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Gravitational-wave memory (GW memory) is a lingering effect of some gravitational waves: things "don't return", e.g., the displacement of test masses by the passing of the wave (the gravitational wave strain) persists rather than return to no relative displacement. A gravitational wave might have this no-return effect, or might return space somewhat back to as before but not completely. Such a memory effect can be linear or nonlinear. A GW event with this "memory" property is termed a memory burst. The phrase burst gravitational wave may be taken to mean this, but also may be used for any GW event. (In context, either of these might be shortened to the word burst).
Some detectors are less likely to "see" the memory effects of gravitational waves. It is thought that pulsar timing arrays might be able to sense a memory burst from a binary SMBH merger.