Concept:In a stretched string fixed at both ends, stationary (standing) waves are established only when the length of the string allows for nodes at both boundaries.
Formula:$$ L = n \frac{\lambda}{2} $$
Solution:- The simplest standing wave (fundamental harmonic) consists of a single loop, which is exactly half a wavelength (\( \lambda/2 \)).
- Therefore, the length of the string \( L \) must always be an integral multiple (\( n = 1, 2, 3, ... \)) of a half wavelength to form stable standing wave patterns.
Why other options are incorrect:Options A, C, and D are incorrect because standing waves also form at lengths like 1.5\( \lambda \) or 2.5\( \lambda \), which are integral multiples of a half wavelength, but not integral multiples of full, double, or triple wavelengths.
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