Question

A middle C string on a piano has a fundamental frequency of 262 Hz and the A note has a 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 tensions in the two strings

Answer 1

Because $f_1$ = 262 Hz for the C string we can use Equation to find the frequencies $f_2$ and $f_3$
$f_2=2f_1=524Hz$
$f_3=3f_1=786Hz$
Using Equation for the two strings vibrating at their fundamental frequencies gives
$f_{1A}=\frac{1}{2L}\sqrt{\frac{T_A}{\mu }}\Rightarrow f_{1C}=\frac{1}{2L}\sqrt{\frac{T_C}{\mu }}$
$\therefore$ $\frac{f_{1A}}{f_{1C}}=\sqrt{\frac{T_A}{T_C}}\Rightarrow \frac{T_A}{T_C}={\left(\frac{f_{1A}}{f_{1C}}\right)}^2={\left(\frac{440Hz}{262Hz}\right)}^2=2.82$