What is the wave speed on a stretched string in terms of the tension F_T and linear density mu?

• A. v = sqrt(mu / F_T)
• B. v = sqrt(F_T / mu)
• C. v = F_T * mu
• D. v = F_T / mu

Answer: (B)

The speed of a transverse wave on a stretched string depends on the tension pulling the string tight and the inertia of the string itself. For a small segment, increasing the tension makes the restoring force stronger and wave propagation faster, while increasing the mass per length μ makes the segment harder to accelerate, slowing the wave. From the wave equation for small transverse displacements, the speed satisfies v^2 = F_T / μ, so the wave speed is v = sqrt(F_T / μ). This also matches units: tension has units of N and μ is kg/m, so N/(kg/m) = m^2/s^2, whose square root gives m/s. Higher tension → faster waves; higher linear density → slower waves.