He-3 films adsorbed on the atomically flat surface of graphite provide a model system for the study of two-dimensional magnetism on a triangular lattice. We have made a study of the regime in which the T = 0 ground state of the second He-3 layer is a fully polarized ferromagnet. NMR, using broadband SQUID detection, at a range of low fields above the spin-flop transition, and over a wide temperature range 0.3-200 mK, has enabled us to disentangle the influence of sample finite size effects and magnetic field on the spin-wave spectrum. We demonstrate that the spin-wave spectrum is governed by a different effective exchange constant than that determining the high temperature magnetism. This is understood in terms of frustrated atomic ring exchange.