The Wave Model of Light

Light is a wave, and not the kind that travels through water or air. It is a ripple in the electric and magnetic fields themselves, and it can race across totally empty space. Wiggle an electric charge back and forth and you send a disturbance outward at an astonishing fixed speed. That self carrying ripple is what your eyes catch when they see colour.

Light is an electromagnetic wave

Picture an electric charge being shaken up and down. As it moves it makes a changing electric field, and a changing electric field always creates a changing magnetic field alongside it. That new magnetic field is also changing, so it creates another electric field, and the two keep handing the baton back and forth as the wave races away.

• The wave is transverse: the electric and magnetic fields point across the direction the wave travels, not along it.
• It needs no medium. Because each field keeps regenerating the other, the wave carries itself, which is how sunlight crosses the vacuum of space to reach us.
• Every electromagnetic wave travels at the same speed in a vacuum, $c = 3.0 \times 10^{8}$ m/s, whether it is radio, red light or X rays.

The red curve is the electric field and the blue dashed curve is the magnetic field. They are at right angles to each other and both at right angles to the direction the wave travels. That side to side wiggle is what makes light a transverse wave.

The wave equation links speed, frequency and wavelength

Every wave obeys one short rule that ties together how fast it goes, how many wiggles pass each second, and how long each wiggle is. For light that rule is speed equals frequency times wavelength.

The speed $v$ is in metres per second, the frequency $f$ is in hertz (wiggles per second), and the wavelength $\lambda$ is the length of one full wiggle in metres. For light in a vacuum the speed is locked at $c$, so $c = f\lambda$. This means colours with a shorter wavelength must have a higher frequency, because the product always has to equal the same $c$.

Standing waves on a string

Now send a wave down a string that is tied down at both ends. The wave races to the far end, bounces back, and the returning wave overlaps the one still coming. When they line up just right the string settles into a frozen looking pattern called a standing wave.

A standing wave forms when a travelling wave superposes with its own reflection. The string is pinned at both ends, so those ends can never move: they are nodes. In between sit the antinodes, the points that swing with the biggest movement. Only certain wavelengths fit neatly between the fixed ends, and they are given by $\lambda = \dfrac{2L}{n}$, where $L$ is the length of the string and $n = 1, 2, 3, \ldots$ counts the antinodes (the harmonic number).

The two dashed curves, red and blue, show the two extremes the string swings between. Where they cross are the nodes, which never move, and the fat middles are the antinodes, where the string moves most. This pattern has three antinodes, so it is the third harmonic with $n = 3$.

See it for yourself

Send a single pulse or a continuous wave down the string and watch it reflect off the fixed end. Set the end to Fixed, turn on Oscillate, and tune the frequency until the reflected wave lines up with the incoming one and a steady standing wave appears, with still nodes and big swinging antinodes.

How to actually solve one

These questions come in two flavours, and both are quick once you spot which is which.

1. For a light question, reach for $c = f\lambda$. Make sure the wavelength is in metres (nanometres are $\times 10^{-9}$ m), then rearrange for whatever is missing.
2. For a standing wave on a string fixed at both ends, use $\lambda = \dfrac{2L}{n}$, where $n$ is the number of antinodes.
3. Count antinodes carefully: $n = 1$ is the fundamental with the longest wavelength, $n = 2$ is the second harmonic, and so on.
4. If you need the speed of the wave on the string, combine the two ideas with $v = f\lambda$.

Try one: a string of length $0.90$ m is fixed at both ends and vibrates in its third harmonic (three antinodes). Find the wavelength of the standing wave.

Lock it in with active recall

Cover the answer and say each one out loud before you flip. Rate yourself honestly — the cards you find hard come back sooner, the ones you know are spaced further out.

See the recipe in action in the Worked Examples tab, then test yourself in Try It.