The equation of a wave is given by
`y = a sin omega [(x)/v -k]`
where ` omega ` is angular velocity and v is the linear velocity . The dimensions of k will be

Options

A. `[T^(-2)]`
B. `[T^(-)]`
C. [T]
D. [LT]

Explanation

To find the dimensions of \( k \) in the wave equation \( y = a \sin\left(\omega \left(\frac{x}{v} - k\right)\right) \), we can follow these steps:

### Step 1: Understand the equation
The equation involves a sine function, which means the argument of the sine function must be dimensionless. This means that the terms inside the sine function must have the same dimensions.

### Step 2: Identify the terms
In the expression \( \frac{x}{v} - k \), we have:
- \( x \): This represents displacement.
- \( v \): This represents linear velocity.
- \( k \): We need to find the dimensions of this term.

### Step 3: Determine the dimensions of \( x \) and \( v \)
- The dimension of displacement \( x \) is given by \( [L] \) (length).
- The dimension of velocity \( v \) is given by \( [LT^{-1}] \) (length per time).

### Step 4: Find the dimensions of \( \frac{x}{v} \)
Now, we can find the dimensions of \( \frac{x}{v} \):
\[ \frac{x}{v} = \frac{[L]}{[LT^{-1}]} = \frac{L}{LT^{-1}} = T \]
This means \( \frac{x}{v} \) has the dimensions of time \( [T] \).

### Step 5: Set the dimensions equal
Since \( \frac{x}{v} \) and \( k \) are being subtracted in the sine function, they must have the same dimensions:
\[ [k] = [\frac{x}{v}] = [T] \]

### Conclusion
Thus, the dimensions of \( k \) are:
\[ \text{Dimensions of } k = [T] \]

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