The wave equation: v = fλ

How are speed, frequency and wavelength linked?

In one time period $T$, a wave moves forward by one wavelength $\lambda$. So its speed is

with $v$ in m/s, $f$ in hertz and $\lambda$ in metres.

What decides what?

• The medium (and its temperature) decides the speed: about $340\ \text{m/s}$ in air, $1500\ \text{m/s}$ in water.
• The source decides the frequency: a tuning fork of $256\ \text{Hz}$ always vibrates $256$ times a second.
• The wavelength follows from the two: $\lambda = \dfrac{v}{f}$.

So in the same air, a higher note has a shorter wavelength. And when a sound goes from air into water, its frequency stays the same but it speeds up, so its wavelength grows.

You should find that frequency times wavelength always gives the speed you chose. Doubling the frequency halves the wavelength; raising the speed to water's stretches every wave.

Watch out. All sounds in the same air travel at the same speed. A high note is not faster than a low one; it just has shorter waves.

A worked example

A tuning fork of $288\ \text{Hz}$ sounds in air, and then in water at $1480\ \text{m/s}$. Find the wavelength in each.

1. In air. $\lambda = \dfrac{340\ \text{m/s}}{288\ \text{Hz}} = 1.18\ \text{m}$.
2. In water. The frequency stays $288\ \text{Hz}$: $\lambda = \dfrac{1480\ \text{m/s}}{288\ \text{Hz}} = 5.14\ \text{m}$.
3. So the waves are about four times as long in water, because sound is about four times as fast there.

Practise with new numbers

Try it in Python

The program finds the wavelength of the notes of a scale.

Remember these

• $v = f\lambda$.
• The medium decides the speed; the source decides the frequency.
• Going into a new medium, the frequency stays the same and the wavelength changes.

Check yourself

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

1. A wave has $f = 160\ \text{Hz}$ and $\lambda = 2.125\ \text{m}$. What is its speed?

2. A sound of $2000\ \text{Hz}$ travels in air at $340\ \text{m/s}$. What is its wavelength?

3. A sound goes from air into water. Which of $f$, $v$ and $\lambda$ stay the same?

4. Two whistles of $1000\ \text{Hz}$ and $3000\ \text{Hz}$ blow together far away. Which sound arrives first?