Standing Wave Calculator
A standing wave is the shimmering, seemingly motionless pattern that appears when a wave bounces back on itself and the two combine, and it is exactly what happens on a plucked guitar string, a violin string or any string fixed at both ends, so this calculator works out the harmonic frequencies and wavelengths such a string can produce. Enter the length of the string, the speed at which waves travel along it, and the harmonic number you are interested in, and it returns the frequency and wavelength of that harmonic together with the fundamental frequency, updating as you type. The reason a fixed string can only vibrate at certain special frequencies is that both ends must stay still, forming nodes, so only whole numbers of half-waves fit neatly along its length. The simplest pattern, a single hump, is the fundamental or first harmonic, the lowest note the string plays, and its frequency is the wave speed divided by twice the length. Every other allowed frequency is a whole-number multiple of that fundamental: the second harmonic is twice as high, the third three times, and so on, with the wavelength of the nth harmonic being twice the length divided by n. This neat ladder of frequencies is the physics behind every stringed instrument, because it is the particular mix of these harmonics that gives a guitar, a piano and a cello their distinctive tone, and it is why pressing a string to shorten it raises the pitch. The same idea, with slightly different rules, governs pipes, drums and microwave cavities. That makes the tool genuinely useful for physics and music students learning about waves, resonance and harmonics and checking homework, and for anyone exploring how instruments make their notes. The formulas and a worked example are explained clearly below.
How it works
For a string of length L fixed at both ends, the nth harmonic frequency is n times the wave speed divided by twice L. The wavelength of the nth harmonic is twice L divided by n. The fundamental, the first harmonic, is the wave speed divided by twice L, and all the harmonics are whole-number multiples of it.
Worked example
For a 1 metre string with a wave speed of 200 metres per second, the fundamental is 200 divided by 2, which is 100 hertz, with a wavelength of 2 metres. The second harmonic is 200 hertz, the third 300 hertz, and so on.