A wave is propagating on a long stretched string along its length taken as the positive x-axis. The wave equation is given as
`y=y_0e^(-(t/T-x/lamda)^(2))`
where `y_0=4mm, T=1.0s and lamda=4cm`.(a) find the velocity of wave. (b) find the function `f(t)` giving the displacement of particle at x=0. (c) find the function g(x) giving the shape of the string at t=0.(d) plot the shape g(x) of the string at t=0 (e) Plot of the shape of the string at t=5s.

Explanation

(a). The wave equation is
`y=Ae^(-(t/T-x/lambda)^(2))`
This may be expressed as
`y=Ae^(-1/lambda^(2)((x-lambda)/T^(1))^(2)`
This is the form `f(x-vt)`
Therefore,the velocity of wave `v=lambda//T=8.0//1.0=8 cm//s`
(b). substituting `t=0` in given Eq. (i),we get
`f(t)=Ae^(-(t//T)^(2)`
(c ) substituting `t=0` in given Eq.(i), we get
`g(x)=aE(-(X//LAMBDA)^(2))`
(d). At `t=0`, the displacement is maximum at the originwhile at `t=5 s`, the displacement is maximum at distance `x=vt=8xx5=40 cm`
The function `g(x)` at `t=0` and `5 s` are plotted in `s-5.2` (a) and (b)