Physics - Sound Wave Question with Solution | TestHub

For string fixed at both the ends, resonant frequency are given by

$\mathrm{v}=\frac{\mathrm{n}\mathrm{v}}{2\mathrm{L}}$

Where symbols have their meaning. It is given that 315 Hz and 420 Hz are two consecutive resonant frequency, let these nth and (n+1)th harmonics.

$315=\frac{\mathrm{n}\mathrm{v}}{2\mathrm{L}}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\dots \left(\mathrm{i}\right)$

$420=\frac{\left(\mathrm{n}+1\right)\mathrm{v}}{2\mathrm{L}}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\dots \left(\mathrm{i}\mathrm{i}\right)$

⟹Eq.(i)÷Eq.(ii)

$⟹\frac{315}{450}=\frac{\mathrm{n}}{\mathrm{n}+1}⟹\mathrm{n}=3$

From Eq. (i), lowest resonant frequency

${\mathrm{v}}_0=\frac{\mathrm{v}}{2\mathrm{L}}=\frac{315}{3}=105\mathrm{ }\mathrm{H}\mathrm{z}$