Question

At $t=0$, a transverse wave pulse travelling in the positive $x$ direction with a speed of $2\text{m/s}$ in a long wire is described by the function, $y=\frac{6}{x^2},$ given that $x\ne 0$. Transverse velocity of a particle at $2\text{m}$ and at $2\text{s}$ will be

• A. $3\text{m/s}$ upwards
• B. $3\text{m/s}$ downwards
• C. $8\text{m/s}$ upwards
• D. $8\text{m/s}$ downwards

Answer 1

The correct option is B $3\text{m/s}$ downwards

The equation of transverse wave on wire at $t=0$ is,
$y(x,t=0)=\frac{6}{x^2}$
Since, wave speed, $v=2\text{m/s}$
At time $t$, equation of wave will be
$y(x,t)=\frac{6}{(x-2t{)}^2}$
$[$wave is travelling along positive $x$ axis$]$
For particle velocity,
$\frac{\partial y}{\partial t}=6(-2)(x-2t{)}^{-3}(-2)$
$\Rightarrow v_y=\frac{24}{(x-2t{)}^3}$
At $x=2\text{m};t=2\text{s},$
$v_y=\frac{24}{(2-4{)}^3}=-3\text{m/s}$
$i.e.$ particle will be moving downward with speed $3\text{m/s}$.

$\begin{array}{c}\text{Why this question?}\\ \text{Tips: For a transverse wave, the particle velocity is}\\ \text{obtained by partial differentiation of wave equation}\\ \text{w.r.t time i.e.}{v}_y=\frac{\partial y}{\partial t}\end{array}$