What you need to understand about waves is less complex than what you need to understand gravity. Basically you just need to understand the wave equation to understand waves. This requires "only" partial differential equations, the calculus of several variables.
To understand the background of what the wave equation is applied to, though, you need an understanding of linearized gravity. This is harder ro come by.
The wave equation is usually taught along with electromagnetism, and presented along with vector calculus. An informal but still more rigorous than the average popularization text would be "Div, Grad, Curl and all that".
Usually, gravity is presented via tensor methods, which are one step beyond the vector calculus methods I mentioned previously. Tensors are usually taught at the graduate level, though they might be itroduced at the late undergraduate level. Perhaps there is some novel approach that avoids this, but I don't know what it might be. The reason for tensors is to be able to handle arbitrary coordinates. Coordinates in which the wave equation for gravitational waves manifestly applies exist, but are rather specialized and may not "intuitive". While these coordinates make the wave nature of gravitational waves manifest, they make other concepts like measuring distance, or doing physics much trickier. F=ma isn't going to cut it for what is needed to understand physics in arbitrary coordinates, one will need a different approach to physics to understand how physics works in these specialized coordinates.
The coordinate system used to describe gravitational waves can be described informally in a fairly simple manner, though. One simply crates a lattice of dust particles in free space, before a wae passes, that are evenly spaced. One then assigns constant coordinates to each of these particles. While the particles have been given constant coordinates, the metric changes as they pass, and this change has physical consequences. The metric, though , is a rather abstract concept.
The problem as I see it is in understanding how to do physics in these specialized coordinates. Basically, if one has (some approximation of) a rigid ruler, when the points of a free-falling body have constant coordinates, the points on the rigid body will not be constant. One usually expects the points on a rigid body to have constant coordinates, but this is not the usual approach for gravitational waves. Figuring out how things happen from the perspective of a rigid (or nearly rigid) ruler is a more challenging problem, one that is usually not presented in introductory texts.
As a side-note, may one be concerned about what "approximately rigid" might mean, and why do we even have to bring up the idea that rigidity might be approximate. These are reasonable concern, but I'm afraid my answers would get beyond the level at which I'm trying to keep this post.
You may be looking for popularizations, rather than a serious mathematical treatment. Certainly there are a number of them around, for instance LIGO tries to popularize them. I don't like any of the popularziations that much, though - but there are a number around. They're well meaning, but may leave the reader confused about key points, if they try to apply them to answer personal questions.
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