(a) Taking equation of a palne progressive wave as `y=alphasin.(2pi)/(gamma)(vt-x)`. Write down the expression for the paticle velocity. Show that the particle veloity at a point = `"wave velocity" xx "slope of the displacement curve at the point"`.

Explanation

To solve the problem step by step, we will derive the expression for particle velocity from the given wave equation and then show the relationship between particle velocity, wave velocity, and the slope of the displacement curve.

### Step 1: Write down the wave equation
The equation of the plane progressive wave is given as:
\[ y = \alpha \sin\left(\frac{2\pi}{\gamma}(vt - x)\right) \]

### Step 2: Find the particle velocity
The particle velocity \( v_p \) is defined as the rate of change of displacement \( y \) with respect to time \( t \). Therefore, we differentiate \( y \) with respect to \( t \):
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