The wave equation connects three fundamental properties of any wave: speed (v), frequency (f), and wavelength (λ). It states that wave speed equals frequency multiplied by wavelength: v = f × λ. Mastering this equation and rearranging it is one of the most-tested calculation skills in KS3 and GCSE physics.
What do speed, frequency, and wavelength mean?
Before we predict anything with the equation, let us be precise about what each quantity actually measures.
Wave speed (v) is how fast the wave travels through a medium, measured in metres per second (m/s). For a given medium at a given temperature, wave speed is approximately constant — it does not depend on the source.
Frequency (f) is the number of complete wave cycles that pass a fixed point every second, measured in hertz (Hz). One hertz means one complete wave per second. For sound, frequency corresponds to pitch: a high-frequency sound is a high-pitched note. For light, frequency corresponds to colour.
Wavelength (λ) is the distance between two successive identical points on a wave — for example, crest to crest or compression to compression — measured in metres (m). The Greek letter lambda (λ) is the standard symbol.
Period (T) is the time for one complete wave cycle, in seconds. Period and frequency are reciprocals: T = 1/f. At KS3 the period is sometimes introduced; it becomes more prominent at GCSE.
Here is a helpful picture. Imagine waves as cars coming off a production line at regular spacing. If you double how quickly they are produced (frequency doubles) but the speed stays the same, the cars must be half as far apart (wavelength halves). That inverse relationship between f and λ, at constant v, is exactly what the equation encodes.
What is the wave equation?
The wave equation is:
v = f × λ
where v is wave speed in m/s, f is frequency in Hz, and λ is wavelength in m.
A quick units check: Hz × m = (1/s) × m = m/s ✓
Rearranging to find the other quantities:
- To find frequency: f = v ÷ λ
- To find wavelength: λ = v ÷ f
What do you think will happen to wavelength if I double the frequency of a wave while keeping its speed constant? Predict first — write your answer down — then check it by substituting into λ = v ÷ f. You should find that doubling f exactly halves λ, confirming the inverse relationship.
How do you use the wave equation? Step-by-step worked examples
Example 1: Finding wavelength (sound in air)
A sound wave travels through air at 340 m/s. Its frequency is 170 Hz. Find the wavelength.
- Write the formula: v = f × λ
- Rearrange for λ: λ = v ÷ f
- Substitute values: λ = 340 ÷ 170
- Calculate: λ = 2 m
Check: a wave at 170 Hz (a fairly low note) having a 2 m wavelength is consistent with wavelengths of audible sound, which range roughly from 2 cm to 17 m. ✓
Example 2: Finding frequency (light)
Green light travels at 3 × 10⁸ m/s. Its wavelength is 550 nm (550 × 10⁻⁹ m). Find the frequency.
- Write: v = f × λ
- Rearrange: f = v ÷ λ
- Substitute: f = 3 × 10⁸ ÷ (550 × 10⁻⁹)
- Calculate: f ≈ 5.45 × 10¹⁴ Hz
Check: visible light frequencies range from about 4 × 10¹⁴ Hz to 7 × 10¹⁴ Hz. ✓
Example 3: Finding wave speed
A water wave has a frequency of 5 Hz and a wavelength of 3 m. Find the wave speed.
- Write: v = f × λ
- No rearrangement needed.
- Substitute: v = 5 × 3
- Answer: v = 15 m/s
Example 4: Finding wavelength with unit conversion
A radio wave has a frequency of 100 MHz (100 × 10⁶ Hz = 1 × 10⁸ Hz) and travels at 3 × 10⁸ m/s. Find the wavelength.
- Convert MHz to Hz first: 100 MHz = 1 × 10⁸ Hz
- λ = v ÷ f = 3 × 10⁸ ÷ 1 × 10⁸ = 3 m
This is a typical FM radio wavelength — which explains why radio aerials are roughly 0.75–1.5 m long (a quarter-wavelength is the optimal aerial length).
How does wavelength vary across the electromagnetic spectrum?
All electromagnetic waves travel at the same speed in a vacuum (3 × 10⁸ m/s). As frequency increases, wavelength decreases — exactly as the wave equation predicts.
| Wave type | Approximate frequency | Approximate wavelength | Speed in vacuum |
|---|---|---|---|
| Radio waves | ~10⁶ Hz (1 MHz) | ~300 m | 3 × 10⁸ m/s |
| Microwaves | ~10¹⁰ Hz (10 GHz) | ~3 cm | 3 × 10⁸ m/s |
| Infrared | ~10¹³ Hz | ~30 μm | 3 × 10⁸ m/s |
| Visible light | ~5 × 10¹⁴ Hz | ~600 nm | 3 × 10⁸ m/s |
| Ultraviolet | ~10¹⁶ Hz | ~30 nm | 3 × 10⁸ m/s |
| X-rays | ~10¹⁸ Hz | ~0.3 nm | 3 × 10⁸ m/s |
| Gamma rays | ~10²⁰ Hz | ~3 pm | 3 × 10⁸ m/s |
Notice that multiplying any frequency by its corresponding wavelength gives approximately 3 × 10⁸ m/s — a satisfying confirmation of v = f × λ right across the spectrum.
What is the relationship between period and frequency?
Period (T) and frequency (f) are reciprocals of each other:
- T = 1/f
- f = 1/T
If a wave has a frequency of 50 Hz, its period is T = 1/50 = 0.02 s (20 milliseconds). If a wave has a period of 0.001 s, its frequency is f = 1/0.001 = 1 000 Hz = 1 kHz.
Because f = 1/T, the wave equation can also be written as v = λ/T. Units check: metres ÷ seconds = m/s ✓. This form is useful when you are given the period rather than the frequency.
What are the common mistakes to avoid with the wave equation?
Knowing where students go wrong is almost as useful as knowing the equation itself.
- Mistake 1 — Units: frequency must be in Hz, wavelength in metres, speed in m/s. Convert before substituting: 1 nm = 10⁻⁹ m; 1 cm = 0.01 m; 1 MHz = 10⁶ Hz; 1 GHz = 10⁹ Hz.
- Mistake 2 — Swapping λ and v: λ is the Greek letter lambda (wavelength, in metres); v is speed. They look different on paper but students sometimes confuse them in a hurry. Write the full equation before substituting.
- Mistake 3 — Forgetting to convert MHz or GHz: 100 MHz is NOT 100 Hz. Always check the prefix and multiply accordingly before substituting.
- Mistake 4 — Thinking frequency changes in a new medium: when a wave moves from one medium to another, its speed changes and its wavelength changes — but its frequency stays the same. Frequency is set by the source, not by the medium.
Frequently asked questions
Does frequency change when a wave moves from one medium to another?
No — frequency is determined by the source that generates the wave. When a wave crosses a boundary between two media (for example, light entering glass from air), the speed changes because the new medium transmits the wave differently. The wavelength therefore also changes to maintain v = f × λ. However, the number of wave cycles arriving at the boundary each second must equal the number leaving — so frequency is unchanged. This is why refraction (the bending of waves at a boundary) occurs: the change in speed causes the change in direction when waves arrive at an angle.
What is the speed of sound in air?
Sound travels at approximately 340 m/s in air at room temperature (around 20 °C). This is roughly one million times slower than the speed of light, which is why you see lightning before you hear thunder — light from the strike reaches you almost instantly, while sound takes about three seconds per kilometre. The speed of sound increases with temperature (warmer air molecules transmit vibrations more quickly) and is much higher in liquids and solids — about 1 480 m/s in water and around 5 000 m/s in steel — because the particles are more closely packed.
What is the speed of light?
The speed of light in a vacuum is approximately 3 × 10⁸ m/s (300 000 km/s). Every type of electromagnetic wave — radio, microwave, infrared, visible light, ultraviolet, X-ray, and gamma ray — travels at this speed in a vacuum. When electromagnetic waves enter a denser medium such as glass or water, they slow down. Visible light slows to about 2 × 10⁸ m/s in glass, which causes it to refract (bend) when it enters or leaves the glass at an angle. The ratio of the speed in a vacuum to the speed in a medium is called the refractive index of that medium.
How do I remember and rearrange the wave equation?
Draw a formula triangle: write v at the top, and f and λ side by side at the bottom. To find any quantity, cover it with your finger — the remaining symbols show how to calculate it. Cover v → multiply f × λ; cover f → divide v by λ; cover λ → divide v by f. Always check units before substituting, and always sense-check your answer. A useful mnemonic for the equation is "Very Fast Lamborghinis" — v = f × λ. Worked example as a check: f = 1 Hz, λ = 340 m gives v = 340 m/s, which is exactly the speed of sound in air. ✓
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