Give the definition,$SI$ unit,and dimensional formula of the time period,angular frequency,and frequency of a wave.

(N/A) Time period: The time taken by any string element to complete one full oscillation is called the Time period $(T)$.
In the equation of displacement of a wave,$y(x, t) = a \sin (kx - \omega t + \phi)$. If $\phi = 0$,then observing the motion of an element at $x = 0$,we get $y(0, t) = a \sin(-\omega t) = -a \sin(\omega t)$. This represents simple harmonic motion $(SHM)$ with amplitude $a$ and time period $T$. Since the sine function repeats at $2\pi$ intervals,$\omega T = 2\pi$,which gives $\omega = \frac{2\pi}{T}$.
Angular frequency: The angular frequency of oscillations of particles of the medium in a wave is called the angular frequency of the wave. Its symbol is $\omega$,its $SI$ unit is $\text{rad } s^{-1}$,and its dimensional formula is $[M^0 L^0 T^{-1}]$.
Frequency: The number of oscillations performed by a particle of the medium in one second is known as the frequency of oscillation. Its symbol is $\nu$ or $f$. Since $f = \frac{1}{T}$,the $SI$ unit of frequency is $s^{-1}$ or $\text{Hz}$ (Hertz),and its dimensional formula is $[M^0 L^0 T^{-1}]$.