AkCalculators

Wavelength — λ = v/f and wavelength in a medium λ = λ₀ / n explained

Wavelength is the spatial period of a wave — the distance between successive identical points (e.g., crest-to-crest). Two very useful relations are:

λ = v / f    (general wave relation)

λ = λ₀ / n    (wavelength in a medium)

The first expresses wavelength in terms of wave speed v (m/s) and frequency f (Hz). The second shows how free-space (vacuum) wavelength λ₀ shortens inside a medium with refractive index n. Below we explain both, show worked examples, and give measurement and unit tips.

1. λ = v / f — origin and use

Given wave crests spaced λ apart and moving at speed v, the number of crests passing per second equals f = v / λ. Rearranged, λ = v / f. Applies to mechanical waves, sound waves, water waves and electromagnetic waves (use phase velocity in the medium).

2. Units and unit consistency

Always use SI units internally: v in m/s, f in Hz and λ in meters. This tool includes unit selectors for common units (Hz/kHz/MHz, m/cm/mm/nm) and converts to SI under the hood. Return values are presented in the units you selected for convenience.

3. Solving λ = v / f

• Compute λ: λ = v / f
• Compute f: f = v / λ
• Compute v: v = f × λ

4. Examples — wave relation

Example A — sound: v ≈ 343 m/s, f = 440 Hz → λ = 343 / 440 ≈ 0.7795 m (≈ 77.95 cm).

Example B — visible light (in vacuum): f = 5×10¹⁴ Hz, v ≈ 3×10⁸ m/s → λ ≈ 600 nm (red/orange region).

5. λ = λ₀ / n — light in a medium

Electromagnetic waves in a medium propagate slower than in vacuum according to v = c / n, where c is the speed of light in vacuum and n is refractive index. Vacuum wavelength λ₀ = c / f. Inside the medium the wavelength shortens to λ = v / f = (c / n) / f = λ₀ / n.

6. Examples — medium relation

Example C — light entering glass: Vacuum λ₀ = 600 nm, glass n = 1.5 → λ = 600 / 1.5 = 400 nm inside glass.

Example D — compute n: If vacuum λ₀ = 500 nm and measured λ in medium = 333.33 nm, then n = λ₀ / λ = 500 / 333.33 ≈ 1.5.

7. Measurement & practical tips

Measure wavelength directly (e.g., water waves) by spacing between crests. For sound, measure frequency with a detector and compute λ via v/f. For light, use diffraction gratings to measure λ₀ or interferometry for precise measurements. When measuring λ in a medium you may measure frequency (unchanged across boundary) and wave speed (or use λ = λ₀ / n if λ₀ known).

8. Dispersion & wavelength dependence

Refractive index n often depends on wavelength (dispersion). That means λ inside medium varies with λ₀ differently across spectrum. Use material dispersion data when designing optics — e.g., crown and flint glasses have different dispersion used for achromats.

9. Common mistakes

• Mixing units — convert cm/mm/nm to meters first (done automatically by this tool).
• For light, don't mix phase and group velocities without care — this calculator handles simple phase wavelength relations.
• Ensure frequency is the same when using λ = λ₀ / n — frequency does not change across boundary; wavelength and speed do.

10. Summary

Use λ = v / f for basic wave contexts and λ = λ₀ / n for light in media. This calculator supports both relations, unit conversions, step-by-step derivations and CSV export for lab documentation.