The Wave Equation: v = f λ

One equation for every wave

Drop a stone in a pond, pluck a guitar string, switch on a radio — completely different events, yet a single equation describes them all. It is called the wave equation:

v = f λ

In words, wave speed equals frequency times wavelength. Here v is the speed in metres per second (m/s), f is the frequency in hertz (Hz), and λ (the Greek letter "lambda") is the wavelength in metres (m).

If those three words are new, the lesson on wavelength, frequency and amplitude is a good place to refresh them. This page is about putting the numbers to work.

Why the equation has to be true

Think about what frequency and wavelength actually mean.

• Wavelength (λ) is the length of one complete wave — one crest to the next crest.
• Frequency (f) is how many of those waves pass you each second.

So in one second, the number of waves passing is f, and each one is λ long. The total length of wave that streams past you in that second is therefore f × λ. But "length passing per second" is exactly what we mean by speed. That is the whole reason v = f λ works — it is just length-per-wave multiplied by waves-per-second.

Speed is set by the medium

Here is the idea students most often miss. The speed of a wave is fixed by the material it travels through, not by whatever made the wave.

Sound travels through air at about 340 m/s whether it is a flute or a foghorn. Light travels through a vacuum at 3 × 10⁸ m/s no matter the colour. This means that in a given material, f and λ are locked in a trade-off: if you increase the frequency, the wavelength must shrink to keep the product v the same. A high-pitched whistle and a low rumble travel side by side at the same speed — the whistle just packs its waves closer together.

Rearranging the equation

You can solve for whichever quantity is missing:

To find	Use
Speed	v = f λ
Frequency	f = v / λ
Wavelength	λ = v / f

A simple memory trick: cover the quantity you want in the triangle with v on top and f × λ on the bottom.

Worked example 1: finding speed

A water wave at the beach has a frequency of 0.5 Hz (one wave every 2 seconds) and a wavelength of 6 m. How fast is it moving?

v = f λ = 0.5 Hz × 6 m = 3 m/s

A gentle walking pace — about right for waves rolling into shore.

Worked example 2: finding wavelength

A radio station broadcasts at 100 MHz, which is 100,000,000 Hz. Radio waves travel at the speed of light, 3 × 10⁸ m/s. What is the wavelength?

λ = v / f = (3 × 10⁸) ÷ (1 × 10⁸) = 3 m

That is why FM aerials are around a metre or two long — they are tuned to wavelengths of a few metres.

Worked example 3: finding frequency

A sound wave in air (v = 340 m/s) has a wavelength of 0.85 m. What is its frequency?

f = v / λ = 340 ÷ 0.85 = 400 Hz

That is close to the note G above middle C — a pitch you can hum.

Try it yourself! 🧪

Measure the speed of a wave on the floor.

1. Lay a long rope or a Slinky along a smooth floor and have a friend hold the far end still.
2. Flick your end once and time how long the pulse takes to reach the other end. Speed = length ÷ time. This is v.
3. Now shake your end steadily and count how many waves you make per second — that is f.
4. Measure the distance between two crests — that is λ.
5. Multiply f × λ and compare with the v you timed in step 2. They should roughly match.

You have just verified the wave equation with your own hands. The same v = f λ also explains how a wave bends when it slows down, which you can read about in reflection and refraction of light.