Wavelength, Frequency and Amplitude

The three numbers that describe any wave

Waves are everywhere. Sound waves carry music to your ears, light waves bring colour to your eyes, ripples cross a pond, and radio waves carry your favourite shows. As different as these all seem, every single wave can be described by just three measurements: its wavelength, its frequency, and its amplitude. Learn these three, and you can describe any wave in the universe. Let's meet them one at a time.

If you would like to revisit what a wave actually is before we dig into the numbers, see waves and vibrations.

Picturing a wave

Imagine a wavy line drawn across a page, like a row of smooth hills and valleys. The top of each hill is called a crest, and the bottom of each valley is called a trough. The flat, middle line that the wave wobbles above and below is called the rest position. With this picture in mind, the three measurements are easy to see.

Wavelength: how long one wave is

The wavelength is the length of one complete wave. The easiest way to measure it is from one crest to the next crest (or from one trough to the next trough). It is a distance, so we measure it in metres (m), and scientists give it the symbol λ, the Greek letter "lambda".

• A wave with crests close together has a short wavelength.
• A wave with crests spread far apart has a long wavelength.

For light, wavelength decides the colour: red light has a longer wavelength than blue light. For sound, a longer wavelength goes with a lower note.

Frequency: how often waves pass

The frequency tells you how many whole waves pass a point each second. We measure it in a unit called the hertz (Hz). If 50 complete waves pass you every second, the frequency is 50 Hz.

• Many waves passing per second = high frequency.
• Few waves passing per second = low frequency.

For sound, frequency sets the pitch — how high or low a note sounds. A high frequency makes a high, squeaky note; a low frequency makes a deep, booming note. Humans can hear roughly 20 Hz up to 20,000 Hz. (Sound above that is called ultrasound.)

There is a neat link between wavelength and frequency: if a wave is travelling at a steady speed, then short wavelengths and high frequencies go together, while long wavelengths and low frequencies go together. We will see exactly why in a moment.

Amplitude: how big the wave is

The amplitude is the height of the wave measured from the rest position up to a crest (not all the way from trough to crest — just from the middle line to the top). It tells you how much energy the wave is carrying.

• A tall wave (big amplitude) carries lots of energy.
• A small wave (small amplitude) carries little energy.

What amplitude controls depends on the wave:

• For sound, amplitude sets the loudness. Big amplitude = loud; small amplitude = quiet.
• For light, amplitude sets the brightness. Big amplitude = bright; small amplitude = dim.

Importantly, amplitude is completely separate from frequency. You can have a loud low note or a quiet low note, a loud high note or a quiet high note. Loudness and pitch are two different things.

The wave equation: tying it together

These measurements are connected by one of the most useful equations in physics, the wave equation:

wave speed = frequency × wavelength v = f × λ

Here v is the speed of the wave in metres per second (m/s), f is the frequency in hertz, and λ is the wavelength in metres. This single equation works for sound waves, water waves, light, and radio waves alike.

It also explains the link we noticed earlier. If a wave's speed stays the same, then frequency and wavelength must trade off against each other: raise the frequency and the wavelength must shrink; lower the frequency and the wavelength must grow. Their product always equals the speed.

Worked example 1: finding wave speed

Problem. A water wave has a wavelength of 2 metres, and 3 complete waves pass a post each second. How fast is the wave moving?

v = f × λ Frequency f = 3 Hz, wavelength λ = 2 m v = 3 × 2 = 6 m/s

The wave travels at 6 metres per second.

Worked example 2: finding wavelength

Problem. A sound in air has a frequency of 170 Hz. Sound travels at about 340 m/s in air. What is its wavelength?

We rearrange the wave equation to make λ the subject:

v = f × λ, so λ = v ÷ f λ = 340 ÷ 170 = 2 m

The sound wave is 2 metres long. Notice that a higher-frequency sound — say 680 Hz — would give λ = 340 ÷ 680 = 0.5 m, a shorter wavelength, just as the equation predicts.

Why it matters

Once you know these three numbers, you can predict how a wave behaves. Engineers tune radio stations by choosing frequencies. Doctors pick high-frequency ultrasound to see fine detail inside the body. Musicians change frequency to play different notes and change amplitude to play softly or loudly. The whole electromagnetic spectrum — radio, microwaves, light, X-rays — is just one family of waves sorted by wavelength and frequency.

Try it yourself! 🧪

Experiment 1 — See frequency and amplitude on a string. Tie one end of a long skipping rope or spring to a fixed point such as a door handle, and hold the other end. Shake your hand slowly up and down — you make long waves with a low frequency. Now shake faster — the waves get closer together: higher frequency, shorter wavelength. Next, shake gently to make short humps (small amplitude), then shake harder to make tall humps (big amplitude). You have just controlled frequency and amplitude with your own hand, and seen that they are independent.

Experiment 2 — Hear pitch change with water. Line up several identical glasses and fill them with different amounts of water — a little in the first, more in each one after. Gently tap each glass with a metal spoon. The glass with the least water makes the highest pitch (fastest vibration, highest frequency); the one with the most water makes the lowest. Then tap one glass softly, then harder: the pitch stays the same but it gets louder — that is amplitude changing while frequency stays put. You have made a musical instrument that shows the difference between pitch and loudness.

(Tap glasses gently so they do not crack, and have an adult help with anything glass.)

Wavelength, frequency, amplitude, and the equation that joins them — with these four ideas, the whole world of waves makes sense.