Calculate wave celerity in meters per second from wavelength and frequency, period, or water depth, including shallow and deep water classification.
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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 โ celerity (m/s)
- ฮป โ wavelength (m)
- f โ frequency (Hz)
- T โ period (s), where f = 1/T
- ฯ โ angular frequency, 2ฯ/T (rad/s)
- k โ wavenumber, 2ฯ/ฮป (rad/m)
- h โ water depth (m)
- g โ gravitational acceleration, 9.80665 m/sยฒ
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 ร 10โธ m/s |
| Sound in air (20 ยฐC) | 343 m/s |
| Sound in fresh water (about 20 ยฐC) | โ 1480 m/s |
| Deep ocean swell (T = 12 s) | โ 18.7 m/s |
| Tsunami in 4 km ocean | โ 198 m/s (713 km/h) |
| Shallow-water wave, h = 2 m | โ 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 โ โ(gยทh) |
| 0.05 โ 0.5 | Intermediate | Use full dispersion |
| > 0.5 | Deep | c โ gยทT / (2ฯ) |
Worked Examples
FM radio wave. ฮป = 3.0 m, f = 100 MHz. c = 3.0 ร 100,000,000 = 3.0 ร 10โธ m/s. That is close to the vacuum speed of light; the rounded wavelength and frequency give an approximate comparison.
Ocean swell, deep water. T = 10 s. Using c โ gT/(2ฯ) = 9.81 ร 10 / 6.283 โ 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 โ โ(9.81 ร 4000) โ 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 describes wave-energy transport. For deep-water gravity waves group speed is half the phase speed; the relationship depends on the wave and medium.
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.
