Wave Speed Calculator

The universal wave equation

Every travelling wave obeys the relationship v = f x lambda, where v is speed in metres per second, f is frequency in hertz (cycles per second), and lambda is wavelength in metres (distance between successive crests). Multiplying how many full cycles pass a point each second by the length of each cycle gives how far the wave front advances per second. This single equation applies to ripples on water, sound in air, seismic shaking, light, radio waves, X-rays, and every other form of wave energy.

Period, wavenumber, and angular frequency

Three derived quantities complete the standard wave description. The period T = 1/f is the time for one full cycle, measured in seconds. The wavenumber k = 1/lambda (sometimes written 2*pi/lambda for the angular wavenumber) counts spatial cycles per metre and is the spatial analogue of frequency. Angular frequency omega = 2*pi*f, in radians per second, is the rate of phase change and appears in the standard wave function y = A*sin(omega*t - k*x). These quantities are all derivable from frequency and wavelength once wave speed is known, and this calculator displays all of them automatically.

Why wave speed depends on the medium, not the source

A common misconception is that turning up the frequency makes a wave travel faster. In reality the speed of a mechanical wave is set almost entirely by the medium it moves through, its density, tension, temperature, and elasticity, not by the source producing it. When you raise the frequency, the wavelength shortens by exactly the same proportion so that their product, the speed, stays constant. Sound travels at roughly 343 m/s in air at 20 C regardless of pitch, about 1,480 m/s in water, and around 5,000 m/s in steel. Electromagnetic waves travel at the speed of light, approximately 299,792,458 m/s in a vacuum, and slow when they enter glass, water, or other transparent materials.

Sound in air: temperature dependence

For sound in dry air, the Newton-Laplace formula gives a close approximation: v = 331.3 x sqrt(1 + T_C / 273.15) m/s, where T_C is the temperature in Celsius. At 0 C this yields about 331.3 m/s; at 20 C about 343 m/s; at 40 C about 355 m/s. The speed rises by roughly 0.6 m/s per degree Celsius. Humidity has a smaller but measurable effect (water vapour is less dense than dry air and slightly raises the speed), but the temperature term dominates in most practical situations.

String waves and electromagnetic waves

For a taut string or rope, the wave speed is v = sqrt(F / mu), where F is the tension in newtons and mu is the linear mass density in kg/m. Tightening a guitar string (increasing F) raises both the wave speed and the audible pitch. For electromagnetic waves in a transparent medium with refractive index n, the wave speed is v = c / n. Glass has n of about 1.5, so light travels at about 200,000 km/s inside it rather than 300,000 km/s. Diamond, with n of about 2.42, slows light to about 124,000 km/s, which is why it sparkles so vividly.

Rearranging to find frequency or wavelength

Because the three quantities are linked by one equation, knowing any two lets you find the third. Use the "Solve for" selector on General mode to choose your unknown: lambda = v / f gives wavelength when speed and frequency are known; f = v / lambda gives frequency when speed and wavelength are known. This is how musicians, radio engineers, and oceanographers move between the quantities they can measure most easily. A radio station broadcasting at 100 MHz emits waves about 3 metres long, because lambda = (3 x 10^8) / (100 x 10^6) = 3 m.