Question

A sinusoidal wave trsvelling in the positive direction on a stretched string has amplitude `2.0 cm`, wavelength `1.0 cm` and velocity `5.0 m//s`. At `x = 0` and ` t= 0` it is given that `y = 0` and `(dely)/(delt) lt 0`. Find the wave function `y (x, t)`.

Answer 1

Correct Answer - A::B::D
We start with a general equation for a rightward moving wave,
`y(x, t) = A sin (kx - omegat + Phi) `
The ampitude given is
`A = 2.0 cm = 0.02 m`
A wavelength is given as
`lambda = 1.0 m`
:. Angular wave number,
`k =(2pi)/( lambda) = (2pi) m^(-1)`
Angular frequency,
`omega = vk = (10pi) rad//s`
` :. y(x, t) = (0.02) sin [ 2pi (x - 5.0t) + Phi]`
We are given that for `x = 0`, `t = 0`,
`y =0`
and `(dely)/(delt) lt 0`
i.e. `0.02 sin Phi = 0` (as `y = 0`)
and `-0.2 pi cos Phi lt 0`
From these conditions, we conclude that
` Phi = 2npi` , where `n = 0,2,4,6`......
Therefore,
`y(x, t) = (0.02 m) sin [(2pi m^(-1)) x - (10pi rad s^(-1)) t] m`