If you're already comfortable with the Wave Equation, then it's somewhat straightforward. You start with some spacetime geometry (i.e. some "known metric") and

If you're already comfortable with the Wave Equation, then it's somewhat straightforward. You start with some spacetime geometry (i.e. some "known metric") and then study small perturbations that obey the wave equation. The key is finding distributions of matter that, by Einstein's Equation, cause such perturbations to arise. In differential geometry, the device called a "metric" let's you define things like angles and distances on some space. This is generally denoted as g with the Greek letters mu and nu as subcripts. Just to give you a feel of what's going on, I'll leave out the subscripts, but what OP's saying is that g = k + h which says that g, the space-time metric, decomposes into some known, usually easy to work with metric k, plus some perturbations on top, h. In the math you put restrictions on h to make precise the "small perturbations" part. Anyway, what Gravity Waves boil down to are situations (read distributions of matter) that cause h to obey the wave equation. In other words, gravitational waves are spacetimes that have a component that waves "on top" of a given reference "background" spacetime. Of course, you could cherry pick your background such that anything looks like waves on top, but typically we choose the background to be Minkowskian (i.e. flat). FWIW, the above perturbative approach is called Linearized Gravity and is kind of a (nice) hack to restrict study of the really hairy non-linear Einstein Equation to cases that are manageable.