Vibrations Produced in a String

Exactly the same standing-wave reasoning used for air columns (sections 6.7.2, 6.7.3) applies to a stretched STRING of length l, linear density (mass per unit length) m, held under tension T between two rigid supports. Since BOTH ends are rigidly fixed, both must always be NODES (with the usual $\pi$-radian phase change on reflection at each, from section 6.3.1), and the possible standing-wave patterns are built up exactly as for a pipe open at both ends -- an antinode-node-antinode... structure, but here bounded by nodes at both ends instead of antinodes.

If the string is plucked exactly at its centre, it settles into the FUNDAMENTAL mode: a single loop, with an antinode at the centre and nodes only at the two fixed ends -- HALF a wavelength fills the string's length, $\lambda=2l$, giving fundamental frequency (from $v=n\lambda$ and the wave speed on a stretched string, $v=\sqrt{T/m}$) $n=\dfrac{v}{2l}=\dfrac{1}{2l}\sqrt{\dfrac{T}{m}}$ -- the FIRST HARMONIC.

If instead the string's exact centre is held motionless (touched lightly) while it is plucked at a point midway along one of the resulting halves, TWO equal loops form, with an additional node exactly at the centre -- a FULL wavelength fills the string, $\lambda_1=l$, giving $n_1=\dfrac{v}{l}=\dfrac{1}{l}\sqrt{\dfrac{T}{m}}=2n$ -- the SECOND HARMONIC, or FIRST OVERTONE, exactly twice the fundamental frequency. …