Question

The wave function $y = \dfrac{2}{{{{\left( {x - 3t} \right)}^2} + 1}}$ is a solution for a linear wave equation. $x$ and $y$ are in cm. Find its speed.
A) $3{\text{cm}}{{\text{s}}^{ - 1}}$
B) ${\text{4cm}}{{\text{s}}^{ - 1}}$
C) ${\text{5cm}}{{\text{s}}^{ - 1}}$
D) ${\text{6cm}}{{\text{s}}^{ - 1}}$

Answer 1

The given wave function satisfies the wave equation and so the velocity of the wave will be the ratio of the angular velocity of the wave to its wavenumber. The angular velocity is the coefficient of the time $t$ whereas the wavenumber is the coefficient of the position $x$ .

Formula used:
-The velocity of a wave is given by, $v = \dfrac{\omega }{k}$ where $\omega$ is the angular velocity or angular frequency of the wave and $k$ is the wavenumber of the wave.

Complete step by step answer.
Step 1: Express the general representation of the wave function to match with the given wave function.
The general representation of the wave function is given by, $y = A\sin \left( {kx - \omega t + \phi } \right)$ where $A$ is the amplitude of the wave, $k$ is the wavenumber of the wave, $x$ is its position at the time $t$ , $\omega$ is the angular frequency of the wave and $\phi$ is the initial phase of the wave.
The given wave function is $y = \dfrac{2}{{{{\left( {x - 3t} \right)}^2} + 1}}$ .
Since the angular frequency $\omega$ is the coefficient of the time $t$ we have $\omega = 3{\text{rad}}{{\text{s}}^{ - 1}}$ .
Also, since the wavenumber is the coefficient of the position $x$ we have $x = 1{\text{cm}}$ .
Step 2: Express the relation for the velocity of the wave.
The velocity of the wave can be expressed as $v = \dfrac{\omega }{k}$ ------- (1)
Substituting for $\omega = 3{\text{rad}}{{\text{s}}^{ - 1}}$ and $x = 1{\text{cm}}$ in equation (1) we get, $v = \dfrac{3}{1} = 3{\text{cm}}{{\text{s}}^{ - 1}}$ .
So the speed of the wave is obtained as $v = 3{\text{cm}}{{\text{s}}^{ - 1}}$ .

So the correct option is A.

Note: Here the amplitude of the wave cannot be determined with certainty by comparing the given wave function and its general form. The given wavefunction does not contain a sine term. This however does not change the fact that the wavenumber is the coefficient of $x$ and the angular frequency is the coefficient of $t$ . Wavenumber refers to the number of waves present in a given distance. The wave function corresponds to the displacement of the wave.