This paper studies the numerical stability of method of boundary integral type, in which the free surface is tracked explicitly. The focus is on two-dimensional motions, periodic in the horizontal direction, so that issues of boundary conditions for the free surface can be avoided. The case considered is rather special, but analysis has provided a definitive answer in this case, and the treatment of this case might partially clarify the numerical issues in the more realistic problems. The results are presented in detail. A calculation of an overturning wave illustrates the resolution made possible by a fully stable numerical method. A careful study of an overturning wave by method of this class was presented previously.