Write the definition of the speed of a progressive wave and derive $v = \frac{\omega}{k}$ and $v = \frac{\lambda}{T}$.

(N/A) Wave speed: The distance covered by a wave in unit time is called wave speed. Its $SI$ unit is $m/s$.
To find the speed of a progressive wave,we consider the motion of a point on the wave (e.g.,a crest). As shown in the figure,a point on the wave maintains its displacement relative to the wave pattern as it moves.
If the wave covers a displacement $\Delta x$ in time $\Delta t$,the wave speed is given by:
$v = \frac{\Delta x}{\Delta t}$
For a progressive wave represented by $y(x, t) = A \sin(kx - \omega t + \phi)$,the phase of the wave remains constant for a specific point on the wave pattern:
$kx - \omega t = \text{constant}$
As the wave moves,for a point to maintain the same phase at a later time $(t + \Delta t)$,its position must change to $(x + \Delta x)$:
$k(x + \Delta x) - \omega(t + \Delta t) = kx - \omega t$
$kx + k\Delta x - \omega t - \omega\Delta t = kx - \omega t$
$k\Delta x = \omega\Delta t$
$\frac{\Delta x}{\Delta t} = \frac{\omega}{k}$
Since $v = \frac{\Delta x}{\Delta t}$,we get $v = \frac{\omega}{k}$.
We know that $\omega = \frac{2\pi}{T}$ and $k = \frac{2\pi}{\lambda}$.
Substituting these values:
$v = \frac{(2\pi / T)}{(2\pi / \lambda)} = \frac{\lambda}{T}$.