Standing Waves In Strings

Question

A string fixed at both ends and one metre long vibrates in its fundamental mode, so what is the wavelength of the standing wave produced on it?

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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: $L = \dfrac{\lambda}{2}$. Rearranging, $\lambda = 2L = 2 	imes 1 = 2$ 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 $n = 1, 2, 3, \ldots$, 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.

Standing Waves In Strings

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