Question

The displacement of a progressive wave is represented by ${\text{y = A sin(}}\omega {\text{t - kx)}}$ where $x$ is the distance and t is time. The dimensions of $\dfrac{\omega }{k}$ are same as those of the
A. velocity
B. wave number
C. wavelength
D. frequency

Answer 1

Waves help in the transfer of energy without transporting the matter. Waves are disturbances that transport energy from one point to another point through a medium without transporting the matter. Waves need oscillating or vibrating sources to transfer energy from one location to another location in a medium.

Complete step by step answer:
Imagine that a transverse wave is travelling in positive x-direction. Remember that harmonic waves are sinusoidal waves. The displacement of the particle in a medium is given by $y$. It is a function of $x$ and $t$. It is represented as:
${\text{y}}\left( {{\text{x,t}}} \right) = {\text{Asin}}\left( {{\text{kx - }}\omega {\text{t + }}\phi } \right)$
Where ${\text{k = }}\dfrac{{2\pi }}{\lambda }$
$\omega = \dfrac{{2\pi }}{{\text{T}}}$
Here, $k$ is called the wavenumber and the equation represents a travelling wave of some shape.
$\left( {\omega {\text{t - kx}}} \right)$ represents an angle. Angles have no dimensions.
$[\omega ] = \dfrac{1}{{[{\text{t]}}}} = [{{\text{T}}^{ - 1}}]$
$\Rightarrow [{\text{k}}] = \dfrac{1}{{[{\text{x}}]}} = [{{\text{L}}^{ - 1}}]$
$\Rightarrow \left[ {\dfrac{\omega }{{\text{k}}}} \right] = \dfrac{{[{{\text{T}}^{ - 1}}]}}{{[{{\text{L}}^{ - 1}}{\text{]}}}} = [{\text{L}}{{\text{T}}^{ - 1}}] \\\ \therefore \left[ {\dfrac{\omega }{{\text{k}}}} \right] = {\text{velocity}}$

Therefore, the correct answer is option A.

Note: Waves have some common properties like wavelength, frequency, amplitude and speed. How far a wave can move in a particular medium is the amplitude of the wave. The frequency describes how often the wave occurs and the speed tells us how quickly a wave can move. The distance between any two consecutive wave crests or two wave troughs is said to be wave length of the wave. If the waves are closer then it means frequency is higher and more energy is carried by the waves. If waves interact with other waves, the interaction is called wave interference. Wave interference may happen when two waves travelling in two opposite directions meet.