A string fixed at both ends has length L = 1.0 m; speed of waves on the string is 50 m/s. What is the fundamental frequency?

• A. 50 Hz
• B. 12.5 Hz
• C. 75 Hz
• D. 25 Hz

Answer: (D)

For a string fixed at both ends, standing waves occur only at specific wavelengths that fit an integer number of half-wavelengths along the length. The longest wavelength that fits is twice the length, lambda = 2L, which corresponds to the fundamental mode. Since v = f lambda, the fundamental frequency is f1 = v / (2L).

Plugging in v = 50 m/s and L = 1.0 m gives f1 = 50 / (2 × 1) = 25 Hz. This is the lowest frequency that can form a stable standing wave on the string. Higher modes occur at integer multiples of this fundamental: the second harmonic is 50 Hz, the third is 75 Hz, and so on. The option 12.5 Hz doesn’t correspond to a standing-wave mode for this setup, so 25 Hz is the correct fundamental frequency.