  We analytically explore the effect of falling matter on a spherically
symmetric wormhole supported by a spherical shell composed of exotic matter
located at its throat. The falling matter is assumed to be also a thin
spherical shell concentric with the shell supporting the wormhole, and its
self-gravity is completely taken into account. We treat these spherical thin
shells by Israel's formalism of metric junction. When the falling spherical
shell goes through the wormhole, it necessarily collides with the shell
supporting the wormhole. To treat this collision, we assume the interaction
between these shells is only gravity. We show the conditions on the parameters
that characterize this model in which the wormhole persists after the spherical
shell goes through it.
