Question

Let velocity of a sound wave be ‘v’ and $'\omega '$ be angular velocity. The propagation constant of the wave is
A. $\sqrt {\dfrac{\omega }{v}} $
B. $\sqrt {\dfrac{v}{\omega }} $
C. $\dfrac{\omega }{v}$
D. $\dfrac{v}{\omega }$

Answer 1

Hint: Time period of anything is the amount of time required for it to complete one oscillation. Usually when we have angular velocity we can get frequency from it and when we get frequency the inverse of that frequency gives us the time period. We can also get wavelength from that frequency and then propagation constant.

Formula used:
$y = A\sin \left( {kx - \omega t} \right)$

Complete step by step solution:
There are two kinds of waves normally. One will be a transverse wave and the other will be a longitudinal wave. In case of transverse waves the direction of propagation of waves is perpendicular to the direction of vibration of the particles of the medium. Transverse wave has crests and troughs.
In case of the longitudinal wave particles of the medium of propagation vibrates in the direction of propagation of the wave. This longitudinal wave has compressions and rarefactions. In case of longitudinal waves particles vibrate in to and fro motion. Transverse waves can propagate without medium too whereas longitudinal waves require medium to propagate. Example for transverse wave is light whereas example for the longitudinal wave is sound.
Generally the wave equation can be represented as
$y = A\sin \left( {kx - \omega t} \right)$
Where A is the amplitude of the wave and ‘k’ is the angular wave number and ‘$\omega $’ is the angular velocity.
Angular wave number ‘k’ can also be called a propagation constant.
The velocity of sound(v) is given as
$v = \dfrac{\omega }{k}$
Hence from the above given velocity term we can get propagation constant.
$ \Rightarrow v = \dfrac{\omega }{k}$
$\therefore k = \dfrac{\omega }{v}$

So, the correct answer is “Option C”.

Note: There is another way to get the propagation constant. Since we have got the angular velocity we will find out the frequency from it and then we will find out the wavelength from that frequency as we have velocity too. After finding wavelength angular wave number is nothing but 2 pi upon the wavelength.