Enter the wavelength (m) and the frequency (hz) into the Celerity Calculator. The calculator will evaluate the Celerity.
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Celerity Formula
Celerity is the phase speed of a wave: the distance a crest travels per second. The calculator uses one of three formulas depending on the inputs you have.
c = λ × f
c = λ / T
ω² = g·k·tanh(k·h), c = ω / k
- c — celerity (m/s)
- λ — wavelength (m)
- f — frequency (Hz)
- T — period (s), where f = 1/T
- ω — angular frequency, 2π/T (rad/s)
- k — wavenumber, 2π/λ (rad/m)
- h — water depth (m)
- g — gravitational acceleration, 9.80665 m/s²
The first two formulas apply to any non-dispersive wave or to a single frequency component. The third is the linear gravity-wave dispersion relation for surface water waves; the calculator solves it numerically for k, then returns c. Surface tension, currents, and finite-amplitude effects are ignored.
Reference Values
Use these to sanity-check a result before trusting it.
| Wave type | Typical celerity |
|---|---|
| Light in vacuum | 2.998 × 10⁸ m/s |
| Sound in air (20 °C) | 343 m/s |
| Sound in water | ≈ 1480 m/s |
| Deep ocean swell (T = 12 s) | ≈ 18.7 m/s |
| Tsunami in 4 km ocean | ≈ 198 m/s (713 km/h) |
| Shallow-water wave, h = 2 m | ≈ 4.4 m/s |
For water waves, the depth-to-wavelength ratio (h/L) tells you which regime you are in.
| h / L | Regime | Approximation |
|---|---|---|
| < 0.05 | Shallow | c ≈ √(g·h) |
| 0.05 – 0.5 | Intermediate | Use full dispersion |
| > 0.5 | Deep | c ≈ g·T / (2π) |
Worked Examples
FM radio wave. λ = 3.0 m, f = 100 MHz. c = 3.0 × 100,000,000 = 3.0 × 10⁸ m/s. That matches the speed of light, as expected for a radio wave in air.
Ocean swell, deep water. T = 10 s. Using c ≈ gT/(2π) = 9.81 × 10 / 6.283 ≈ 15.6 m/s. The wavelength is about 156 m, so this holds in any water deeper than roughly 78 m.
Tsunami in shallow water. h = 4000 m, treated as shallow because tsunami wavelengths exceed 100 km. c ≈ √(9.81 × 4000) ≈ 198 m/s.
FAQ
Is celerity the same as wave speed? Yes for phase celerity, which is what this calculator returns. It differs from group velocity, which is the speed at which wave energy travels and is generally slower in dispersive media.
Why does the water-wave mode need depth? Surface gravity waves are dispersive. In water shallower than about half a wavelength, the bottom slows the wave. Period alone is not enough.
Can I use this for light or sound? Yes, in the wavelength-and-frequency or wavelength-and-period modes. Just enter values consistent with the medium you care about.
