The transverse displacement `y(x, t)` of a wave on a string is given by `y(x, t)= e ^(-(ax^(2) + bt^(2) + 2sqrt((ab))xt)`. This represents as :
Wave moving along + x-axis with a speed `sqrt((a)/(b))`
Wave moving along - x-axis with speed `sqrt((b)/(a))`
Standing wave of frequency `sqrt (b)`
Standing wave of frequency `(1)/(sqrt (b))`

Options

A. Wave moving along + x-axis with a speed `sqrt((a)/(b))`
B. Wave moving along - x-axis with speed `sqrt((b)/(a))`
C. Standing wave of frequency `sqrt (b)`
D. Standing wave of frequency `(1)/(sqrt (b))`

Answer

To solve the problem, we need to analyze the given wave equation:

\[ y(x, t) = e^{-(a x^2 + b t^2 + 2\sqrt{ab} x t)} \]

### Step 1: Identify the form of the wave equation

We need to express the wave equation in a recognizable form. A typical wave equation can be expressed as a function of \( x \) and \( t \) in the form of \( f(x \pm vt) \).

The given equation can be rewritten as:

\[ y(x, t) = e^{-\left(\sqrt{a} x + \sqrt{b} t\right)^2} \]

This indicates that the wave can be expressed as a function of \( \sqrt{a} x + \sqrt{b} t \).

### Step 2: Determine the wave speed

From the rewritten form, we can identify the coefficients of \( x \) and \( t \):

- Coefficient of \( x \) is \( \sqrt{a} \)
- Coefficient of \( t \) is \( \sqrt{b} \)

The speed \( v \) of the wave can be calculated using the formula:

\[ v = \frac{\text{Coefficient of } t}{\text{Coefficient of } x} = \frac{\sqrt{b}}{\sqrt{a}} = \sqrt{\frac{b}{a}} \]

### Step 3: Determine the direction of wave propagation

Next, we need to determine the direction of the wave. The signs of the coefficients of \( x \) and \( t \) indicate the direction:

- If both coefficients have the same sign, the wave travels in the negative \( x \)-direction.
- If the coefficients have opposite signs, the wave travels in the positive \( x \)-direction.

In our case, both coefficients \( \sqrt{a} \) and \( \sqrt{b} \) are positive, which means the wave is traveling in the negative \( x \)-direction.

### Conclusion

Based on the analysis, we conclude that the wave is moving along the negative \( x \)-axis with a speed of \( \sqrt{\frac{b}{a}} \).

### Final Answer

The correct option is:
**Wave moving along - x-axis with speed \( \sqrt{\frac{b}{a}} \)**.

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