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^(-1)]`
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 will follow these steps:

### Step 1: Understand the wave equation
The wave equation is given as:
\[ y = a \sin \left( \omega \left( \frac{x}{v} - k \right) \right) \]
Here, \( \omega \) is the angular velocity, \( v \) is the linear velocity, \( x \) is the position, and \( k \) is a constant we need to find the dimensions of.

### Step 2: Identify the argument of the sine function
The argument of the sine function, \( \omega \left( \frac{x}{v} - k \right) \), must be dimensionless because the sine function takes a dimensionless argument.

### Step 3: Analyze the terms inside the sine function
We can express the argument as:
\[ \omega \left( \frac{x}{v} - k \right) \]
This means that both \( \omega \frac{x}{v} \) and \( \omega k \) must also be dimensionless.

### Step 4: Find the dimensions of \( \frac{x}{v} \)
- The dimension of \( x \) (position) is \( [L] \).
- The dimension of \( v \) (velocity) is \( [L][T]^{-1} \).

Thus, the dimension of \( \frac{x}{v} \) is:
\[ \frac{[L]}{[L][T]^{-1}} = [T] \]

### Step 5: Find the dimensions of \( \omega \)
- The angular velocity \( \omega \) is defined as \( \frac{2\pi}{T} \), where \( T \) is time.
- Therefore, the dimension of \( \omega \) is:
\[ [\omega] = [T]^{-1} \]

### Step 6: Set up the equation for dimensionless argument
Since \( \omega \frac{x}{v} \) is dimensionless, we have:
\[ [\omega] \cdot \left[\frac{x}{v}\right] = [T]^{-1} \cdot [T] = 1 \quad (\text{dimensionless}) \]

### Step 7: Analyze \( \omega k \)
For \( \omega k \) to be dimensionless, we can write:
\[ [\omega] \cdot [k] = [T]^{-1} \cdot [k] = 1 \]
This implies:
\[ [k] = [T] \]

### Conclusion
The dimensions of \( k \) are:
\[ [k] = [T] \]

### Final Answer
The dimensions of \( k \) are \( [T] \).

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