The displacement of a progressive wave is represented by y = A sin(wt – k x ), where x is distance and t is time. Write the dimensional formula of (i) ω and (ii) k.

The correct option is T⁻¹ , L⁻¹.

To find the dimensional formulas of $\omega$ and $k$, we use the principle of dimensional homogeneity. In any trigonometric function, such as the sine function in the wave equation:
$y=A\sin (\omega t-kx)$
the argument of the function (the angle) must be a dimensionless quantity. Therefore, the term $(\omega t-kx)$ is dimensionless.

This implies that both individual terms, $\omega t$ and $kx$, must be dimensionless because we can only subtract quantities with the same dimensions.

Step 1: Finding the dimensions of $\omega$
Since $\omega t$ is dimensionless:
$\left[\omega t\right]=\left[M^0{L}^0{T}^0\right]$
Given that $t$ represents time, its dimensional formula is $\left[T\right]$. Substituting this in:
$\left[\omega \right]\left[T\right]=1$
$\left[\omega \right]=\frac{1}{\left[T\right]}=\left[T^{-1}\right]$
Thus, the dimensional formula of $\omega$ is T⁻¹.

Step 2: Finding the dimensions of $k$
Since $kx$ is dimensionless:
$\left[kx\right]=\left[M^0{L}^0{T}^0\right]$
Given that $x$ represents distance, its dimensional formula is $\left[L\right]$. Substituting this in:
$\left[k\right]\left[L\right]=1$
$\left[k\right]=\frac{1}{\left[L\right]}=\left[L^{-1}\right]$
Thus, the dimensional formula of $k$ is L⁻¹.

Combining the two results, the dimensional formulas of $\omega$ and $k$ are T⁻¹ , L⁻¹ respectively.