The wave travels along a string whose equation is `y=(p^(3))/(p^(2)+(px-qt)^(2))`, where p = 2 unit and q = 0.5 units.
Find the direction of propagation of wave.

Options

A. along + y - axis
B. along - x-axis
C. along + x-axis
D. None of these

Explanation

To determine the direction of propagation of the wave described by the equation \( y = \frac{p^3}{p^2 + (px - qt)^2} \), we will analyze the equation step by step.

### Step 1: Identify the wave equation form
The given wave equation can be analyzed to see if it matches the standard form of a wave equation, which typically looks like:
\[ y = f(kx - \omega t) \]
or
\[ y = f(\omega t + kx) \]
where \( k \) is the wave number and \( \omega \) is the angular frequency.

### Step 2: Rewrite the equation
The equation can be rewritten as:
\[ y = \frac{p^3}{p^2 + (px - qt)^2} \]
We can observe that the term \( (px - qt)^2 \) is present in the denominator.

### Step 3: Analyze the term \( (px - qt) \)
The term \( (px - qt) \) suggests a relationship between \( x \) and \( t \). Specifically, it indicates that the wave depends on both position \( x \) and time \( t \).

### Step 4: Determine the direction of propagation
In the term \( (px - qt) \):
- If we have \( (kx - \omega t) \), the wave travels in the positive x-direction.
- If we have \( (kx + \omega t) \), the wave travels in the negative x-direction.

In our case, we have \( (px - qt) \):
- Here, \( p \) corresponds to \( k \) (wave number) and \( q \) corresponds to \( \omega \) (angular frequency).
- Since the term is of the form \( (px - qt) \), it indicates that the wave is traveling in the positive x-direction.

### Conclusion
Thus, the direction of propagation of the wave is along the positive x-axis.

### Final Answer
The wave travels in the positive x-direction.

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