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Interference

To get more quantitative about this "addition of amplitudes," we make the following assumption, which is crucial for the arguments to follow and is even valid for the most important kinds of waves, namely $EM$ waves, under all but the most extreme conditions:

LINEAR SUPERPOSITION OF WAVES:
 

For water waves this is not perfectly true (water waves are very peculiar in many ways) but to a moderately good approximation the amplitude (height) of the surface disturbance at a given position and time is just the sum of the heights of all the different waves passing that point at any instant. This has some alarming implications for sailors! If you are sailing along a coastline with steep cliffs, the incoming swells are apt to be reflected back out to sea with some efficiency; if the reflected waves from many parts of the shoreline happen to interfere constructively with the incoming swells at the position of your boat, you can encounter "freak waves" many times higher than the mean swell height. Experienced sailors stay well out from the coastline to avoid such unpredictable interference maxima.