Stationary Waves in a Stretched String Fixed at Both Ends: Fundamental Mode and Harmonics

Consider a string of length $L$, rigidly fixed at both ends -- the situation in every stringed instrument and in the standard sonometer experiment. Because both ends are physically clamped, both ends of the string must be nodes of the stationary wave at all times: no stationary wave can be sustained on the string unless this boundary condition is satisfied at both ends simultaneously. Since successive nodes of a stationary wave are always spaced $\lambda/2$ apart, this condition can only be met if the string's length $L$ is an exact whole-number multiple of $\lambda/2$: $L = \frac{n\lambda_n}{2}, \qquad n=1,2,3,\dots \;\Rightarrow\; \lambda_n = \frac{2L}{n}$ Using the wave-speed relation $v=f\lambda$, and noting that $v=\sqrt{T/\mu}$ is fixed once the string's tension and mass per unit length are fixed, the allowed (resonant) frequencies are $f_n = \frac{nv}{2L} = n\,f_1, \qquad f_1 = \frac{v}{2L}$ The lowest of these, $f_1$ (the case $n=1$, a single loop with one antinode at the centre and nodes only at the two fixed ends), is called the fundamental frequency or the first harmonic -- it is this frequency, generally, that is heard most strongly and that is usually taken to define the note the string is sounding. The mode $n=2$ (two loops, an extra node exactly at the midpoint), of frequency $f_2=2f_1$, is called the second harmonic or the first overtone; $n=3$ (three loops), of frequency $f_3=3f_1$, is the third harmonic or second overtone; and so on for every positive integer $n$. Because every positive integer value of $n$ gives a physically allowed mode, a stretched string fixed at both ends supports the complete harmonic series -- every integer multiple of the fundamental is present -- which is exactly why a sonometer or any stringed instrument can be made to sound a rich, well-defined musical note built from a fundamental plus a full ladder of overtones. Si …