Question

A travelling harmonic wave is represented by the equation $y\left( {x,t} \right) = {10^{ - 3}}\sin \left( {50t + 2x} \right)$, where $x$ and $y$ are in meter and $t$ is in seconds. Which of the following is a correct statement about the wave? The wave is propagating along the?
(A) negative $x$ axis with speed $25\,m{s^{ - 1}}$
(B) the wave is propagating along the positive $x$ axis with speed $25\,m{s^{ - 1}}$
(C) the wave is propagating along the positive $x$ axis with speed $100\,m{s^{ - 1}}$
(D) the wave is propagating along the negative $x$ axis with speed $25\,m{s^{ - 1}}$

Answer 1

The solution can be determined by comparing the general wave equation with the given wave equation, then by using the velocity of the wave formula the velocity of the wave can be determined. The velocity is the ratio of the angular frequency and the wave number.

Formula Used: The velocity of the wave is given by,
$v = \dfrac{\omega }{k}$
Where, $v$ is the velocity of the wave, $\omega$ is the angular frequency of the wave and $k$ is the wave number.

Complete step by step answer:
Given that,
The travelling harmonic wave is represented by the equation $y\left( {x,t} \right) = {10^{ - 3}}\sin \left( {50t + 2x} \right)$, where $x$ and $y$ are in meter and $t$ is in seconds.
The general wave equation is given by,
$y = a\sin \left( {\omega t + kx} \right)\,...............\left( 1 \right)$
By equating the equation (1) with the given equation, then
$a\sin \left( {\omega t + kx} \right) = {10^{ - 3}}\sin \left( {50t + 2x} \right)$
From the above equation, then the value of the $\omega$ and $k$ is,
$\omega = 50$ and $k = 2$
Now,
The velocity of the wave is given by,
$v = \dfrac{\omega }{k}\,.................\left( 2 \right)$
By substituting the value of the angular frequency of the wave and the wave number in the above equation (2), then the equation (2) is written as,
$v = \dfrac{{50}}{2}$
By dividing the terms in the above equation, then the above equation is written as,
$v = 25\,m{s^{ - 1}}$
The value of the velocity is positive then the wave propagating along the negative $x$ axis with speed $25\,m{s^{ - 1}}$.
Hence, the option (D) is the correct answer.

Note: The velocity of the wave propagating is directly proportional to the angular frequency of the wave and inversely proportional to the wave number of the wave. If the angular frequency is increased, then the velocity also increases. If the wave number is increasing, the velocity of the wave decreases.