Question

When a wave traverses a medium the displacement of a particle located at $x$ at a time is given by $y = a \sin (bt - cx)$, where $a$, and $b$ are constants of the wave, which of the following is a quantity with dimensions?

• A. $\frac{y}{a} $
• B. $bt$
• C. $cx$
• D. $\frac{b}{c} $

Answer 1

Answer: (D)

Given, $y=a \sin (b t-c x)$
Comparing the given equation with general wave equation $y=a \sin \frac{2 \pi t}{T}-\frac{2 \pi x}{\lambda}$
we get $b=\frac{2 \pi}{T}, c=\frac{2 \pi}{\lambda}$
(a) Dimensions of $\frac{y}{a}=\frac{\text { metre }}{\text { metre }}=\frac{L}{L}$
(b) Dimensions of $b t=\frac{2 \pi}{T} . t=\frac{T}{T}$
(c) Dimensions of $c x=\frac{2 \pi}{\lambda} \cdot x=\frac{L}{L}$
(d) Dimensions of $\frac{b}{c}=\frac{2 \pi}{T} \frac{2 \pi}{\lambda}=\frac{\lambda}{T}=L T^{-1}$