Fundamental Frequency and Overtones

Clamping a string rigidly at both its ends, at $x=0$ and $x=L$, and setting it vibrating (as when a guitar string is plucked) can only produce standing waves whose displacement is forced to vanish at both boundaries, $y(x=0,t)=0$ and $y(x=L,t)=0$. Since consecutive nodes of any stationary wave pattern are spaced $\lambda_n/2$ apart, fitting this pattern exactly between the two fixed boundaries requires $n(\lambda_n/2)=L$ for some positive integer $n$, giving the set of allowed (quantised) wavelengths $\lambda_n=2L/n$ -- so not every wavelength can form a valid standing wave on this string; only this discrete set is permitted by the boundary conditions. The corresponding natural (allowed) frequencies follow directly from $f_n=v/\lambda_n=nv/(2L)$. The lowest of these, corresponding to $n=1$, is called the fundamental frequency, $f_1=v/(2L)$; because every higher allowed frequency turns out to be an exact integer multiple of this fundamental, $f_n=nf_1$, the full sequence of natural frequencies forms a complete harmonic series, $f_1:f_2:f_3:\ldots=1:2:3:\ldots$. The second natural frequency, $f_2 …