Write difinition of wave speed and derive `v = (omega )/(k).`

Explanation

Wave speed: "The distance covered by wave in unit time the call wave speed".
It IS unit is `m/s.`
To find the speed of the progressive wave speed of particle on crest is required to find. For this, suppose the positions of the crest on wave are shoen in dark point in figure.

Each pont maintains its displacement on wave
The displacement of points on string is changed, ut displacement of points on wave pettern is not changed
If wave completes `Delta x` displacemenet in `Delta t` time, then
`therefore v = (Delta x )/(Delta t)`
Here, `kx - omega t=` constnat
Thus, the position of point must be changed with time such that phase remains constant.
Thus, `kx - omega t =k ( x + Delta x) - omega (t + Delta t)`
`kx - omega t = kx + k Delta x - omega t - omega Delta t `
`therefore k Delta x = omega Delta t =0`
`If Delta x to 0 and Delta t to 0` then,
`k dx = omega dt`
`therefore (dx)/(dt) = (omega)/(k)`
`therefore v = (omega)/(k) [ because (dx)/(dt)=` wave speed]
Now `omega = (2pi)/(T) and k = (2pi)/(lamda),`
`v = ((2pi)/(T))/((2pi)/(lamda))`
`therefore v = (lamda)/(T)`
but `1/F` frequency v
`therefore c = v lamda`
Mechanical speed is influenced by the inertial and elastic properties of medium.
In the same of different speeds can be passed.