Q81 / 120 · VITEEE 2009 · Physics

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?

  • A
    $\frac{y}{a}$
  • B
    $b t$
  • C
    $\quad c x$
  • Correct answer
    D
    $\frac{b}{c}$

Solution

Here, $y=a \sin (b t-c x)$ Compar ing this equation with geneal 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 of $\frac{y}{a}=\frac{[L]}{[L]}=$ Dimensionless (b) Dimensions of $b t=\frac{2 \pi}{T} \cdot t=\frac{[T]}{[T]}$ $=$ Dimensionless (c) Dimensions of $c x=\frac{2 \pi}{\lambda} \cdot x=\frac{[L]}{[L]}$ $=$ Dimensionless (d) Dimensions of $\frac{b}{c}=\frac{\frac{2 \pi}{T}}{\frac{2 \pi}{\lambda}}=\frac{\lambda}{T}=\left[L T^{-1}\right]$

Hence, the correct answer is option: 'D'.

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