Question

The ratio of angular frequency and angular wave number is
(A) particle velocity
(B) wave velocity
(C) energy
(D) wavelength per unit oscillation

Answer 1

Use the formula of the angular frequency and the formula of the angular wave number. Divide both the formula to find the ratio of the both by simplifying the obtained relation. Compare the obtained relation with the options given, to find the answer.
Useful formula:
(1) The formula of the angular frequency is given by
$\omega = 2\pi f$
Where $\omega$ is the angular frequency and $f$ is the frequency of the wave.
(2) The formula of the wave number is given by
$k = \dfrac{{2\pi }}{\lambda }$
Where $k$ is the wavenumber of the wave and $\lambda$ is the wavelength.
(3) The formula of the wave velocity is given by
$v = \lambda f$
Where $v$ is the velocity of the wave, $\lambda$is the wavelength and the $f$ is the frequency.

Complete step by step solution:
The angular frequency is defined as the ratio of the angular displacement and the time taken. Wavelength is the length of the one wave. Its reciprocal value provides the value for the wavenumber. It is also known as the propagation number or the angular wavenumber.
Let us consider the formula of the angular frequency of the wave.
$\omega = 2\pi f$ ………………………… (1)
Let us consider the formula of the wave number.
$k = \dfrac{{2\pi }}{\lambda }$ …………………………… (2)
Let us divide the equation (1) and (2) to find the ratio of the angular frequency to the angular wave number.
$\dfrac{\omega }{k} = \dfrac{{2\pi f}}{{\dfrac{{2\pi }}{\lambda }}} = \lambda f$
$\dfrac{\omega }{k} = v$
Hence the ratio of the angular frequency and the angular wave number is the wave velocity.

Thus the option (B) is correct.

Note: In the above solution, don't confuse the angular frequency with that of the regular frequency. This is because the frequency is the reciprocal of the time period of the wave where the angular frequency is the ratio of the angular displacement and the time taken.