Fundamental Frequency of a String

A string clamped at both ends cannot move at its ends, and that single boundary condition does all the work. Only standing waves with a node at each clamp can survive; the longest one that fits is a single half-wave, so $\lambda = 2L$. Substitute that into the universal wave equation $v = f\lambda$ and you get $f = v/2L$. Nothing else is going on here — this page is the wave equation plus a boundary. The higher modes fit two half-waves, three, four, and so on, giving $f_n = nv/2L$: a series of exact integer multiples. That integer relationship is why a string produces a pitched note rather than a noise, and why two strings an octave apart share so many partials that we hear them as almost the same note.

A worked case from a guitar. The standard scale length is 25.5 in, about 648 mm, and the low E string sounds 82.4 Hz. The wave speed along that string is therefore $v = 2 \times 0.648 \times 82.4 \approx 107$ m/s. Press the string at the twelfth fret and the vibrating length halves to 324 mm, which doubles the frequency to 164.8 Hz — one octave up. Every fret position on the neck is this equation solved for $L$.

Marin Mersenne published the laws of the vibrating string in Harmonie universelle in 1636, decades before Newton: frequency varies inversely with length, inversely with the square root of the string's mass per unit length, and directly with the square root of tension. The last two are bundled into $v$ here, through $v = \sqrt{T/\mu}$, and the page keeps $v$ as an input rather than burying that relation so you can see which quantity you are actually changing. The same $f = v/2L$ also gives the fundamental of a pipe open at both ends, where $v$ is the speed of sound: a 0.65 m open pipe sounds near 264 Hz, roughly middle C.

The dominant error is entering the speed of sound in air. The $v$ in this equation is the speed of the transverse wave travelling along the string itself, typically 100 to 300 m/s, and it has nothing to do with the 343 m/s at which the resulting sound reaches your ear. The string sets the frequency; the air merely carries it. Two further traps: a pipe closed at one end holds a quarter wave, not a half, so its fundamental is $v/4L$ — an octave lower than an open pipe of the same length, which is why a stopped organ rank saves so much pipe. And $L$ is the vibrating length between the fixed points, from nut or fret to bridge, not the whole length of the string as it lies on the instrument.