Question

How does the fundamental frequency of a vibrating string depend on the radius of the cross-section of the string and the mass density material of the string?  

Answer 1

Imagine a string that is stretched L distances apart between two rigid supports. Let T be the string's tension, r be its cross-sectional radius and ρ be the material's mass density. Then, the mass of the string M = (πr²L) ρ, so that its linear density, i.e., mass per unit length, m = M/L = πr²ρ.

When T and L are constant, the fundamental frequency (n) of a vibrating string is inversely proportional to the square root of its linear density, according to the law of mass of the string.

`n ∝ 1/sqrtm`

∴ `n ∝ 1/(sqrt(πr^2ρ))`

∴ `n ∝ 1/r`   when L, T and ρ are constant, and

`n ∝ 1/sqrtρ`   when L, T and r are constant.