The interpretation of the bispectrum and bicoherence for non-linear interactions of continuous spectra

The authors extend the discussion of the bispectrum and bicoherence to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension. They evaluate the bispectral density in terms of the dielectric function of the driven mode and the dimensionless coupling constant, and point out that if the latter can be assumed to be real, the phase of the bispectrum can be used to distinguish resonant from non-resonant modes; the authors then evaluate the power in the driven mode separately for the resonant and non-resonant cases, to give results in a form suitable for comparison with experiment. They then consider the bicoherence, and show that this statistic is of limited value for continuous spectra, since its maximum value is depressed in an ill-defined way. On the other hand, the normalisation inherent in the definition does enhance weak regions of the spectrum and can show up interactions too weak to be readily detected in the bispectrum itself.