Since gravitational waves are a type of propagation of energy of some sort, they ought to induce their own gravitational field. I'm assuming this extra gravitational force / curvature is independent from the wave itself, so there ought to be an observer that would 'feel' or 'observe' the wave pass, and in addition 'feel' or 'observe' a secondary attractive force towards the 'densest' part of the wave. I'm not sure if it is possible to discern these two effects without experiencing them as a whole but I'm assuming one could somehow.
Regarding the secondary attractive force, could this be used / amassed to a sufficient degree to allow two (or more) gravitational waves to orbit each other, with the extremal case being a gravitational standing wave arranged in a loop (held together by its own mass-energy)? Is such a system capable of collapsing into a black hole given enough energy in the wave(s) and sufficiently small orbit?
Answer
A theorized object called a geon, "an electromagnetic or gravitational wave which is held together in a confined region by the gravitational attraction of its own field energy", would seem like a match for what you're talking about. The wiki article mentions that exact solutions involving geons have been found (one is discussed in this paper), though it's not clear if they would be stable. For numerical simulations of various scenarios involving strong gravitational waves whose behavior is highly nonlinear, see this pdf. And here is a pdf of John Wheeler's original 1955 paper proposing them, which contains an image of a toroidal electromagnetic geon (he later notes that gravitational wave geons would be spherical rather than toroidal) consisting of two electromagnetic waves traveling in opposite directions, held to their distorted paths by their own gravity:
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