Tension of a string, based on Hz, string-weight and string-length

Question

I’m not a physics student. Instead I have an electric bass guitar :)

• I know my strings vibrating-length: 0.800 Meter
• I know the total weight of this length: 0.016575 Kilo
• I know my strings resonant frequency: 55.0 Hz
• I know the wave-velocity: 88.0 Meter per Second

From that, how do I calculate the tension (in Kilo or Newton)?

Answer 1

The frequency of a standing wave on a guitar string is given by

$$f = \frac{v}{2L}$$

where $v$ is the velocity, and $L$ is the length of the string. It can be shown by using the wave equation (which I'll skip, as it is a more complex derivation) that the velocity of a wave on a string is related to the tension in the string and the mass per unit length, which can be written as:

$$v = \sqrt{\frac{T}{\rho}}$$

where $\rho = m/L$, is the mass per unit length. Combining these, we can write the frequency as:

$$f = \frac{\sqrt{\frac{T}{m/L}}}{2L}$$

Solving for tension, we have:

$$T=4mLf^2$$