Answer: remain stationary
Stationary waves (so called because they seem to be immobile) occur when two waves interfere with the same frequency, amplitude but with different direction, along a line with a phase difference of half wavelength.
In this kind of waves there are two types of points:
The nodes, which are points that remain motionless or stationary and do not vibrate. They are due to the destructive interference of both waves when they meet.
The antinodes, which are points that vibrate with a maximum vibration amplitude. They are due to the non-destructive interference of both waves.
According to this explanation and comparing it with the description, when this two waves pass through each other, the point P will become a node, hence it will remain stationary.