Three letters. One relationship. It applies to every wave in the syllabus, from ripples on a pond to gamma rays from a distant star. Learn it once, use it forever.
The Key Idea
If you know how often a wave repeats (frequency) and how long each repeat is (wavelength), you know how fast it travels. Multiply them. That is it.
The wave equation
v = f λ
SECTION 01
Where the equation comes from.
This is one of those formulae that looks like a definition you must memorise. It is not. It comes straight from the basic definition of speed, and you can derive it in two lines.
Speed is distance divided by time:
speed = distance ÷ time
For one complete wave, the distance covered is one wavelength (λ), and the time taken is one period (T):
v = λ ÷ T
But period and frequency are reciprocals: T = 1 ÷ f. So 1 ÷ T equals f. Substitute:
v = λ × f = f λ
The wave equation is just speed = distance ÷ time, applied to one cycle of a wave. If you remember that, you can rebuild the formula from scratch in any exam.
SECTION 02
Rearranging for what you need.
In Cambridge papers you will be asked for v, sometimes f, sometimes λ. The same equation rearranged three ways:
For wave speed
v = f λ
For frequency
f = v ÷ λ
For wavelength
λ = v ÷ f
The Examiner's Trap: The Prefix Panic
Cambridge rarely gives you raw Hertz (Hz) or Metres (m). To use this equation, you must convert to base units first or your answer will be off by millions. Memorize these conversions before Paper 4:
kHz (kilohertz) → multiply by 1,000 ($10^3$)
MHz (megahertz) → multiply by 1,000,000 ($10^6$)
cm (centimetres) → divide by 100 ($10^{-2}$)
nm (nanometres) → multiply by $10^{-9}$ (vital for light waves)
SECTION 03
One equation, every wave.
The same formula works for:
Water waves in a ripple tank, where v might be 0.3 m/s
Sound waves in air, where v is roughly 340 m/s
Light and all electromagnetic waves in a vacuum, where v is exactly 3 × 10⁸ m/s
The wave equation does not care what the wave is made of. It only cares that the wave is repeating.
Worked Example
A sound wave in air has a frequency of 680 Hz and a wavelength of 0.50 m. Calculate the speed of sound in air.
Step 1 · Identify what you know
Frequency f = 680 Hz. Wavelength λ = 0.50 m. The wave is a sound wave. Speed v is unknown.
Step 2 · Pick the right form of the equation
We need v, so use the equation in its original form:
v = f λ
Step 3 · Substitute and solve
v = 680 × 0.50 = 340 m/s
The speed of sound in air is 340 m/s, which is the value to remember.
Practice questions.
A water wave travels at 1.5 m/s and has a frequency of 5 Hz. Find its wavelength.
Answer: λ = v ÷ f = 1.5 ÷ 5 = 0.3 m.
Light travels at 3 × 10⁸ m/s. Red light has a wavelength of 700 nm (7 × 10⁻⁷ m). Find its frequency.
Answer: f = v ÷ λ = (3 × 10⁸) ÷ (7 × 10⁻⁷) ≈ 4.3 × 10¹⁴ Hz.
A radio station broadcasts at 100 MHz. The wave travels at 3 × 10⁸ m/s. Find the wavelength.
Answer: First convert 100 MHz = 10⁸ Hz. λ = v ÷ f = (3 × 10⁸) ÷ 10⁸ = 3 m.
Two waves travel at the same speed. Wave A has a wavelength of 0.4 m and Wave B has a wavelength of 0.8 m. How do their frequencies compare?
Answer: Wave A has twice the frequency of Wave B (since f and λ are inversely proportional at fixed v).