Wave Speed Calculator

Wave speed is frequency times wavelength

Three numbers describe any repeating wave, and the calculator turns any two of them into the third. Frequency counts how many cycles pass a fixed point each second, in hertz. Wavelength measures the distance from one crest to the next, in metres. Speed says how far the pattern travels in that second. Multiply the first two and you get the third, because cycles per second times metres per cycle leaves metres per second.

A 1,000 Hz tone in air at 20 C runs at 343.2 m/s, so each cycle stretches 343.2 / 1,000 = 0.343 m. Run the same tone in fresh water, where the speed is 1,482 m/s, and the wavelength grows to 1.482 m. The tone did not change; the road under it did.

The period sits in the same family. One cycle of a 1,000 Hz tone takes 1 / 1,000 = 0.001 s, or 1 ms. NASA's classroom handout on the speed of sound builds the same relationship from a pipe and a tuning fork.

Nothing in the formula cares what kind of wave it is. Sound, ripples on a pond, a wave down a jump rope and a radio signal all obey v = f x wavelength. Only the speed constant changes.

The medium sets the speed, not the wave

Send the same 1,000 Hz tone through air, water and steel and it keeps its frequency everywhere, but its speed and its wavelength change with the material. Stiffer and lighter materials carry the disturbance faster, which is why steel outruns water and water outruns air.

The jump from air to steel is a factor of about 17, and from air to light about 874,000. That spread is why the calculator makes you choose a medium instead of guessing one speed. A NOAA teaching module lists the same ladder: roughly 1,000 ft/s in air, 4,800 ft/s in water and 6,000 ft/s in solid rock.

One caution on the light row. Radio waves, microwaves and visible light all travel at the vacuum speed in air to within 0.03 percent, so for frequency to wavelength work you can use 299,792,458 m/s and stay accurate to about a millimetre per metre.

Worked example: counting seconds after lightning

The oldest wave speed measurement most people make is thunder. Lightning arrives as light, which crosses a kilometre in 3 microseconds, so the flash is effectively instant. The sound then covers the distance at the speed of sound.

Count 3 seconds between the flash and the thunder. At 343 m/s that is 3 x 343 = 1,029 m, close to the old rule of one mile per five seconds and one kilometre per three seconds. Count 9 seconds and the strike was 9 x 343 = 3,087 m, about 3 km. At 30 seconds you are 10 km away and out of the immediate danger zone most safety guides use.

The same math runs the other way for anyone timing an echo. Clap 343 m from a canyon wall and the sound makes a 686 m round trip, so the echo returns after 686 / 343 = 2.0 s. Ultrasonic tape measures and parking sensors do this dozens of times a second with a 40 kHz pulse whose wavelength is 343 / 40,000 = 0.0086 m.

Cold days stretch the wait. At 0 C the speed drops to 331 m/s, so 3 seconds spans 993 m instead of 1,029 m, a 3.5 percent error if you ignore the temperature.

Air temperature moves the number

Sound is pressure passing through moving molecules, and warmer molecules move faster, so the wave moves faster. For air the working formula is v = 331.3 x square root of (1 + T / 273.15), with T in degrees C. A simpler engineering approximation is v = 331.3 + 0.606 x T, which is accurate to within 0.2 percent from -20 C to 40 C.

Every degree C is worth about 0.6 m/s, which is 0.18 percent. For thunder counting that is noise, but for an ultrasonic flow meter or a distance sensor it is the dominant error term, which is why those devices carry a temperature sensor on the board.

Humidity also nudges the speed up, because water vapor is lighter than the nitrogen and oxygen it displaces, but the effect stays under about 0.5 percent for normal weather. Altitude matters mostly through temperature: the standard atmosphere cools 6.5 C per km up to 11 km, so sound slows as you climb even though the pressure is not the direct cause.

Light and radio waves: the vacuum speed and refractive index

Electromagnetic waves do not need a medium, and in a vacuum they all travel at c = 299,792,458 m/s. That figure is exact because the SI definition of the metre fixes the metre from the second and this speed. The NIST constants pages list it with zero uncertainty.

In transparent materials light slows by the refractive index n, so v = c / n. Crown glass with n = 1.52 carries light at 299,792,458 / 1.52 = 197,231,880 m/s, about 66 percent of the vacuum speed. Water with n = 1.33 gives 225,408,000 m/s. Diamond with n = 2.42 slows light to 124 million m/s, the steepest slowdown of any common clear material.

Radio work leans on the same numbers. A 2.4 GHz Wi-Fi signal has a wavelength of 299,792,458 / 2,400,000,000 = 0.125 m, so a quarter wave antenna element is about 3.1 cm. An FM station at 100 MHz runs a 3.0 m wavelength, which is why FM antennas are metres long and Wi-Fi antennas are centimetres.

Worked example: sonar and the ocean floor

A depth sounder pings a pulse of sound straight down and times the echo. Seawater at 20 C carries sound at 1,522 m/s, and the pulse has to make a round trip, so depth = 1,522 x time / 2.

An echo returns 3.94 s after the ping. The depth is 1,522 x 3.94 / 2 = 2,998 m, about 3 km. Trimble that number against a chart and the check works: a 3,000 m column of seawater should echo back in 2 x 3,000 / 1,522 = 3.94 s.

Real sonar adds two corrections. First, seawater is layered: near the surface it is warm and fast, and it cools with depth, so the pulse bends rather than travelling in a straight line. Second, salinity adds about 1.3 m/s per part per thousand over fresh water, and pressure adds roughly 1.7 m/s per 100 m of depth. Oceanographers pack all of this into tables, and the honest answer is that a single number like 1,522 m/s is a surface value, not a column average.

Fresh water runs 1,482 m/s at 20 C, about 2.6 percent slower, so the same 3.94 s echo in a lake reads 2,920 m. Multiply a lake depth by the seawater speed and you will over report depth by about 80 m over 3 km, which matters for anyone casting off a shelf.

How to sanity check a wave speed answer

Three checks catch most mistakes. First, check the units: frequency in hertz and wavelength in metres give metres per second, but a frequency in kHz with a wavelength in cm gives m/s only if you convert both. 100 kHz at 1.36 cm is 100,000 x 0.0136 = 1,360 m/s, a plausible seawater figure; 100 x 1.36 would be nonsense.

Second, check the magnitude against the ladder. Sound in ordinary materials lands between 300 and 7,000 m/s, with gases at the bottom, liquids around 1,400 to 1,600, and metals on top. A sound speed of 34 m/s means someone slipped a decimal, and 34,000 m/s means the same.

Third, check which value is fixed. For a driven system, a string, a loudspeaker, an antenna, the frequency is set by the driver and the wavelength adjusts to the medium. For a plucked string or a struck pipe, the length sets the wavelength and the pitch adjusts. Reading the situation backwards is the most common error, and the wavelength calculator works the same relationship from the other side.