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Question
A string of length L fixed at both ends vibrates in its fundamental mode at a frequency ν and a maximum amplitude A. (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-direction. Write the equation describing the standing wave.
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Solution
Given:
Length of the string = L
Velocity of wave is given as:
\[V = \sqrt{\frac{T}{m}}\]
(a) \[\text{Wavelength}, \lambda = \frac{\text{Velocity}}{\text{Frequency}}\]
\[\Rightarrow \lambda = \frac{\sqrt{\frac{T}{m}}}{\frac{1}{2L}\sqrt{\frac{T}{m}}} = 2L\]
\[\text{ Wave number,} K = \frac{2\pi}{\lambda} = \frac{2\pi}{2L} = \frac{\pi}{L}\]
(b) Equation of the stationary wave:
We have `y = Asin (2pix)/lambda * cos(omegat + phi)`
Given, At t = 0
`x = L/2`
y = 0
∴` 0 = Asin (2piL/2)/(2L)cos(0 + phi)`
or cos Φ = 0
or Φ = π/2
Now,
`y = Asin (2pix)/(2L)cos(2 pi f t + pi/2)`
= `Asin(pix)/Lsin(2 pi f t)`
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