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A wave disturbance in a medium is descri...

A wave disturbance in a medium is described by `y(x, t) = 0.02 cos(50pit + (pi)/(2))cos(10pix)` where `x and y` are in meter and `t` is in second`

A

The wavelenghts of wave is `0.2 m`

B

An antinode occurs at `x=0.3 m`

C

The speed of the wave is `5.0 m//s`

D

A displacement node occurs at `x=0.4 m`

Text Solution

Verified by Experts

The correct Answer is:
D
`y (x , t) = 0.02 cos (50 pi t + pi/2) cos (10 pi x) `...(1)
This is an equation of a stationary wave, having terms in displacement (x) and time (t) .
(i) At the node amplitude = 0 ` :. Cos ( 10 pix ) = 0 `
For this ` :. 1- pi x = pi/2, (3pi)/2,` ...
` :. x = 1/20 i.e. x = 0.05 m, ......... 0.15` m .
(ii) At the antinode, amplitude is maximum
`:. cos (10 pi x) = pm 1. " For this " 10 pi x = 0, pi, 2pi, .....`
` x = 0, " or " 10 pi x = pi`
` :. x = 1/10 " or " x = 0.1 m, 0.2 m ... 0.3 `m
(iii) From the given equation `50 pi t = omegat = 2 pi n t`
` :. n = 25`
` :. v = n lambda = 25 xx 0.2 = 5 ` m/s
(iv) ` lambda = 2` ( Distance between two nodes or antinodes )
` = 2 xx 0.1 = 0.2` m.
Thus (d) is wrong . (a) , (b) and (c ) are correct.
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