Interference: in phase and out of phase

Before we go too far into the mathematics, we need to demonstrate the basis for it all: interference. Adding waves together often comes down to the question, "In some particular region of space, do the waves reinforce each other -- acting in the same direction -- or do they undermine each other?" In other words, when the waves meet, are they in phase or out of phase.

You can see the effect of these two situations if you play with the Desmos app show in the picture below. Click on the link to activate it.

Image and applet courtesy of Desmos