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 scenario, we start with the provided information about the wave shape at a specific time \( t_0 \).

### Step-by-Step Solution:

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

2. **General Form of Wave Equation**:
The general equation for a wave propagating in the positive x-direction is:
\[ g(x, t) = A \sin\left(kx - \omega t\right) \]
where \( k \) is the wave number and \( \omega \) is the angular frequency.

3. **Relate Wave Parameters**:
We know that the wave speed \( v \) is related to the wavelength \( \lambda \) and the period \( T \) by:
\[ v = \frac{\lambda}{T} \]
From this, we can express \( T \) as:
\[ T = \frac{\lambda}{v} \]

4. **Compare Given Wave Shape with General Form**:
The given wave shape does not have a time-dependent term. To incorporate time, we consider the wave's propagation from \( t_0 \) to a later time \( t \). We can express the wave at time \( t \) as:
\[ g(x, t) = A \sin\left(\frac{x^2}{a^2} - \frac{v}{a}(t - t_0)\right) \]

5. **Substituting the Wave Parameters**:
We can rewrite the wave equation using the relationship between \( k \) and \( a \):
\[ g(x, t) = A \sin\left(\frac{x^2}{a^2} - \frac{v}{a}(t - t_0)\right) \]

6. **Final Wave Equation**:
The final wave equation at any time \( t \) is:
\[ g(x, t) = A \sin\left(\frac{x - vt + vt_0}{a}\right) \]
This can be simplified to:
\[ g(x, t) = A \sin\left(\frac{x - vt + vt_0}{a}\right) \]

### Final Result:
Thus, the wave equation at any instant \( t \) is:
\[ g(x, t) = A \sin\left(\frac{x - v(t - t_0)}{a}\right) \]