Given the equation for a wave on a string
`y=0.03 sin(3x-2t)`
where y and x are in meters and t is in seconds.
(a). At t=0, what are the values of the displacement at x=0,0.1 m,0.2 m, and 0.3m ?
(b). At x=0.1 m what are the values of the displacement at t=0,0.1 s, and 0.2 s?
(c ) what is the equation for the velocity of oscillation of the particles of the string?
(d). what is the maximum velocity of oscillation?
(e). what is the velocity of propagation of the wave?

Explanation

(a). `At t=0, y=(0.03 sin 3 x)m`
for `x=0, y=0.03 sin(0)=0`
`x=0.1 m y=0.03sin(0.3 rad)`
`0.03 sin((0.3xx180^@)/pi)=0.03xxsin(17.2^@)`
`=0.03xx0.2957=8.87xx10^(-3) m`
similarly for
`x=0.2 m, y=0.03 sin(0.6 rad)=1.69xx10^(-2)` and for `x=0.3 m, y=0.03sin(0.9 rad)=2.35xx10^(-2) m`
(b). At `x=0.1 m, y=0.03 sin(0.03-2t)`
At `t=0, y=0.03 sin (0.3 rad)=8.87xx10^(-3) m`
`t=0.1 s, y=0.03sin(0.3-0.2)=0.03sin(0.1 rad)`
`=2.99xx10^(-3) m`
`t=0.2 s, y=0.03sin(0.3-0.4)=-2.99xx10^(-3) m`
(c ). Velocity of particle,
`(dy)/(dt)=(d)/(dt)[0.03sin(3x-2t)]=-0.06 cos(3x-2t)`
`=-6xx10^(-2) cos(3x-2t)m//s`
(d). Maximum velocity or the velocity amplitude is `6xx10^(2) m//s`
(e). comparing `y=0.03sin(3x-2t)` with the wave equation `y Asin kx-omegat` , we have
`k=2pi//lambda=3`,and `omega 2pif 2`
Therefore, `v=f lambda= (omega)/(k)=(2)/(3)=0.667 m//s`