelength but they are out of phase with each other. They can only add destructively.

Example: Draw the superposition of these waves.

Adding many sine waves together can produce some interesting waveforms. These waves are starting to look like sawtooth and triangle, but they need more contributing sine waves.

Interference is even more complex and interesting in higher dimensions.

Take a look at these 2-D ripple tank simulations:
Dipole source: Two wave sources interfere.
Intersecting Planes: Two wave planes interfere at 90°.
Beats: Two wave sources with slightly different frequencies.

Click to see the waveform recorded by your microphone.
Click to see the spectrum of frequencies recorded by your microphone.

Consonance and Dissonance

Some of what makes music sound good depends on the cultural background of the listener. Listeners used to the western music tradition generally prefer sound waves that have a simple ratio between wavelengths.

Question: Guess which of these combinations of sound waves will sound better to your ears?

The first pair of frequencies has a harmonious 1:2 ratio.

I don't hate the second combination, but it does have a slight sour, tense, or sad sound to me. Of course, good music might find a place for that feeling.

Example: Draw the superposition of these waves. When waves have similar frequencies they produce a pattern called beats.

You can decide if you hate the beats sound by clicking

frequency =
     volume =

frequency =
     volume =
   analyser =

The pattern in the graph above is sent as an electrical wave into your speakers. In the speaker, magnets convert the electrical wave into vibrations. The vibrations spread through the air as an audible wave of pressure.

Music theory explores what groups of notes sound best together. It's interesting that you can evoke emotions by adjusting the ratio of paired note frequencies.

For example, the major scale is a collection of notes that can sound nice together partly because they have a simple ratio between frequencies. Minor chords are collections of notes that might feel sad or tense partly because of the ratio between frequencies.


Waves tend to spread out in every direction possible within their medium. Diffraction is when a wave spreads out after some of the wave hits a barrier. This explains how sound can be heard through an open door even when there is no line of sight.

Diffraction can be explained by thinking of every point on a wavefront as the source of spherical waves. These spherical waves interfere with each other to produce different patterns.

You can see examples of diffraction in these 2-D ripple tank simulations:

Single Slit: Waves spread out in a circle as they pass through a gap.
Double Slit: Notice the pattern of constructive and destructive interference.
Half Plane: Waves curve around a wall.
Obstacle: Waves of the right wavelength can almost ignore a small barrier.

Example: Draw the pattern of the waves that would pass through the gap as the waves propagate to the right.
Example: Draw the pattern of the waves that would pass through the gap as the waves propagate to the right.

This is the famous quantum mechanical double slit experiment.
Notice the alternating constructive and destructive interference pattern.

gap separation =

The Doppler Effect

The Doppler effect is heard when a source of sound moves past an observer as a "vvvVVVRRROOOMMMmmm". The observed sound drops in frequency very quickly as the source zooms past.

Moving towards a wave source
  • frequency seems higher
  • wavelength seems shorter
  • Moving away from a wave source
  • frequency seems lower
  • wavelength seems longer
  • A sonic boom occurs when the source of sound is moving at the speed of sound. Can you produce a sonic boom in the simulation above?

    Light can also be Doppler shifted, but it is harder to notice because light moves so fast. A light source near the speed of light will noticeably shift in color.

    The convention is to say that light is blueshifted as it moves towards you and redshifted as it moves away.

    Astronomers use this color shifting to calculate the relative motion of far away objects. In 1929 Edwin Hubble published the observation that almost all galaxies are redshifted. This was the initial evidence that the universe is expanding.

    Redshifted light from stars expanding away from us is actually the main reason why the sky is dark at night.

    The Doppler effect is used in laser cooling to produce temperatures near absolute zero.

    Radar guns use the Doppler effect to measure relative speed. They bounce radio waves off moving objects to measure the frequency shift of the reflected waves. The frequency shift indicates the relative speed.

    Question: Imagine you are driving towards a cop with a radar gun. How will your motion affect how they see your reflected radio waves?

    The cop will observe your light at a higher frequency and a shorter wavelength.


    Rigid solids oscillate. They swing back and forth at a consistent rhythm. Springs, bridges, buildings, wine glasses, and musical instruments all oscillate. We also see oscillation in lasers, atoms, electric circuits, orbits, swing sets, and pendulums.

    Systems that oscillate have a natural frequency for their vibrations. When no external forces are applied they will always oscillate at the same frequency.

    An applied force that matches the natural frequency can lead to larger oscillations more easily. This applied force is said to be resonant with the natural frequency of the system. The force pushes left as the system moves left, and right as the system moves right. This way it doesn't fight the natural motion of the system.

    length =

    frequency = Hz

    Simulation: Adjust the frequency of the force on the pendulum to find a resonant frequency. (use the default length)

    A frequency around 0.5 Hz has a good resonance.

    Simulation: Adjust the length of the pendulum. How does the length affect the resonant frequency?

    A shorter length has a higher resonant frequency and a longer period.
    A longer length has a lower resonant frequency and a shorter period.

    The period of a pendulum can be calculated with this equation.

    Standing Waves

    A standing wave is a wave that oscillates in time, but does not move through space. Standing waves occur in most musical instruments in the form of vibrating strings or columns of air. You can produce a standing wave if you shake a string at just the right frequency while someone else holds the other end still.

    Standing waves occur when two waves moving in opposite directions interfere with each other. This often occurs with reflected waves in a cavity, like a pipe. You can see a standing wave clearly in a Rubens' Tube.

    In a bounded region the waves reflect off the walls. For most wavelengths the numerous reflected waves interfere destructively to produce a superposition of zero. When the wavelength fits evenly into a closed region the reflected waves resonate with each other to produce a standing wave. The wave looks like an alternating constructive then destructive interference pattern.

    This simulation calculates the superposition of 64 reflections in a bounded region. Only certain wavelengths resonate to produce a standing wave.

    1.0 m
    wavelength = m

    Simulation: for at least 5 wavelengths that resonate to produce standing waves.

    Extra Credit: Can you figure out an equation that produces those results?
    Use λ for the resonate wavelengths
    L for the length of the region
    n for any positive integer [1, 2, 3, 4, 5, ...]

    Standing wave resonance in a bounded region occurs when the wavelength is even multiples of twice the length of the region.

    $$2L = n \lambda $$ $$n = 1,2,3,4,5 \, \dots $$ \( \lambda \) = resonance wavelengths [m]
    \( L \) = length of bounded region [m]
    \( n \) = positive integers

    The locations where the amplitude stays at zero are called nodes. Some of them are marked in the diagram above with a ⚫. The locations between the nodes with the maximum amplitude are called anti-nodes.

    Question: Count the nodes and anti-nodes in the bottom standing wave.

    There are 8 nodes if you count the edges.

    There are 7 anti-nodes.