Write the relation between angular frequency, angular wave number and wave velocity?

The relation between angular frequency, angular wave number, and wave velocity can be derived from the fundamental properties of a wave. Let us break it down step by step:

Definitions:

Angular frequency (ω): It represents how fast a wave oscillates in radians per second.
Angular wave number (k): It describes the spatial variation of the wave and is defined as the number of radians per unit distance (radians per meter).
Wave velocity (v): It is the speed at which the wave propagates through a medium.
Basic wave relation:

For a wave traveling in one dimension, the displacement of the wave can be written as: y(x, t) = A * sin(kx - ωt + φ) Here:
k is the angular wave number.
ω is the angular frequency.
x is the position.
t is time.
φ is the phase constant.
Relation between ω, k, and v:

The wave velocity (v) is related to the angular frequency (ω) and angular wave number (k) as: v = ω / k
Explanation:

The wave velocity can be interpreted as the ratio of the angular frequency (how fast the wave oscillates in time) to the angular wave number (how fast the wave varies in space).
This formula highlights that the velocity of the wave depends on both its temporal oscillation and spatial variation.
Units check:

ω has units of radians per second (rad/s).
k has units of radians per meter (rad/m).
v has units of meters per second (m/s), which is consistent when ω is divided by k.
Final Relation: The relation between angular frequency (ω), angular wave number (k), and wave velocity (v) is:

ω = k * v
or equivalently,
v = ω / k

This is the fundamental equation connecting these three quantities in wave motion.