A string of length L fixed at both ends vibrates in its fundament… - Vidyadip

Question

A string of length $L$ fixed at both ends vibrates in its fundamental mode at a frequency $v$ and a maximum amplitude $A.$

Find the wavelength and the wave number $k.$
Take the origin at one end of the string and the $X-$axis along the string. Take the $Y-$axis along the direction of the displacement. Take $t = 0$ at the instant when the middle point of the string passes through its mean position and is going towards the positive $y-$directian. Write the equation describing the standing wave:

✓

Answer

Fundamental frequency$\text{v}=\frac{1}{2\text{l}}\sqrt{\frac{\text{T}}{\text{m}}}$
$\Rightarrow\sqrt{\frac{\text{T}}{\text{m}}}=\text{v}2\text{l}$ $\Big[\sqrt{\frac{\text{T}}{\text{m}}}=$ velocity of wave$\Big]$

Wavelength, $\lambda=\frac{\text{velocity}}{\text{frequency}}=\frac{\text{v}2\text{l}}{\text{v}}=2\text{l}$

and wave number $\text{K}=\frac{2\pi}{\lambda}=\frac{2\pi}{2\text{l}}=\frac{\pi}{\text{l}}$

Therefore, equation of the stationary wave is

$\text{y}=\text{A}\cos\Big(\frac{2\pi\text{x}}{\lambda}\Big)\sin\Big(\frac{2\pi\text{Vt}}{\text{L}}\Big)$
$=\text{A}\cos\Big(\frac{2\pi\text{x}}{2}\Big)\sin\Big(\frac{2\pi\text{Vt}}{\text{2L}}\Big)$
$\text{v}=\frac{\text{V}}{2\text{L}}$ $\Big[$because $\text{v}=\big(\frac{\text{v}}{2\text{l}}\big)\Big]$

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