The problem of spontaneous shear rupture of a single circular asperity on an infinite fault plane is studied. Initially, the fault plane is broken everywhere except at a circular asperity. An applied displacement at infinity results in a stress concentration along the bounding edge of the asperity. The frictional stress on the broken part of the fault plane is taken to be a constant. Once a point on the asperity breaks, the stress there drops to the same value as on the 'main' fault surface. The rupture is started by relaxing the shear stress at a point on the asperity edge and is then allowed to propagate spontaneously, using a critical stress level fracture criterion. The rupture process is calculated numerically. It is found that for asperities of constant strength, the rupture first propagates around the edge of the asperity an then inward, a phenomenon best described by the well-known term of classic military maneuver: 'the double encircling pincer movement.' In the appendix, the expressions for the far-field seismic radiation due to the rupture of such an asperity are derived. It is shown that the nth Cartesian component of the far-field displacement at (x,t) for P, SV, and SH waves, using the notation of Aki and Richards (1980), is given by un(x,t) =(Dni/4&pgr;&rgr;c2R)∫∫S0&tgr;i3{&xgr;,t -<(R-&xgr;⋅&ggr;)/c>}dS(&xgr;). Thus the far-field pulses can be directly found from the stress drops on the fault plane. This formula is also true for 'crack' or 'dislocation' problems. The directivity function Dni for displacement for the asperity problem is found to be that for the double couple, modified by some factor. In particular, the fault plane is a nodal plane for SV waves. For the rupturing of asperities on a finite fault, these directivity functions are applicable only to the initial part of observed pulses at a receiver, provided the receiver is not located on the fault plane outside the broken part of the main crack edge, in which caseit is inapplicable, for all time.