The displacement of A particle at x = 0 of a stretched string carrying a wave in the positive X-direction is given by `f(t)=Ae^(-t^2)`. The wave speed is V. Write equation of the wave.

Options

A. `f(x,t) = Ae^(-(t + x)^2)`
B. `f(x,t) = Ae^(-(t - x/v)^2)`
C. `f(x,t) = Ae^((t + x/v)^2)`
D. `f(x,t) = Ae^(-(t + x/v)^2)`

Answer

To solve the problem, we need to derive the equation of the wave based on the given displacement function at \(x = 0\) and the wave speed \(V\).

### Step-by-Step Solution:

1. **Identify the given function**: The displacement of the particle at \(x = 0\) is given by:
\[ f(t) = A e^{-t^2} \]
This represents the wave function at the origin.

2. **Understand the wave equation**: The general form of the wave equation for a wave traveling in the positive x-direction is:
\[ y(x, t) = A \sin(\omega t - kx) \]
where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(k\) is the wave number.

3. **Relate the given function to the wave equation**: Since the displacement at \(x = 0\) is given by \(f(t)\), we can express the wave function at any point \(x\) and time \(t\) as:
\[ y(0, t) = A e^{-t^2} \]
This implies that the wave function must also be of the form \(A e^{-(t - \frac{x}{V})^2}\) to account for the wave traveling in the positive x-direction.

4. **Determine the wave parameters**:
- The wave speed \(V\) relates to the angular frequency \(\omega\) and wave number \(k\) by the equation:
\[ V = \frac{\omega}{k} \]
- Here, we need to express \(\omega\) and \(k\) in terms of the given function.

5. **Find the angular frequency**: The term \(e^{-t^2}\) suggests that we can relate it to a Gaussian wave packet. The angular frequency \(\omega\) can be derived from the argument of the exponent:
\[ \omega = 1 \quad \text{(since the exponent is } -t^2\text{)} \]

6. **Find the wave number**: The wave number \(k\) can be expressed as:
\[ k = \frac{\omega}{V} = \frac{1}{V} \]

7. **Write the wave equation**: Now substituting \(\omega\) and \(k\) back into the wave equation, we get:
\[ y(x, t) = A e^{-(t - \frac{x}{V})^2} \]

### Final Wave Equation:
Thus, the equation of the wave is:
\[ y(x, t) = A e^{-(t - \frac{x}{V})^2} \]