Sines and cosines can be used to express general periodic functions as an infinite sum known as a Fourier series. We’ve learned that the periodicity of a sine or cosine function is useful in that we only need to know its behavior on a finite domain to determine how it will behave on any domain. On the left side of this demonstration, you can see exactly how periodicity works for sine and cosines. Recall that they are invariant under the shift $x\longrightarrow x+2\pi$. You can adjust the values of $M$ and $t$ for the equations $\sin(Mx+2\pi t)$ and $\cos(Mx +2\pi t)$. In the second half of the demonstration, we have two waves traveling in opposite directions. When they intersect, their amplitudes interfere constructively or destructively; the resultant wave is displayed below. Try adjusting each of their frequencies and moving time forward. Notice how a very complicated wave can be described by a superposition of two or more simple periodic functions.