The authors present a description of solution space of dispersion relation for capillary and dilational waves using complex analysis. By allowing the wavenumber, frequency and rheological parameters in the dispersion relation to be complex, it is demonstrated that both types of resonance phenomena, local peaks (troughs) in damping rates and mixed-mode behaviour, can be attributed to the occurence of a ‘root coalescence point’ in the resulting fully complex solution space. Using this complex description, the authors demonstrate how approximate values of surfactant rheological parameters over a specific frequency range can be determined from measurements of the capillary wave mode.