The displacement of a progressive wave is represented by y = A ( … - Vidyadip

Question

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

✓

Answer

We have to apply principle of homogeneity to solve this problem. Principle of homogeneity states that in a correct equation, the dimensions of each term added or subtracted must be same, i.e., dimensions of $\text{LHS}$ and $\text{RHS}$ should be equal. According to the problem, $\text{y}=\text{A}\sin(\omega\text{t}-\text{kx})$ Here $y = [L]$ hence $\text{A}\sin(\omega\text{t}-\text{kx})=[\text{L}]$ Here $A = [L]$, which peak value of $y$ So, $\omega\text{t}-\text{kx}$ Should be dimensionless,

$[\omega\text{t}]=\text{constant}$

$\Rightarrow[\omega]=[\text{T}^{-1}]$

$[\text{Kx}] = \text{Constant}$

$\Rightarrow[\text{k}]=[\text{L}^{-1}]$

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