"do these waves behave like ripples in water, so a single event generates multiple repetitive, concentric waves permeating through space?"
No they don't. There is a law known as [Huygens' principle](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_princi...) which says that when a disturbance at a particular point creates a wave, that wave only propagates on an outwards-expanding sphere that is centered at that point of disturbance, and does not produce any effect on the interior of that sphere. This was originally formulated for light waves, but it also holds for other kinds of waves, such as sound waves or, in this case, gravitational waves. What this means is that when you look at something that's far away you see a sharp image of exactly what happened there a short time ago (the time it had taken the light to reach you), whereas if the principle did not hold, each light source would have a small "echo" after it which would blur the image.
However, one of the reasons Huygens' principle holds is that the waves are propagating over three dimensions. In contrast, water waves only propagate over two dimensions, so Huygens' principle fails. That is why ripples continue to emanate from a spot even long after the disturbance there is over. More generally, Huygens' principle holds whenever the number of dimensions is odd and fails whenever the number of dimensions is even.
[Note: I may be wrong on why Huygens' principle fails for water waves. Water waves are actually pretty complicated compared to other kinds of waves and I am not knowledgeable in all the subtleties.]
No they don't. There is a law known as [Huygens' principle](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_princi...) which says that when a disturbance at a particular point creates a wave, that wave only propagates on an outwards-expanding sphere that is centered at that point of disturbance, and does not produce any effect on the interior of that sphere. This was originally formulated for light waves, but it also holds for other kinds of waves, such as sound waves or, in this case, gravitational waves. What this means is that when you look at something that's far away you see a sharp image of exactly what happened there a short time ago (the time it had taken the light to reach you), whereas if the principle did not hold, each light source would have a small "echo" after it which would blur the image.
However, one of the reasons Huygens' principle holds is that the waves are propagating over three dimensions. In contrast, water waves only propagate over two dimensions, so Huygens' principle fails. That is why ripples continue to emanate from a spot even long after the disturbance there is over. More generally, Huygens' principle holds whenever the number of dimensions is odd and fails whenever the number of dimensions is even.
[Note: I may be wrong on why Huygens' principle fails for water waves. Water waves are actually pretty complicated compared to other kinds of waves and I am not knowledgeable in all the subtleties.]