d'Alembert's formula

And using the duality property, we can express our solution as some general function f of t plus x over c plus another general function g of t minus x over c, where these general functions f of t and g of t are the inverse transforms of our a and b functions of omega.

This very generalized form of a wave solution is known as d'Alembert's solution, and it represents any wave as a superposition of a traveling wave going left and a traveling wave going right.

If we now express our functions p and q as Fourier series, converting them to an infinite sum of sinusoids with a suitable fundamental frequency, we will recover precisely the same general solution as we found before.

Let's see how the idea of harmonics comes in with the wave equation.