Question

I have a taut string and I flick one end to make a wave travel through it. If I increase the tension in the string by 20% and flick it again, what changes and by how much?

Answer 1

Answer: The speed of the wave propagation and the frequency of the fundamental harmonic would increase by 9.5%.

Explanation: The speed of wave propagation through a string is given by

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

where $\rho[/\tex] is the linear density of the material the string is made of and [tex]T$ is the tension. The fundamental harmonic of the string is the one that has its wavelength equal to $2L$, where $L$ is the length of the string so we have for its frequency

$f=\frac{v}{\lambda}=\frac{v}{2L}=\frac{1}{2L}\sqrt{\frac{T}{\rho}}.$

Now if the tension increases by 20% we would have $T'=1.2T$ and also

$v'=\sqrt{\frac{T'}{\rho}}=\sqrt{\frac{1.2T}{\rho}}=\sqrt{1.2}\sqrt{\frac{T}{\rho}}=1.095v$

and, similarly

$f'=\frac{1}{2L}\sqrt{\frac{T'}{\rho}}=1.095f.$

This means that the speed of the wave propagation and the frequency of the fundamental harmonic would increase by 9.5%.