Fore a transverse wave on a string, the string displacement is describrd by
`y(x,t)=f(x-at)`
where f represent a function and a is a negative constant. Then which of the following is//are correct statement(S)?

Options

A. the shape of the string at time `t=0` is given by `f(x)`
B. the shape of wave form does not change as it moves along the string
C. waveform moves in +ve x-direction
D. the speed of waveform is a

Explanation

To solve the problem, let's analyze the given wave function and the statements one by one.

### Given:
The displacement of the transverse wave on a string is described by:
\[ y(x,t) = f(x - at) \]
where \( a \) is a negative constant.

### Step 1: Analyze the wave function at \( t = 0 \)
At \( t = 0 \):
\[ y(x, 0) = f(x - a \cdot 0) = f(x) \]
This means that the shape of the wave at \( t = 0 \) is given by the function \( f(x) \).

**Conclusion:** The first statement is correct.

### Step 2: Analyze the shape of the wave along the string
The function \( f(x - at) \) indicates that the shape of the wave does not change as time progresses. The function \( f \) remains the same regardless of the value of \( t \).

**Conclusion:** The second statement is also correct.

### Step 3: Determine the direction of wave travel
The direction of wave travel is determined by the signs of the coefficients of \( x \) and \( t \) in the argument of the function.

The argument \( x - at \) can be rewritten as:
- Coefficient of \( x \): \( +1 \) (positive)
- Coefficient of \( t \): \( -a \)

Since \( a \) is a negative constant, \( -a \) becomes positive. Thus:
- Coefficient of \( t \): \( -a \) (positive)

Both coefficients are positive, which means the wave travels in the negative direction.

**Conclusion:** The third statement is incorrect.

### Step 4: Calculate the speed of the wave
The speed \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\text{coefficient of } t}{\text{coefficient of } x} \]
Here:
- Coefficient of \( t \) is \( -a \)
- Coefficient of \( x \) is \( +1 \)

Thus, the speed of the wave is:
\[ v = \frac{-a}{1} = -a \]
Since \( a \) is negative, \( -a \) will be positive.

**Conclusion:** The fourth statement is correct.

### Final Summary of Statements:
1. **First statement:** Correct
2. **Second statement:** Correct
3. **Third statement:** Incorrect
4. **Fourth statement:** Correct