Core answer: Wavelength, frequency and wave speed obey one relation: v = fλ (speed = frequency × wavelength). For light in vacuum v is fixed at 299,792,458 m/s, so a 100 MHz FM radio wave is exactly 3 m long; red light at 700 THz is about 430 nm. Fix the medium first — the same sound wave is 4.3× longer in water than in air.
The universal wave equation
Every periodic wave — sound, light, radio, water ripples — obeys v = fλ. Two consequences:
- In a given medium, higher frequency means shorter wavelength, in exact inverse proportion.
- When a wave crosses into a new medium, the frequency stays constant (the source sets it) and the wavelength changes with the speed.
| Wave | Medium | Speed | Example |
|---|---|---|---|
| Light / radio | vacuum | 299,792,458 m/s | 100 MHz FM → λ = 3.0 m |
| Light | water / glass | ~225,000 / ~200,000 km/s | refraction, lenses |
| Sound (20 °C) | air | 343 m/s | 440 Hz (A4) → λ = 0.78 m |
| Sound | water | ~1,480 m/s | sonar |
| Sound | steel | ~5,900 m/s | rail "ping" heard early through the track |
Worked examples
Example 1 — Wi-Fi antenna size. 2.4 GHz Wi-Fi has λ = c/f = 3×10⁸ / 2.4×10⁹ = 0.125 m = 12.5 cm. A quarter-wave antenna is ~3.1 cm — exactly the stub on your router. 5 GHz Wi-Fi halves that to ~1.6 cm.
Example 2 — Concert hall acoustics. A 100 Hz bass note in air has λ = 343/100 ≈ 3.4 m. Rooms smaller than half a wavelength cannot support the standing wave cleanly, which is why small rooms have "boomy" uneven bass.
Example 3 — Sonar. A ship's sonar pings at 20 kHz in seawater (v ≈ 1,480 m/s): λ = 7.4 cm. The echo delay tells distance: an echo after 2 s means the object is 1,480 × 2 / 2 = 1,480 m away (divide by 2 for the round trip).
Frequency bands you actually use
| Band | Frequency | Wavelength | Use |
|---|---|---|---|
| AM radio | 530–1,600 kHz | 190–570 m | long-distance broadcast |
| FM radio | 88–108 MHz | 2.8–3.4 m | local broadcast |
| Wi-Fi / Bluetooth | 2.4 / 5 GHz | 12.5 / 6 cm | short-range data |
| Visible light | 430–750 THz | 400–700 nm | violet to red |
| X-ray | >30 PHz | <10 nm | medical imaging |
Common mistakes and myths
- "Frequency changes when entering a new medium" — no. Frequency is locked by the source; speed and wavelength adjust. This is why light keeps its color (frequency) underwater while bending.
- Mixing up nm and MHz for light — optical people speak in wavelength (nm), radio people in frequency (Hz); convert with f = c/λ.
- Assuming sound is instant — 343 m/s is slow: thunder 3 s after lightning is 1 km away; a 100 m stadium hears the crowd's roar with a noticeable delay from the far end.
- Using air's sound speed for underwater sonar — water is ~4.3× faster; using 343 m/s makes distance estimates 4× too small.
- Forgetting temperature drift — sound speed in air rises ~0.6 m/s per °C; precision acoustics must correct for it.