When a wave traverses a medium, the displacement of a particle located at $x$ at a time $t$ is given by $y=a…
When a wave traverses a medium, the displacement of a particle located at $x$ at a time $t$ is given by $y=a \sin (b t-c x)$, where $a, b$ and $c$ are constants of the wave, which of the following is a quantity with dimensions?
$\frac{y}{a}$
$b t$
$c x$
$\frac{b}{c}$
Solution
Given, $y=a \sin (b t-c x)$
Comparing the given equation with general wave equation
$y=a \sin \left(\frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}\right)$
we get
$b=\frac{2 \pi}{T}, c=\frac{2 \pi}{\lambda}$
(a) Dimensions o
$\text { of } \begin{aligned}
\frac{y}{a} & =\frac{\text { metre }}{\text { metre }}=\frac{[\mathrm{L}]}{[\mathrm{L}]} \\
& =\text { Dimensionless }
\end{aligned}$
(b)
$\begin{aligned}
\text { Dimensions of } b t & =\frac{2 \pi}{T} \cdot t=\frac{[\mathrm{T}]}{[\mathrm{T}]} \\
& =\text { Dimensionless }
\end{aligned}$
(c) Dimensions of
$\begin{aligned}
c x & =\frac{2 \pi}{\lambda} \cdot x=\frac{[\mathrm{L}]}{[\mathrm{L}]} \\
& =\text { Dimensionless }
\end{aligned}$
(d) Dimensions of
$\begin{aligned}
\frac{b}{c} & =\frac{2 \pi}{T} / \frac{2 \pi}{\lambda} \\
& =\lambda / T=\left[\mathrm{LT}^{-1}\right]
\end{aligned}$
Thus, option (d) has dimensions.