We find new behaviour in a lattice with second-neighbour interactions. First we find solitary waves with oscillatory spatial decay, this is something that cannot occur in lattices with nearest-neighbour interactions alone. This is found by standard asymptotic analysis, which leads to an exact curve in parameter space where this behaviour starts. A number of highly accurate quasi-continuum approximations are derived and solved. One of these suggests a possible method for subsonic solitary waves to cease existing. An alternative method of approximation elegantly reveals the differences in shape between subsonic and supersonic solitary waves. This is based on the weak form of the differential - delay equation for the travelling wave, and is derived using the calculus of variations.