A new third-order solution for bichromatic bi-directional water waves in finite depth is presented. Details of the complete solution can be found in a companion paper. In the present paper, we give the form of the solution, and focus the attention on the nonlinear dispersion relation, which incorporates the effect of an ambient current, with the option of specifying zero net volume flux. The dispersion relation is verified against classical expressions from the literature, and limitations and problems with these classical expressions are identified. Finally, third-order resonance curves for unidirectional and short-crested finite amplitude waves and their three-dimensional perturbations are calculated. Dominant class I and class II wave instabilities (determined by classical stability
analyses) are shown to be located close to these resonance curves.