Obtain the equation of frequency of oscillations in string tied at both ends.
Obtain the equation of frequency of oscillations in string tied at both ends.
Let a string of length $\mathrm{L}$ is tied at two ends and it has tension.
Its one end is tied at $x=0$ and another at $x=\mathrm{L}$.
For node, displacement $\sin k \mathrm{~L}=0$ at any point. (Here $x=\mathrm{L}$ )
$\therefore k \mathrm{~L}=n \pi \quad($ where $n=1,2,3, \ldots, n)$
$\therefore \frac{2 \pi \mathrm{L}}{\lambda}=n \pi$
$\therefore \mathrm{L}=\frac{n \lambda}{2} \quad \ldots$ (1) $\quad$ where $n=1,2,3, \ldots$
$\therefore \lambda=\frac{2 \mathrm{~L}}{n} \quad \ldots$ (2) $\quad n=1,2,3, \ldots$
and $v=\lambda v$ where $v$ is speed of wave and $v$ is frequency.
$\therefore \frac{v}{v}=\lambda=\frac{2 \mathrm{~L}}{n}$
$\therefore \quad \mathrm{v}=\frac{n v}{2 \mathrm{~L}} \quad \ldots$ (3) $\quad n=1,2,3, \ldots$
Equation $(2)$ is for wavelength of stationary wave and equation (3) is for its natural frequency.
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