  We discuss the notion of resonance, as well as the existence and uniqueness
of periodic solutions for a forced simple harmonic oscillator. While this topic
is elementary, and well-studied for sinusoidal forcing, this does not seem to
be the case when the forcing function is general (perhaps discontinuous). Clear
statements of theorems and proofs do not readily appear in standard textbooks
or online. For that reason, we provide a characterization of resonant
solutions, written in terms of the relationship between the forcing and natural
frequencies, as well as a condition on a particular Fourier mode. While our
discussions involve some notions from $L^2$-spaces, our proofs are elementary,
using this the variation of parameters formula; the main theorem and its proof
should be readable by students who have completed a differential equations
course and have some experience with analysis. We provide several examples, and
give various constructions of resonant solutions. Additionally, we connect our
discussion to notions of resonance in systems of partial differential
equations, including fluid-structure interactions and partially damped systems.
