A middle `C` string on a piano has a fundamental frequency of `262 Hz`, and the A note has fundamental frequency of `440 Hz`. (a) Calculate the frequencies of the next two harmonics of the `C` string. (b) If the strings for the `A` and `C` notes are assumed to have the same mass per unit length and the same length, determine the ratio of tension in the two strings.

Explanation

(a) Because `f_(1) = 262 Hz` for the `C` string, we can use Equation to find the frequencies `f_(2)` and `f_(3)`,
`f_(2) = 2f_(1) = 524 Hz`
`f_(3) = 3f_(1) = 786 Hz`
Using Equation for the two strings vibrating at their fundamental frequencies gives
`f_(1//A) = (1)/(2L) sqrt((T_(A))/(mu)) rArr f_(1C) = (1)/(2L)sqrt((T_(C))/(mu))`
= `:. (f_(1A))/(f_(1C)) = sqrt((T_(A))/(T_(C))) rArr (T_(A))/(T_(C)) = ((f_(1A))/(f_(1C)))^(2) = ((440Hz)/(262Hz))^(2) = 2.82`.