What determines the frequency of a vibrating string?

• A. Length, mass, and tension.
• B. Length and color.
• C. Only tension.
• D. Ambient temperature.

Answer: (A)

The pitch of a vibrating string is set by how fast waves travel along it and how many half-wavelengths fit in its length. The wave speed on a string under tension is v = sqrt(T/μ), where T is the tension and μ is the mass per unit length. For the fundamental mode, the frequency is f = v/(2L) = (1/(2L)) sqrt(T/μ). So increasing tension makes the wave speed and therefore the frequency higher; increasing the mass per length slows the wave and lowers the frequency; and a longer string reduces the frequency because fewer half-wavelengths fit in its length. All three factors—length, mass per unit length, and tension—determine the frequency. Color has no effect, and only varying tension ignores two crucial determinants. Temperature can cause small changes by altering tension slightly, but it isn’t the primary factor in the standard model.