A wave propagates in a string in the positive x-direction with velocity v. The shape of the string at `t=t_0` is given by
`f(x,t_0)=A sin ((x^2)/(a^2))`. Then the wave equation at any instant t is given by

Options

A. `g(x,t)=A sin""[x-v(t-t_0)]^2/(a^2)`
B. `g(x,t)=A sin""[x+v(t-t_0)]^2/(a^2)`
C. `g(x,t)=A sin""[x-v(t+t_0)]^2/(a^2)`
D. `g(x,t)=A sin""[x+v(t+t_0)]^2/(a^2)`

Answer

To derive the wave equation for the given wave propagating in a string, we start with the shape of the string at a specific time \( t = t_0 \).

### Step-by-Step Solution:

1. **Identify the Given Function**:
The shape of the string at time \( t_0 \) is given by:
\[ f(x, t_0) = A \sin\left(\frac{x^2}{a^2}\right) \]

2. **Understand the Wave Propagation**:
The wave is propagating in the positive x-direction with velocity \( v \). The general form of a wave traveling in the positive x-direction can be expressed as:
\[ f(x, t) = A \sin\left(kx - \omega t\right) \]
where \( k \) is the wave number and \( \omega \) is the angular frequency.

3. **Relate Time and Position**:
At \( t = t_0 \), we want the wave function to match the given function. Thus, we can set:
\[ f(x, t_0) = A \sin\left(kx - \omega t_0\right) \]
To ensure that this matches the given function, we need:
\[ kx - \omega t_0 = \frac{x^2}{a^2} \]

4. **Determine Wave Parameters**:
To match the forms, we can equate:
\[ k = \frac{2}{a^2} \quad \text{and} \quad \omega t_0 = 0 \quad \text{(since at } t = t_0, \text{ the term must vanish)} \]
This implies that \( \omega = 0 \) at \( t = t_0 \), but we need to express the wave function for any time \( t \).

5. **Expressing the Wave Equation**:
The wave function can be expressed as:
\[ f(x, t) = A \sin\left(kx - \omega t\right) \]
Substituting \( k = \frac{2}{a^2} \) and using the relationship between angular frequency and wave speed \( \omega = \frac{2\pi}{T} \) (where \( T \) is the period), we can write:
\[ f(x, t) = A \sin\left(\frac{x^2}{a^2} - \frac{v}{a}t\right) \]

6. **Final Form of the Wave Equation**:
Therefore, the wave equation at any instant \( t \) is:
\[ f(x, t) = A \sin\left(\frac{x^2}{a^2} - \frac{vt}{a}\right) \]

### Final Answer:
The wave equation at any instant \( t \) is:
\[ f(x, t) = A \sin\left(\frac{x^2}{a^2} - \frac{vt}{a}\right) \]