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

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 that 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.