why is a compression reflected as a compression from a rigid boundary

When a sound wave gets reflected from a rigid boundary, the particles at the boundary are unable to vibrate. Hence the generation of reflected wave takes place which interferes with the oncoming wave to produce zero displacement at the rigid boundary. At the points where there is zero displacement, the variation in pressure is maximum. This shows that the phase of wave has been reversed but the nature of sound wave does not change i.e. on reflection the compression is reflected back as compression and rarefaction as rarefaction. Let the incident wave be represented by the given equation:

YY = asin(wt−kx)asin⁡(wt−kx),
then the equation of reflected wave takes the form

YY = asin(wt+kx+π)asin⁡(wt+kx+π) = −asin(wt+kx)−asin⁡(wt+kx)
Here ' a ' is basically the amplitude of reflected wave.