In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero (i.e., the forcing function / forcing vector is zero), the equation may still admit a non-trivial solution, known in applied mathematics as an eigenfunction solution, in physics as a "natural mode" solution and in electrical circuit theory as the "zero-input response." This is a concept that spans a wide range of physical disciplines. Common physical examples of "resonant" natural modes would include the resonant vibrational modes of stringed instruments (1D), percussion instruments (2D) or the former Tacoma Narrows Bridge (3D). Examples of "propagating" natural modes would include waveguide modes, optical fiber modes, solitons and Bloch waves. Infinite homogeneous media admit the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. The propagating plane waves we'll study in this article are perhaps the simplest type of propagating waves found in any type of media.