The equation of a standing wave in a string f | vedclass

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Question

(D) The equation of the standing wave is $y = 2A \sin kx \cos \omega t$. The amplitude of a particle at position $x$ is given by $A_p = |2A \sin kx|$.

Vedclass · Accepted Answer

$\sqrt{2}A, \frac{\omega}{2\pi}$

Vedclass · Answer

(D) The equation of the standing wave is $y = 2A \sin kx \cos \omega t$. The amplitude of a particle at position $x$ is given by $A_p = |2A \sin kx|$. $A$ node occurs where $\sin kx = 0$ (e.g.,$x = 0$),and an antinode occurs where $\sin kx = 1$ (e.g.,$x = \frac{\lambda}{4}$). The midpoint between a node $(x = 0)$ and an antinode $(x = \frac{\lambda}{4})$ is $x = \frac{\lambda}{8}$. Substituting $x = \frac{\lambda}{8}$ and $k = \frac{2\pi}{\lambda}$ into the amplitude expression: $A_p = 2A \sin \left( \frac{2\pi}{\lambda} \cdot \frac{\lambda}{8} \right) = 2A \sin \left( \frac{\pi}{4} \right) = 2A \cdot \frac{1}{\sqrt{2}} = \sqrt{2}A$. The frequency of vibration for all particles in a standing wave is the same as the frequency of the wave,which is $f = \frac{\omega}{2\pi}$. Thus,the amplitude is $\sqrt{2}A$ and the frequency is $\frac{\omega}{2\pi}$.

The equation of a standing wave in a string fixed at both ends is given as $y = 2A \sin kx \cos \omega t$. The amplitude and frequency of a particle vibrating at the midpoint between an antinode and a node are respectively:

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