To find the dimensions of $k$, we utilize the Principle of Homogeneity of dimensions, which states that only physical quantities of the same dimensions can be added or subtracted .

1. Analyze the term $(\frac{x}{v} - k)$: Since $k$ is subtracted from the term $\frac{x}{v}$, both terms must have the same dimensional formula.
2. Determine dimensions of $\frac{x}{v}$: $x$ (Length) has dimensions $[L]$. $v$ (Linear Velocity) has dimensions $[LT^{-1}]$ .

• Therefore, the dimensions of $\frac{x}{v}$ are $\frac{[L]}{[LT^{-1}]} = [T]$.

3. Conclusion: Since dimensions of $k$ must equal dimensions of $\frac{x}{v}$, the dimension of $k$ is $[T]$.

Note: The argument of the sine function, $\omega(\frac{x}{v} - k)$, must be dimensionless. Checking: $[\omega] \times [T] = [T^{-1}] \times [T] =$, which confirms the validity of the equation.