Wave Speed Calculator

The wave equation v = fλ links a wave's speed, frequency and wavelength. This calculator solves it for whichever one you need, and adds three modes that work out the speed itself from the medium: sound in air at a given temperature, a wave on a stretched string from its tension and mass, and a water wave from its wavelength and the water depth.

The string mode also gives the fundamental pitch and harmonics if you enter the vibrating length, which is how instrument makers choose string gauges. The water mode uses the full dispersion relation, so it handles everything from wind ripples in the deep ocean to tsunamis and waves breaking on a beach. To convert between frequency, period, wavenumber and photon energy for a single wave, use the wave calculator.

v = fλ

Sound in air: c = √(γRT ÷ M) ≈ 331.3 √(1 + T°C ÷ 273.15) m/s

String: v = √(T ÷ μ), fundamental f₁ = v ÷ 2L

Water: v = √((gλ ÷ 2π) tanh(2πd ÷ λ))

In the sound formula γ = 1.4 for air, R is the gas constant and M the molar mass of dry air. For the string, T is the tension in newtons, μ the mass per meter and L the vibrating length. For water, d is the depth: when it is more than half a wavelength the tanh term is nearly 1 and long waves run faster than short ones; when it is less than a twentieth of a wavelength the speed simplifies to √(gd) and depends only on depth.

Concert A. 440 Hz in air at 343 m/s has a wavelength of 78 cm.

Sound on a cold and a warm day. At −10 °C sound travels at 325.2 m/s; at 30 °C it reaches 349 m/s. Wind instruments go flat in the cold for this reason.

A guitar string. 70 N on a string weighing 0.58 g per meter gives waves at 347.4 m/s; over a 648 mm scale the fundamental is 268.1 Hz, close to middle C.

A tsunami. A 200 km long wave in 4 km of water moves at 198 m/s, about 711 km/h, as fast as a jet airliner.

Ocean swell. A 150 m swell in deep water travels at 15.3 m/s with a period of 9.8 s.

A wave's speed depends on the medium, not on how the wave was made. In general it is the square root of a restoring stiffness divided by an inertia: tension over mass per length for a string, pressure response over density for a gas, gravity times depth for shallow water. Changing the frequency at the source changes the wavelength, not the speed. Sound in air does not depend on pressure at a fixed temperature, because pressure and density rise together, but it does depend on temperature and on the gas: in light helium it is almost three times faster, which is why helium raises the pitch of a voice.

As ocean waves move into shallow water, √(gd) takes over and they slow down. The period stays the same, so the wavelength shortens and the energy piles into a smaller distance, making the wave taller. Eventually the crest outruns the trough and the wave breaks, typically when the height reaches about 0.8 times the water depth. Waves also bend toward shallower water, which is why they arrive nearly parallel to the shore even when the wind blows at an angle. A tsunami, barely noticeable in the open ocean, grows dramatically in the same way.