Question

a guitar string resonates on its fundamental at 240Hz. where must you put your hand on the string so that the fundamental becomes 520Hz?

Answer 1

The frequency of an fundamental mode one can calculate by the expression

(1) $f=c_t/2L$, where $c_t$ the velocity of transverse oscillation of string, and $L$ is its length.

This formula follows from a simple connection of the wavelength with the frequency and speed of wave propagation $f=\frac {c}{\lambda}$ , taking into account that the first resonance is observed at a half-wavelength $L=\frac {\lambda}{2}$ as shown in the figure.

The string on one side is pinched by fixing peg and saddle and cannot move in the transverse direction, and on the other end the guitarist presses it to frets. Assuming that the shear wave velocity is unchanged ($c_t=const$ ) from (1), we obtain the following equation for new length of the string $2 f_1\cdot L_1=2f_2\cdot L_2$ , or $L_2=\frac{f_1}{f_2}\cdot L_1=\frac{240}{520}\cdot L_1=(6/13)L_1$

Answer: To increase the frequency of an fundamental mode guitar string from 240 Hz to 520 Hz one must put your hand on the distance $6/13$ from the saddle.