I think it's actually more complicated than that. The strength of the sun's gravity goes as 1/r^2. Move it to the side a bit, and the direction of the gravitational field changes slightly. This direction change goes as 1/r, by the small angle approximation. So from your description, we'd expect the size of the ripple to go as 1/r^3. But in the case of actual gravitational waves, the field strength of the ripple goes as 1/r. That's why we can detect gravitational waves from distant black holes, but we can't detect their static gravitational fields, which go as 1/r^2.

Electromagnetic waves are similar. The Lienard-Wiechert field formulas [1] have 2 terms. The first term describes the delayed field, and is proportional to 1/r^2, while the second describes waves and is proportional to 1/r.

[1] https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_...