Question

When a wave traverses a medium the displacement of a particle located at \(x\) at a 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?

• A. \(\dfrac{y}{a}\)
• B. \(bt\)
• C. \(cx\)
• D. \(\dfrac{b}{c}\)

Hint:

In any trigonometric function like \(\sin(\cdot)\), the argument must always be dimensionless.

Answer 1

The Correct Option is D

Step 1: Identify the dimensional nature of the sine argument.
In the wave equation \(y = a\sin(bt - cx)\), the quantity inside sine must be dimensionless.
So,
\[ bt - cx \;\; \text{must be dimensionless} \]
Step 2: Check dimensions of each term.
Since \(bt\) is dimensionless,
\[ [b][t] = 1 \Rightarrow [b] = T^{-1} \]
Similarly, \(cx\) is dimensionless,
\[ [c][x] = 1 \Rightarrow [c] = L^{-1} \]
Step 3: Test each option.
(A) \(\dfrac{y}{a}\): Since \(y\) has same dimension as \(a\), ratio is dimensionless.
(B) \(bt\): Dimensionless as proved above.
(C) \(cx\): Dimensionless as proved above.
(D) \(\dfrac{b}{c}\):
\[ \left[\frac{b}{c}\right] = \frac{T^{-1}}{L^{-1}} = LT^{-1} \]
This has dimensions of velocity, so it is a dimensional quantity.
Final Answer: \[ \boxed{\dfrac{b}{c}} \]