Standing Waves In Strings

Standing waves on a string fixed at both ends, treated in NCERT Class 11, Chapter 15 (Waves), must have nodes at both fixed ends. In the fundamental mode there is a single antinode at the centre and one node at each end, so the string holds just half a wavelength: $L=\frac{\lambda }{2}$. Rearranging, $\lambda =2L=2\times 1=2$ m. This lowest-frequency mode is the fundamental or first harmonic. The option 1 m would correspond to the second harmonic, where the whole wavelength fits in the string length. The option 0.5 m corresponds to a higher harmonic with two full wavelengths in the length. The option 4 m wrongly uses $\lambda =4L$, which applies to a pipe closed at one end, not a string fixed at both ends. More generally, the allowed modes obey $L=n\lambda /2$ for $n=1,2,3,\dots$, so the possible wavelengths are ${\lambda }_n=2L/n$ and the fundamental with $n=1$ gives the largest of these. The corresponding frequencies $f_n=nv/(2L)$ form a full harmonic series of all integer multiples, which is what gives a plucked string its rich musical tone. A plausibility check: the fundamental always has the longest wavelength of all modes, and $2L=2$ m being larger than the string length is consistent with only half a wave fitting between the two fixed ends, so the magnitude is entirely reasonable.