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We analytically explore the effect of falling matter on a sphericallysymmetric wormhole supported by a spherical shell composed of exotic matterlocated at its throat. The falling matter is assumed to be also a thinspherical shell concentric with the shell supporting the wormhole, and itsself-gravity is completely taken into account. We treat these spherical thinshells by Israel's formalism of metric junction. When the falling sphericalshell goes through the wormhole, it necessarily collides with the shellsupporting the wormhole. To treat this collision, we assume the interactionbetween these shells is only gravity. We show the conditions on the parametersthat characterize this model in which the wormhole persists after the sphericalshell goes through it.

What wormhole is traversable?: A case of a wormhole supported by a spherical thin shell