Q 11-01-024JEE MainEAMCET 2009 (Engineering)Medium
When a wave traverses a medium, the displacement of a particle located at $x$ at time $t$ is given by $y = a\sin(bt - cx)$, where $a$, $b$ and $c$ are constants of the wave. Which of the following is a quantity with dimensions?
Answer: (D) $\dfrac{b}{c}$
The argument of a sine must be dimensionless, so $bt$ and $cx$ are dimensionless. Also $y$ and $a$ are both lengths, so $y/a$ is dimensionless.
That leaves
$$\left[\frac{b}{c}\right] = \frac{[T^{-1}]}{[L^{-1}]} = [LT^{-1}]$$
which is a velocity. In fact $b/c = \omega/k$ is the speed of the wave.
Solution by Sreeraj P, M.Sc Physics