Derive displacement relation for a progressive wave.

Explanation

Consider a wave-travelling along positives-axis with velocity `barv`. The particle is at the ori (x = 0) at time t vibrate according to the equation.

y(0,t) =`A sin omegat`……………(i)
where A = amplitude of the particle. Consider a particle at P at distance x from O. So P will s vibrating after a time t`(x/v)`. The particle at P will vibrate at time `(t-x/v)` . Displacement of particle at P at time .t. is given by:
`y(x,t) =A sin omega (t-x/v)`.........(ii)
`y(x,t) = A sin (omegat - omega/v x)`
But `omega/v =(2piv)/v =(2pi)/lambda = k [therefore u/v = k]`
where `k =(2pi)/lambda` is called angular wave number or propagation constant.
`therefore y(x,t) =A sin (omegat -kx)`............(iii)
Equation (iii) is the equation of a progressive wave.