One end of a taut string of length `3m` along the x-axis is fixed at `x = 0`. The speed of the waves in the string is `100ms^(-1)`. The other end of the string is vibrating in the y-direction so that stationary waves are set up in the string. The possible wavelength`(s)` of these sationary waves is (are)

Options

A. `y(t) = A sin '(pix)/(6)cos'(50pit)/(3)`
B. `y(t) = A sin'(pix)/(3)cos'(100pit)/(3)`
C. `y(t) = A sin '(5pix)/(6)cos'(50pit)/(3)`
D. `y(t) = A sin '(5pix)/(6)cos250pit`

Answer

A,C,D

Explanation

`V = 100 m//s`
Possible modes of vibration
`l = (2n + 1) (lambda)/(4)`
`lambda = (12)/((2n + 1))m`
`k = (2pi)/(lambda) = (2pi)/(12//(2n + 1)) = ((2n + 1)pi)/(6)`
`omega = vk = 100(2n + 1) (pi)/(6) = ((2n + 1)50pi)/(3)`
If `n = 0 , k = (pi)/(6) , omega = (50 pi)/(3)`
`n = 1 , k = (5pi)/(6) , omega = (250 pi)/(3)`
`n = 7 , k = (5pi)/(2) , omega = 250 pi`