A `100 Hz` sinusoidal wave is travelling in the posotive x-direction along a string with a linear mass density of `3.5 xx10^(-3) kg//m` and a tension of `35 N`. At time `t= 0`, the point `x=0` , has maximum displacement in the positive y-direction. Next when this point has zero displacement, the slope of the string is `pi//20`. Which of the following expression represent (s) the displacement of string as a function of `x`(in metre) and `t` (in second)

Explanation

The speed of wave in string is
`v=sqrt(T)/(mu)=sqrt((35)/(3.5xx10^(-3))=100 m//s`
`lambda=(v)/(f)=(100)/(100)=1 m`
The shape of the string at t=0 s is as shown in figure

Let us assume that
`y=A sin (omegat-kx+theta)`
putting `t=0, x=0 and y=+A`
gives `theta=pi//2`
also `omega=2pif=2pixx100=200pi`
`k+(2pi)/(lambda)=(2pi)/(1)=2pi`
Hence `y=A sin(200pit-2pix+pi/2)`
it is given that
`(dely)/(delx)=(pi)/(20) at t=(T)/(4)=(1)/(400) [:'T=(1)/(f)=(1)/(100)]`
`(dely)/(delx)]_(t=(1)(400))=A2pi cros(200t-2pix+(pi)/(2))=-2Api=(pi)/(20)`
`A=(1)/(40)=0.025`
(i). Amplitude A=0.25 m`
(ii) wave speed =100 m//s`
(iii) wave equation is
`y=0.025sin(200pit-2pix+(pi)/(2))`