Waves: Speed, Frequency & Wavelength
Every wave, from a sound to a radio signal to light, obeys one rule: its speed is the frequency times the wavelength. Solve for any of the three, for sound in air, water or steel, for light, or for any wave you like.
What is a Wave?
A wave carries energy from place to place without carrying the stuff it moves through. Air molecules in a sound wave only jiggle back and forth; the pattern travels on. The frequency is how many times per second each point jiggles, the wavelength the distance between two crests.
The speed is set by the medium, not by the source: a high and a low note reach you at the same moment. Change the frequency and the wavelength changes with it, so that f·λ stays equal to the speed.
The Wave, Step by Step
v = fλKey Relations
In one period T = 1/f a crest moves exactly one wavelength, so the speed is λ/T = fλ.
With T in °C. Warm air carries sound faster: about 0.6 m/s more per degree.
Light of frequency f comes in packets of energy hf. In glass or water it slows down by the refractive index n; the frequency stays, the wavelength shrinks.
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 16 (Waves I), Ch. 17 (Waves II, Table 17-1 speeds of sound), Ch. 33 (Electromagnetic Waves).
- BIPM, The International System of Units (SI), 9th ed., 2019 — exact c = 299 792 458 m/s and h = 6.626 070 15 × 10−34 J·s.
- ISO 16:1975 — standard tuning pitch A4 = 440 Hz.
Frequently Asked Questions
How do you calculate wavelength from frequency?
Divide the wave speed by the frequency: λ = v / f. For the note A (440 Hz) in air at 20 °C: 343 ÷ 440 = 0.780 m. The other way round, f = v / λ, and the speed itself is v = f × λ.
How do you convert the wavelength of light to frequency?
Use the speed of light, c = 299 792 458 m/s: f = c / λ. Green laser light of 532 nm has f = 299 792 458 ÷ 0.000 000 532 = 5.64 × 10¹⁴ Hz, or 564 THz.
How fast does sound travel?
About 343 m/s in air at 20 °C (331 m/s at 0 °C), 1481 m/s in water and about 5960 m/s in steel. Because it covers a kilometre in roughly 3 seconds, you can count the seconds between lightning and thunder and divide by 3 to get the distance in km.
Why does the wavelength change when light enters water but the frequency does not?
The frequency is set by the source: every crest that arrives at the surface must leave it, so the number of crests per second stays the same. The light slows down by the refractive index n, so the wavelength shrinks by the same factor: 532 nm light becomes 532 ÷ 1.333 = 399 nm in water. Its colour stays green.
How do I find the musical note of a frequency?
Count semitones from A4 = 440 Hz: n = 12 × log₂(f ÷ 440). Each semitone multiplies the frequency by 2^(1/12) ≈ 1.0595 and each octave doubles it. Middle C (C4) is 261.63 Hz; 450 Hz is an A4 that is 39 cents (hundredths of a semitone) sharp.
What is the energy of a photon?
E = h × f, with Planck's constant h = 6.626 × 10⁻³⁴ J·s. For light it is handy in electronvolts: E (eV) ≈ 1240 ÷ λ (nm). Green light of 532 nm carries 2.33 eV per photon, an X-ray of 0.1 nm about 12 400 eV.