Question

The displacement of a progressive wave is represented by$y=Asin(\omega t–kx)$,
where 𝑥 is distance and 𝑡 is time. Write the dimensional formula of (𝑖) $\omega$ and (𝑖𝑖) $k$.

Answer 1

Step:1 Find the dimension of ω.

Given, $y=Asin(\omega t-kx)$

$y=\left[L\right]$

Hence, $Asin(\omega t-kx)=\left[L\right]$

Here $A=\left[L\right]$, which is peak value of $y$

We know that the argument for the sine function is always in degrees or radians and therefore dimensionless. And we know that only dimensionally like quantities can be added or subtracted, therefore the quantities 𝜔𝑡 and 𝑘𝑥 must be dimension less, therefore, (𝜔𝑡−𝑘𝑥) should be dimensionless,

Also $\left[\omega t\right]=constant$

$\left[\omega \right]=\left[T^{-1}\right]$

Step:2 Find the dimension of k.

$\left[kx\right]=constant$

$\left[k\right]=\left[L^{-1}\right]$

Final Answer: $\left[\omega \right]=\left[T^{-1}\right],\left[k\right]=\left[L^{-1}\right]$