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Sorry, "echo" wasnt the right term (hence my quotes).

What I am trying to ask is if these behave like concentric water ripples, where from a single event you get first one peak of a wave, followed by many more repeated concentric peaks gradually getting smaller in amplitude? It sounds like there is just a single momentary wavefront without any residual secondary waves? Why is that?

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So if you look at the waveform of the signal, there are in fact smaller ripples after the main event. However, how long these ripples take to settle afterwards to equilibrium is related to how quickly the waves propagate. In the case of ripples on a pond, those travel at about 1 m/s; these gravitational waves travel at the speed of light, roughly 300,000,000 m/s, so we should expect it to settle to equilibrium about 300,000,000 times faster. If it takes 60 seconds on a pond, we would expect the gravitational waves to settle in about 0.0000002 s, or 200 nanoseconds.

Note that this is a _very_ rough estimate, but it should give you an idea of the order of magnitude for the settling time.

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