Standing Waves On A String

A string fixed at both ends supports standing waves with nodes at each end, and the fundamental mode fits exactly half a wavelength into the length, so ${\lambda }_1=2L$. This is the longest possible wavelength the boundary conditions permit and therefore corresponds to the lowest frequency the wire can sound. The fundamental frequency is $f_1=v/{\lambda }_1=v/(2L)$. Substituting $v=180\text{ m/s}$ and $L=0.6\text{ m}$ gives $f_1=\frac{180}{2\times 0.6}=\frac{180}{1.2}=150\text{ Hz}$. The value $300\text{ Hz}$ is actually the second harmonic, using $\lambda =L$ instead of $2L$. The value $75\text{ Hz}$ wrongly uses $\lambda =4L$, which applies to a pipe closed at one end, not a string fixed at both ends. The value $108\text{ Hz}$ has no consistent harmonic basis. This follows the NCERT treatment of normal modes on a string with both ends fixed. As a plausibility check, the fundamental should be the smallest of the allowed harmonic series $f_n=n{f}_1$, and $150\text{ Hz}$ correctly serves as the base from which higher harmonics like 300 Hz and 450 Hz are built.