Consider a wave propagating in the negative x-direction whose frequency is `100 Hz`. At `t = 5 s`,the displacement associated with the wave is given by
`y=0.5 cos (0.1 x)`
where `x` and `y` are measured in centimetres and `t` in seconds. Obtain the displacement (as a function of x) at `t = 10 s`. What is the wavelength and velocity associated with the wave?

Answer

A, B, C, D

Explanation

A wave travelling in negative x-direction can be represented as
`y = (x, t)= A cos (kx + omegat + Phi )`
At `t = 5 s`,
`y(x, t = 5) = A cos (kx - 5omega + Phi)`
Comparing this with the given equation,
We have,
`A =0.5 cm, k = 0.1 cm^(-1)`
and `5omega + Phi = 0`
Now, `lambda =(2pi)/(k)=(2pi)/(0.1) = (20pi) cm`
`omega = 2pif = (200pi) rad//s`
`:. v=(omega)/(k) = (200pi)/(0.1) = (2000pi) cm//s`
From Eq. `(i), Phi = -5omega`
At `t = 10 s`,
`y(x, t = 10) = 0.5 cos (0.1x + 10omega + 5omega)`
Substituting `omega =200pi`,
`y(x, t = 10) =0.5 cos(0.1x + 1000pi)`
`=0.5 cos (0.1x)`