Vibrating String Frequency

This formula gives the fundamental frequency, or lowest resonant pitch, of an ideal stretched string fixed at both ends. It comes from the fact that waves on the string travel at speed \(v=\sqrt{T/\mu}\), and the lowest standing wave that fits on the string has wavelength \(\lambda=2L\), so \(f=v/\lambda\). Increasing tension makes the string wave faster and raises the pitch, while increasing length or mass per unit length lowers the pitch.

Use this when analyzing the pitch of guitar, violin, piano, or other stretched strings, especially when you know the string length, tension, and linear density. It is also useful in physics problems about standing waves, harmonics, and how changing a string's properties changes its tone.

• Using the total string mass instead of mass per unit length μ
• Forgetting that length L is the vibrating length between fixed points, not the full instrument length
• Confusing the fundamental frequency with higher harmonics, which are integer multiples of this value