A wave is represented by -
`y=a sin (At-Bx+C)`
where A, B, C are constants. The Dimensions of A, B, C are

Options

A. `T^(1),L,M^(0)L^(0)T^(0)`
B. `T^(-1),L^(-1),M^(0)L^(0)T^(0)`
C. T,L,M
D. `T^(-1),L^(-1), M^(-1)`

Explanation

To find the dimensions of the constants A, B, and C in the wave equation \( y = a \sin(At - Bx + C) \), we need to analyze the argument of the sine function, which is \( At - Bx + C \). The argument of the sine function must be dimensionless, meaning it has no physical dimensions (i.e., it is a pure number).

### Step-by-Step Solution:

1. **Identify the Argument of the Sine Function**:
The argument of the sine function is \( At - Bx + C \). For the sine function to be valid, this entire expression must be dimensionless.

2. **Set Up the Equation**:
Since \( At \), \( Bx \), and \( C \) must all have the same dimensions, we can express this as:
\[ [At] = [Bx] = [C] \]

3. **Determine the Dimensions of Each Term**:
- For the term \( At \):
- Let \( [A] \) be the dimension of \( A \).
- The dimension of time \( t \) is \( [T] \).
- Therefore, the dimension of \( At \) is:
\[ [A][T] = [A][T^1] \]
- For the term \( Bx \):
- Let \( [B] \) be the dimension of \( B \).
- The dimension of distance \( x \) is \( [L] \).
- Therefore, the dimension of \( Bx \) is:
\[ [B][L] = [B][L^1] \]
- For the constant \( C \):
- Since \( C \) is added to \( At \) and \( Bx \), it must also be dimensionless, which means:
\[ [C] = 1 \text{ (dimensionless)} \]

4. **Equate the Dimensions**:
Since \( [At] = [Bx] = [C] \), we can set up the following equations:
\[ [A][T] = [B][L] = 1 \]

5. **Solve for Dimensions**:
- From \( [A][T] = 1 \):
\[ [A] = [T^{-1}] \]
- From \( [B][L] = 1 \):
\[ [B] = [L^{-1}] \]
- For \( C \):
\[ [C] = 1 \text{ (dimensionless)} \]

### Final Answer:
- The dimensions of the constants are:
- \( [A] = [T^{-1}] \)
- \( [B] = [L^{-1}] \)
- \( [C] = 1 \) (dimensionless)