Wave Speed Calculator
Solve the wave equation v = fλ for whichever term you are missing — speed, frequency, wavelength or period — with units from nanometres to kilometres. Two more modes find the speed of a wave on a stretched string, v = √(F ÷ μ), with its standing-wave harmonics, and the speed of sound in air at any temperature. Free, no sign-up.
Results
343 m/s
Wave speed (v)
440 Hz
Frequency (f)
779.5 mm
Wavelength (λ)
2.273 ms
Period (T)
2.765 krad/s
Angular frequency (ω)
8.06 rad/m
Wavenumber (k)
The working
- λ = v ÷ f = 343 m/s ÷ 440 Hz = 779.5 mm
How to Use the Wave Speed Calculator
- 1. Choose the kind of wave. Any wave uses v = fλ and suits sound, water, light and string waves alike. Wave on a string and Sound in air use the physics specific to those media.
- 2. Say what you want to find and enter the rest. In the general mode, pick speed, frequency, wavelength or period, then type the quantities you know. You can enter a period instead of a frequency when that is what you measured.
- 3. Pick units. Each box has its own menu — m/s, km/h, mph or ft/s for speed; Hz to THz for frequency; nm to km for wavelength; µs up to ks for time.
- 4. Read the results. Every related quantity appears together, including angular frequency and wavenumber, with the formula and your numbers written out and a sketch of the wave with its wavelength marked.
The Wave Equation
For any periodic wave, the speed equals the frequency times the wavelength: v = f × λ. Frequency f is the number of wave crests passing a point each second, in hertz; wavelength λ is the distance between crests, in metres; and the period T = 1/f is the time for one full cycle. Rearranged, f = v ÷ λ, λ = v ÷ f and T = λ ÷ v.
The speed of a wave is set by the medium, not by the source. A louder or higher-pitched note does not travel faster; raising the frequency simply shortens the wavelength. That is why the equation is most useful when the medium fixes v: at a given speed, doubling f halves λ.
| Wave | Speed | Formula |
|---|---|---|
| Any periodic wave | v | v = f × λ |
| Transverse wave on a string | Depends on tension and mass per length | v = √(F ÷ μ) |
| Sound in dry air | Depends on temperature | v = √(γ × R × T) |
| Light or radio in vacuum | 299,792,458 m/s (exact) | c = f × λ |
Worked Examples
- Wavelength of concert A. Sound travels at 343 m/s, and A4 is 440 Hz, so λ = 343 ÷ 440 = 0.7795 m (779.5 mm). The period is 1 ÷ 440 = 2.273 ms, the calculator's starting example.
- A guitar string. A string under 70 N with a mass per length of 0.4 g/m has v = √(70 ÷ 0.0004) = 418.3 m/s. Fixed at both ends and 0.65 m long, its fundamental frequency is v ÷ 2L = 418.3 ÷ 1.3 = 321.8 Hz; the second harmonic is twice that.
- Speed of sound on a warm day. At 20 °C, v = √(1.4 × 287.05 × 293.15) = 343.2 m/s, about 1,236 km/h. At 0 °C it is 331.3 m/s. Count three seconds between lightning and thunder and the strike was about 1 km away.
Speed of Sound in Air and Other Media
Air is treated as an ideal diatomic gas with γ = 1.4 and a specific gas constant R = 287.05 J/(kg·K), which reproduces the standard figure of 331.3 m/s at 0 °C. The speed rises about 0.6 m/s for each degree Celsius, and the straight-line approximation 331.3 + 0.606 θ is shown for comparison. The formula assumes dry air; humid air is slightly faster because water vapour is lighter than the nitrogen and oxygen it replaces.
| Medium | Condition | Speed of sound (m/s) |
|---|---|---|
| Air | 20 °C, dry | 343.2 |
| Fresh water | 20 °C | about 1,481 |
| Sea water | 10 °C, 3% salinity | about 1,500 |
| Copper | solid, longitudinal | 4,664 |
| Steel | stainless 304, longitudinal | 5,912 |
| Aluminium | alloy 1100, longitudinal | 6,313 |
| Diamond | solid | about 12,000 |
What the calculator assumes
The string mode applies to an ideal, perfectly flexible string vibrating with small amplitude; a stiff or heavy string deviates slightly. The sound mode is for dry air near the ground and ignores humidity and altitude pressure effects, which are small compared with temperature. Solid and liquid speeds in the table depend on the exact material and temperature.
Frequently Asked Questions
- What is the formula for wave speed?
- v = f × λ, wave speed equals frequency times wavelength. If you know the period instead of the frequency, use v = λ ÷ T. All three speeds, frequencies and wavelengths must use consistent units, such as metres, seconds and hertz.
- Does frequency change the speed of a wave?
- Not in a given medium. The medium determines the speed; changing the source's frequency changes the wavelength instead. In a dispersive medium, such as light in glass, speed does depend slightly on frequency, which is why a prism separates colours.
- How do you find the speed of a wave on a string?
- Take the square root of the tension divided by the string's mass per unit length: v = √(F ÷ μ). Tighter or lighter strings carry waves faster, which is why tightening a guitar string raises its pitch.
- How fast does sound travel at room temperature?
- About 343 m/s at 20 °C in dry air, which is roughly 1,235 km/h or 767 mph. The speed falls to about 331 m/s at 0 °C and rises by about 0.6 m/s per degree Celsius.
- Why does sound travel faster in solids and liquids than in air?
- Sound passes from particle to particle, and in stiffer materials the restoring forces are much stronger relative to the density. Steel carries sound at around 5,900 m/s, water at around 1,480 m/s, and air at only about 343 m/s.
- How do I use the speed of sound to estimate how far away lightning is?
- Count the seconds between the flash and the thunder and multiply by about 343 m/s. Three seconds is roughly one kilometre, or five seconds about a mile. The flash itself arrives almost instantly, because light is close to a million times faster.