A wave equation is represented as
`r = A sin [ alpha (( x - y)/(2)) ] cos [ omega t - alpha (( x + y)/(2))]`
where ` x` and `y` are in metres and t in seconds . Then ,

Options

A. the wave is a stationary wave.
B. the wave is a progressive wave propagating along ` + x - axis`.
C. the wave is a progressive wave propagating at right angle to the ` + x - axis`
D. all points lying on line ` y = x + ( 4 pi //alpha)` are always at rest.

Explanation

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

\[ r = A \sin \left( \alpha \frac{(x - y)}{2} \right) \cos \left( \omega t - \alpha \frac{(x + y)}{2} \right) \]

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

The wave equation can be expressed in the form of a standing wave, which typically has the form:

\[ y = 2A \sin(kx) \cos(\omega t) \]

In our case, we need to check if the given equation can be manipulated into a similar form.

### Step 2: Check if it represents a standing wave

The given equation can be rewritten as:

\[ r = A \sin \left( \frac{\alpha}{2} (x - y) \right) \cos \left( \left( \omega - \frac{\alpha}{2} \right) t \right) \]

This is not in the standard form of a standing wave equation. Therefore, the first option stating that the wave is a stationary wave is incorrect.

### Step 3: Check if it represents a progressive wave

A progressive wave has the general form:

\[ y = A \sin(\omega t - kx) \]

The given equation does not match this form either, as it contains both sine and cosine terms that depend on both \(x\) and \(y\). Hence, the second option stating that the wave is a progressive wave is also incorrect.

### Step 4: Analyze the third option about propagation direction

The third option suggests that the wave is a progressive wave propagating at a right angle to a certain direction. However, since we have already established that the wave does not fit the form of a progressive wave, this option is also incorrect.

### Step 5: Evaluate the fourth option

For the fourth option, we need to analyze the condition where \(y = x + \frac{4\pi}{\alpha}\):

Substituting \(y\) into the wave equation:

\[ r = A \sin \left( \alpha \frac{(x - (x + \frac{4\pi}{\alpha}))}{2} \right) \cos \left( \omega t - \alpha \frac{(x + (x + \frac{4\pi}{\alpha}))}{2} \right) \]

This simplifies to:

\[ r = A \sin \left( \alpha \frac{-\frac{4\pi}{\alpha}}{2} \right) \cos \left( \omega t - \alpha \frac{(2x + \frac{4\pi}{\alpha})}{2} \right) \]

This results in:

\[ r = A \sin(-2\pi) \cos \left( \omega t - \alpha (x + \frac{2\pi}{\alpha}) \right) \]

Since \(\sin(-2\pi) = 0\), we find that \(r = 0\).

### Step 6: Conclusion

Since \(r = 0\) implies that there is no displacement at that point, and thus the points are at rest. Therefore, the fourth option is correct.

### Final Answer

The correct option is: **All points on the line \(y = x + \frac{4\pi}{\alpha}\) are always at rest.**

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