Source time functions are obtained for the December 19, 1982, Tonga and October 20, 1986, Kermadec earthquakes from deconvolution of P waves, using a model including a significant dip of the ocean bottom. In such a geometry, the amplitudes of subsequent water multiples are large and irregular. As a result, we can model these two events as consisting of a single pulse of short duration (23 s in the case of the 1982 event, and 19 s for the 1986 one), despite complex teleseismic waveshapes. This is in contrast to inversions using flat-layered models, which required very complex source time functions, which were mutually inconsistent at different stations. The anomalously large tsunami generated by the 1982 event is modeled as the result of propagation of the rupture into the sedimentary wedge at the trench, rather than due to a long-lived complex source time function, the latter being also imcompatible with the total lack of excitation of the Earth's normal modes in the low-frequency limit. For the 1986 event, we derive a source duration (19 s) very short in relation to its moment (8¿1027 dyn-cm). This earthquake, a high stress drop rupture featuring a unique combination of strike-slip and thrust motion on a plane dipping oceanwards of the trench, is interpreted as expressing internal deformation of the subducting Pacific slab at a depth of 40 km, under the compressional stresses resulting from the collision. ¿ American Geophysical Union 1989