Question

The equation of a transverse wave travelling along with positive X axis with an amplitude $0.2m$, velocity $360m{s^{ - 1}}$ and wavelength $60m$ can be written as:

A.)$y = 0.2\sin \pi \left[ {6t + \dfrac{x}{{60}}} \right]$
B.)$y = 0.2\sin \pi \left[ {6t - \dfrac{x}{{60}}} \right]$
C.)$y = 0.2\sin 2\pi \left[ {6t - \dfrac{x}{{60}}} \right]$
D.)$y = 0.2\sin 2\pi \left[ {6t + \dfrac{x}{{60}}} \right]$

Answer 1

Hint – Start the solution by describing what a transverse wave is, with a well- labelled diagram. Then describe the general equation of displacement in y axis, then substitute the values given in the problem. Use this method to reach the solution.

Step By Step Answer:
Before moving on the mathematical calculations, let’s start by discussing what a transverse wave is.

Transverse wave – It is a wave that has oscillations in a direction perpendicular to the direction of propagation of the transverse wave. You can understand this quite easily, just imagine a rope tied to a wall from one end and free from the other end. When you shake the rope up and down from the free end, you will notice that the waves are formed and these waves always remain perpendicular to the direction of propagation.

Amplitude ($A$) - It is the maximum wavelength displacement of a wave measured from its equilibrium position.

Crest and Trough – Crest and trough are the highest and lowest displacement of a wave from its equilibrium position.

Wavelength - The distance between two consequent crest and trough.

We know that that the equation for a transverse wave, traveling in X axis is

$y = A\sin 2\pi \left[ {vt - \dfrac{x}{\lambda }} \right]$(Equation 1)
Here,

$A = $Amplitude,
$v = $Velocity,
$\lambda = $Wavelength,
$t = $Time

Given in the question,

$\lambda = 60m$,
$\nu = 360m{s^{ - 1}}$,
$A = 0.2m$.

We also know that

$\nu = v\lambda $
$ \Rightarrow v = \dfrac{\nu }{\lambda }$
$ \Rightarrow v = \dfrac{{360}}{{60}}$
$ \Rightarrow v = 6{s^{ - 1}}$

Substituting the values of $A$ , $v$ and $\lambda $ in equation 1

$ \Rightarrow y = 0.2\sin 2\pi \left[ {6t - \dfrac{x}{{60}}} \right]$

Hence, the correct option in C.

Note – In such types of problems, it is usually very important to know what values we are given. As here we are given the value of frequency of the transverse wave which has the symbol $\nu $. Here we have to use this to find the value of velocity which has the symbol $v$. These two symbols look very similar, but should never be interchangeably used, as it makes the solution incorrect.