The mechanism of the Aleutian islands earthquake of 1946 and the Sanriku earthquake of 1896 is studied on the basis of the data on seismic waves from 5 to 100 s and on tsunamis. These earthquakes generated, despite their relatively small earthquake magnitude, two of the largest and most widespread tsunamis in history. The data obtained at different periods are interpreted in terms of the effective moment, Me. The effective moment at a certain period is defined as a seismic moment of a virtual step function dislocation that explains the observation at this period. The effective moment of the tsunami earthquakes increases rapidly towards 0.5 to 1.0 × 10 29 dyne · cm as the period increases while, for ordinary earthquakes, it is more or less constant. This dependence can be explained in terms of a source deformation having a time constant of about 100 s. The Mc versus ƒ (frequency) diagram provides a diagnostic method of estimating the tsunami potential of earthquakes. If the M e-ƒ diagram for an earthquake has a steep upgrade towards low frequency implying an effective moment exceeding 10 28 dyne · cm at zero frequency, the earthquake has a high tsunami potential. Since the determination of the effective moment at various periods can be made by a simple procedure, this method could be incorporated in the tsunami warning system. The abnormal slow deformation at the source of the tsunami earthquakes may be a manifestation of viscoelasticity of a weak zone beneath the inner margin of the trenches. The weak zone which is implied by large normal-fault earthquakes such as the 1933 Sanriku and the 1929 Aleutian islands earthquakes may be a result of frictional heating at the interface between the oceanic and the continental lithospheres.