A stretched string of length L = 2.0 m and wave speed v = 340 m/s supports a standing wave. What is the fundamental frequency?

A stretched string of length L = 2.0 m and wave speed v = 340 m/s supports a standing wave. What is the fundamental frequency?

• A. 42.5 Hz
• B. 340 Hz
• C. 85 Hz
• D. 170 Hz

Answer: (C)

For a string fixed at both ends, the fundamental mode has a half-wavelength spanning the length, so the fundamental wavelength is twice the length: lambda1 = 2L. With v = f lambda and lambda1 = 2L, the fundamental frequency is f1 = v / (2L). Plugging in L = 2.0 m and v = 340 m/s gives f1 = 340 / (2 × 2.0) = 340 / 4 = 85 Hz. This is the lowest frequency that fits the boundary conditions; higher harmonics occur at f = n v / (2L), giving 170 Hz, 255 Hz, etc. So 85 Hz is the fundamental frequency.

For a string fixed at both ends, the fundamental mode has a half-wavelength spanning the length, so the fundamental wavelength is twice the length: lambda1 = 2L. With v = f lambda and lambda1 = 2L, the fundamental frequency is f1 = v / (2L).

Plugging in L = 2.0 m and v = 340 m/s gives f1 = 340 / (2 × 2.0) = 340 / 4 = 85 Hz. This is the lowest frequency that fits the boundary conditions; higher harmonics occur at f = n v / (2L), giving 170 Hz, 255 Hz, etc. So 85 Hz is the fundamental frequency.