Drop a stone in a pond and ripples spread outwards — but the water itself doesn’t travel to the edge, it just bobs up and down. That’s the secret of every wave: it carries energy from place to place without carrying the matter along with it. This page sets up the vocabulary for describing any wave, and the one equation that ties speed, frequency and wavelength together.
📘 What you need to know
A travelling wave transfers energy from one place to another without transferring matter
Waves are made by oscillating sources, and the oscillations spread away from the source
Wavelength (λ) is the length of one complete wave — e.g. crest to crest
Amplitude (A) is the maximum displacement from equilibrium
Period (T) and frequency (f) are linked by f = 1/T
The wave equation ties it all together: v = fλ = λ/T
What is a travelling wave?
A travelling wave is defined as oscillations that transfer energy from one place to another without transferring matter. That last part is the key idea: the wave moves, and energy moves with it, but the material it travels through mostly stays put — each bit just oscillates around a fixed point.
Every wave starts with an oscillating source. A vibrating guitar string, a loudspeaker cone, a wobbling water surface — the source oscillates, and those oscillations travel outwards, away from it. Depending on the type of wave, the oscillations may need a medium (like air or water) to travel through, or they may cross a vacuum with no particles at all.
The “energy not matter” idea trips people up, so picture a Mexican wave in a stadium. The wave sweeps right around the ground, but no single person runs around the stadium — each just stands up and sits back down. The disturbance travels; the people don’t. Water, air and every other wave medium behaves exactly the same way.
The properties of a wave
Any wave can be described by the same handful of quantities. Two of them — wavelength and amplitude — are easiest to see on a displacement graph.
On a displacement–distance graph, one full wave spans a wavelength λ (crest to crest), and the height from the centre to a crest is the amplitude A.
Here’s each property in plain terms:
Displacement (x) — how far a point on the wave is from its equilibrium position. A vector, so it can be positive or negative. Measured in metres.
Wavelength (λ) — the length of one complete wave, measured between the same point on two consecutive waves (e.g. crest to crest or trough to trough). Measured in metres.
Amplitude (A) — the maximum displacement from equilibrium. Measured in metres.
Period (T) — the time for one complete oscillation to pass a fixed point. Measured in seconds.
Frequency (f) — the number of complete waves passing a fixed point each second. Measured in hertz (Hz).
Wave speed (v) — the distance the wave travels per second. Measured in m s−1.
Watch the axis: a wave graph can plot displacement against distance (where one cycle = the wavelength λ) or against time (where one cycle = the period T). Always check which axis you’re reading before measuring.
Period and frequency
Just as for any oscillation, the period and frequency of a wave are reciprocals of each other:
Frequency and periodf = 1 / T
A wave with a short period (each cycle passes quickly) has a high frequency; a long period means a low frequency. If a full wave passes a point every 0.5 s, then two waves pass each second — that’s 2 Hz.
The wave equation
The single most useful equation for waves links wave speed, frequency and wavelength. Since a wave travels one whole wavelength in one period, its speed is wavelength ÷ period — and because f = 1/T, that’s the same as frequency × wavelength:
The wave equationv = fλ = λ / T
Where v is wave speed (m s−1), f is frequency (Hz), λ is wavelength (m) and T is period (s). This applies to every wave — sound, light, water, all of them.
WE 1
A wave has a frequency of 50 Hz and a wavelength of 6.8 m. Calculate its speed.
Step 1 — use the wave equation v = fλStep 2 — substitute f = 50 Hz, λ = 6.8 mv = 50 × 6.8v = 340 m s⁻¹That’s the speed of sound in air — this could be a 50 Hz sound wave.
Frequency and wavelength are inversely linked
For a wave travelling at a constant speed, the wave equation reveals a trade-off: if the speed v is fixed, then f and λ must balance each other. Squeeze the waves closer together (shorter λ) and more of them pass each second (higher f); stretch them out (longer λ) and fewer pass each second (lower f).
Both waves travel at the same speed. The top one has a short wavelength and high frequency; the bottom has a long wavelength and low frequency. Shorter λ always means higher f at constant speed.
WE 2
A sound wave travels through air at 340 m s⁻¹. If the note being played is middle C, with a frequency of 256 Hz, what is its wavelength?
Step 1 — rearrange the wave equation for λ
λ = v / f
Step 2 — substitute v = 340 m s⁻¹, f = 256 Hzλ = 340 / 256λ = 1.33 mHigher-pitched notes (bigger f) would have shorter wavelengths — same speed, so f and λ trade off.
WE 3
A travelling wave has a period of 4.0 ms and a wavelength of 0.60 m. Calculate its frequency and its speed.
Step 1 — convert the period: 4.0 ms = 4.0 × 10⁻³ sStep 2 — frequency from f = 1/Tf = 1 / (4.0 × 10⁻³) = 250 HzStep 3 — speed from v = fλv = 250 × 0.60 = 150 m s⁻¹f = 250 Hz, v = 150 m s⁻¹Convert milliseconds to seconds first — forget it and the frequency comes out 1000× wrong.
Period T seconds
f = 1/T
Frequency f hertz
× λ
Wave speed v m s⁻¹
The wave quantities at a glance
Quantity
Symbol
What it means
Unit
Displacement
x
Distance of a point from equilibrium (vector)
m
Wavelength
λ
Length of one complete wave
m
Amplitude
A
Maximum displacement from equilibrium
m
Period
T
Time for one wave to pass a point
s
Frequency
f
Waves passing a point per second
Hz
Wave speed
v
Distance travelled by the wave per second
m s−1
💡 Top tips
v = fλ is the equation to know cold — rearrange it for f or λ as needed.
Check the graph’s axis: distance-axis gives wavelength, time-axis gives period. They look identical otherwise.
At constant speed, f and λ are inversely related — one up, the other down.
Measure wavelength or period across one full cycle (crest to crest, or trough to trough).
Watch units: convert ms to s, cm to m, nm to m before substituting.
⚠ Common mistakes
Thinking waves carry matter along with them — they transfer energy, not matter
Reading wavelength off a time-axis graph (that gives the period) or vice versa
Forgetting to convert ms, µs, cm or nm into SI units before using v = fλ
Measuring only half a wave — a wavelength is a complete cycle
Assuming a bigger amplitude means a faster or longer wave — amplitude is independent of v, f and λ
Quick recap: A travelling wave carries energy, not matter, out from an oscillating source. It’s described by displacement, wavelength λ, amplitude A, period T, frequency f and speed v. Period and frequency are reciprocals (f = 1/T), and the wave equation v = fλ = λ/T links speed, frequency and wavelength for every wave.
You’ve now got the language for describing any wave. The next question is how a wave oscillates — whether the particles move up-and-down across the direction of travel, or back-and-forth along it. That difference splits every wave into two families: transverse and longitudinal waves, the next page.
Starting waves? Let’s build a solid base.
Book a free meeting and we’ll work through the wave equation, reading wave graphs and past-paper questions together.
