Light as a wave

One kind of wave, many uses

Your phone uses radio waves. A lamp gives out visible light. A hospital X-ray machine uses waves with a much shorter length. All of these are the same kind of wave, called an electromagnetic wave. They differ only in their wavelength.

An electromagnetic wave is made of an electric field and a magnetic field that change together, at right angles to each other, and travel through space. It needs no medium. In vacuum, all of them travel at the same speed:

The quantities of a wave

• The wavelength $\lambda$ (lambda) is the distance between two crests. We use metre, and also nanometre, $1\ \text{nm} = 10^{-9}\ \text{m}$.
• The frequency $\nu$ (the Greek letter nu) is the number of waves that pass a point in one second. Its unit is the hertz, $1\ \text{Hz} = 1\ \text{s}^{-1}$.
• The wavenumber $\tilde{\nu}$ is the number of waves in one metre. It is $\tilde{\nu} = \dfrac{1}{\lambda}$, in $\text{m}^{-1}$ (or in $\text{cm}^{-1}$, which chemists often use).

The three are linked by the speed:

If the wavelength is long, the frequency is low. This is the main thing to remember. Light of a short wavelength has a high frequency and a large wavenumber.

Example. Red light has a wavelength of $700\ \text{nm}$. We want $\nu$, so $\nu = \dfrac{c}{\lambda} = \dfrac{3.00 \times 10^{8}}{700 \times 10^{-9}} = 4.29 \times 10^{14}\ \text{Hz}$. The wavenumber is $\dfrac{1}{700 \times 10^{-9}} = 1.43 \times 10^{6}\ \text{m}^{-1}$.

The electromagnetic spectrum

Each step to the right in the picture is $1000$ times longer. The part that our eyes can see is very narrow: from about $400\ \text{nm}$ (violet) to $750\ \text{nm}$ (red). Waves a little shorter than violet are ultraviolet. Waves a little longer than red are infrared. We cannot see these, but a camera, a skin burn or a heater can detect them.

You should find that halving the wavelength doubles both the frequency and the wavenumber.

Why we call light a wave

Light shows interference and diffraction. When light from two narrow slits falls on a screen, we see bright and dark bands. Only waves can do this, because crests add up in some places and cancel in others.

The wave picture explained a great deal. But in the years around $1900$ three observations did not fit it: the colours of a glowing hot object, the photoelectric effect, and the sharp lines of atomic spectra. We study them in the next lessons. They led to a new idea.

Practise with new numbers

Try it in Python

This program finds the frequency and the wavenumber of five colours of light.

Remember these

• $c = \nu\lambda$ and $\tilde{\nu} = \dfrac{1}{\lambda}$. In vacuum, $c = 3.00 \times 10^{8}\ \text{m s}^{-1}$ for all electromagnetic waves.
• A shorter wavelength means a higher frequency and a larger wavenumber.
• Visible light is from about $400\ \text{nm}$ to $750\ \text{nm}$, from violet to red.
• Interference and diffraction show that light is a wave.

Check yourself

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

1. An FM radio station broadcasts at $93.5\ \text{MHz}$. What is the wavelength of its waves?

2. Which has the higher frequency, ultraviolet or infrared light?

3. Light has a wavenumber of $2.0 \times 10^{6}\ \text{m}^{-1}$. What is its frequency?

4. Why can light cross empty space when sound cannot?