Mode 1 — Speed from frequency & wavelength
Mode 2 — Frequency from speed & wavelength (inverse)
Mode 3 — Wavelength from speed & frequency (inverse)
Mode 4 — Medium presets (reference values)
Click a card to fill the v value into Mode 2 and Mode 3. The values below are general references, not absolute constants — sound speed changes with the medium’s temperature.
Mode 5 — Period relation: T = 1/f, v = λ/T
Fill in exactly two of the three quantities; the third is computed automatically (plus f = 1/T).
Scope boundaries — read before using
- #1901 wave-velocity → consolidated here. Wave velocity and wave speed use identical physics (v = fλ); velocity only adds the direction of travel. No separate tool — the "wave velocity formula" keyword cluster is covered by this page.
- Not the velocity calculator (#1882). That tool uses v = Δx/Δt for moving objects; its formula is entirely different from v = fλ for periodic waves.
- Not a seismic wave calculator. Shear waves (v = √(G/ρ)) and longitudinal waves via bulk modulus (v = √(B/ρ)) belong to rock mechanics — see the bulk modulus calculator.
- Not the wave number calculator (#1899). That tool computes k = 2π/λ; here k is only a derived quantity. Open wave-number →
- Not the frequency unit converter (#288). The frequency converter changes units (Hz ↔ kHz); its wavelength tab converts λ ↔ f in a fixed medium — it does not solve v. Open frequency-converter →
- Angular form: v = ω/k is covered implicitly — every result shows ω = 2πf and k = 2π/λ as derived quantities.
Verified examples
| Case | Input | Result |
|---|---|---|
| A4 note in air | f = 440 Hz, λ = 0.78 m | v = 343.2 m/s (440 × 0.78 exactly) |
| Inverse: find frequency | v = 343 m/s, λ = 0.78 m | f = 439.74 Hz |
| Sonar in water | v = 1480 m/s, f = 20 kHz | λ = 0.074 m = 7.4 cm |
| Period relation | f = 440 Hz → T = 1/440 s = 2.2727 ms; λ = 0.78 m | v = 343.2 m/s |
| FM radio in vacuum | v = c, f = 100 MHz | λ = 2.99792458 m |
All numbers above were recomputed with 50-digit precision arithmetic (oracle.py) and match this page’s engine.
Methodology
Core formula: v = f × λ, with inverses f = v / λ and λ = v / f. Period relation: T = 1/f so v = λ / T. Derived quantities: angular frequency ω = 2πf and wave number k = 2π/λ (hence v = ω/k is consistent).
Constants: speed of light c = 299,792,458 m/s (exact, SI definition). Sound presets — air 343 m/s (reference 20 °C) and water 1480 m/s (reference ±20 °C) — are general reference values, not natural constants; real sound speed depends on temperature (air ±0.6 m/s per °C), salinity, and pressure.
Validation: inputs must be non-negative numbers; divisors (f, λ, T in denominator position) must be > 0. Results are formatted with at most 6 significant decimals to avoid floating-point artifacts.
Frequently asked questions (FAQ)
What is the wave speed formula?
The formula is v = f × λ: wave speed equals frequency times wavelength. Example: f = 440 Hz and λ = 0.78 m give v = 440 × 0.78 = 343.2 m/s.
What is the difference between wave speed and wave velocity?
In basic physics both use the identical formula v = f × λ. Wave speed is the magnitude (scalar, m/s), while wave velocity adds the direction of travel (vector). This calculator covers both: modes 1–3 solve for the magnitude of v, and the direction of travel is from source to receiver.
How do I compute frequency from speed and wavelength?
Invert the formula: f = v / λ. Example: v = 343 m/s and λ = 0.78 m give f = 343 / 0.78 = 439.74 Hz.
How do I compute wavelength from speed and frequency?
Use λ = v / f. Example: sound in water v = 1480 m/s with f = 20 kHz gives λ = 1480 / 20000 = 0.074 m = 7.4 cm.
How is the period related to wave speed?
Period T = 1/f, so v = λ / T. Example: f = 440 Hz means T = 1/440 s = 2.2727 ms; with λ = 0.78 m, v = 0.78 / (1/440) = 343.2 m/s.
What is the speed of sound in air?
343 m/s is the reference value at 20 °C. The true value depends on temperature (rising about 0.6 m/s per degree Celsius), so the presets in this calculator are reference points, not natural constants.
What is the speed of sound in water?
1480 m/s is the reference value for water at around 20 °C. In seawater it is slightly higher (about 1500–1530 m/s depending on temperature, salinity, and pressure).
What is the speed of light, with a wave example?
The speed of light in vacuum c = 299,792,458 m/s (exact, SI definition). Example: a radio signal at f = 100 MHz in vacuum has λ = 299792458 / 100000000 = 2.99792458 m.
Which units does this calculator support?
Frequency: Hz, kHz, MHz, GHz. Wavelength: m, cm, mm, km, μm, nm. Speed: m/s, km/h, mph, ft/s. Period: s, ms, μs, ns. All conversions use exact factors.
What are angular frequency ω and wave number k?
Angular frequency ω = 2πf (rad/s) and wave number k = 2π/λ (rad/m). Example: f = 440 Hz gives ω = 2764.6 rad/s; λ = 0.78 m gives k = 8.0554 rad/m. Both are shown automatically as derived quantities in every mode.
Does v = f × λ hold for all wave types?
Yes. This formula holds for transverse waves (e.g. light, string waves) as well as longitudinal ones (e.g. sound), as long as the medium is not strongly dispersive. The only thing that changes between media is the value of v.
Why does 440 Hz × 0.78 m give 343.2 m/s?
Because 440 × 0.78 = 343.20 exactly — no coincidence. The A4 note (440 Hz) in air indeed has a wavelength of about 0.78 m, so v = 343.2 m/s, matching the speed of sound in air at 20 °C.
How does this differ from the velocity calculator?
The velocity calculator uses v = Δx/Δt (displacement over time interval) for moving objects. This calculator uses v = f × λ specifically for periodic waves — the formulas and inputs are entirely different.
How does this differ from the wave number calculator?
The wave number calculator computes k = 2π/λ (rad/m). This calculator solves for v, f, or λ themselves; k is only shown as a derived quantity. The two complement each other.
Is wave speed the same in every medium?
No. Wave speed is a property of the medium, not of the wave: sound travels at 343 m/s in air but 1480 m/s in water and about 5000 m/s in steel. Light is actually fastest in vacuum. Always use the v value for your medium.
How do I use the medium presets?
Click one of the preset cards (sound in air, sound in water, or light in vacuum) on the Medium Presets tab. The v value automatically fills into modes 2 and 3 along with its reference note, then you just enter λ or f.