We now would like to show that one can build up the general solution of the wave equation by superimposing certain elementary solutions. Indeed, the elementary solutions being referred to are those discussed in §6. These elementary solutions will form a very convenient “basis” for the vector space of solutions to the wave equation, just as the normal modes provided a basis for the space of solutions in the case of coupled oscillators. Indeed, as we shall see, the elementary solutions are the normal modes for wave propagation. The principal tools needed to understand this are provided by the methods of Fourier analysis, which is very useful in analyzing waves in any number of spatial dimensions. To begin, we will take a somewhat superficial tour of the key results of Fourier analysis. Then we’ll see how to use these results to better understand the solutions to the wave equation.
Fourier analysis, waves, spatial dimension, chapter 8, eight
Torre, Charles G., "08 Fourier Analysis" (2012). Foundations of Wave Phenomena. Book 15.