Some singular, nonlinear, integral equations arising in physical problems

We show that if $Q$, the logarithmic decrement of plane wave intensity with position, is measured as a function of frequency, then the in-phase and quadrature components of the medium’s characteristic response function (in the form of a dielectric function for electromagnetic waves, an elastic response function for acoustic waves, etc.) satisfy a nonlinear, singular, integral equation with the in-phase and quadrature components connected through a causal Hilbert transform pair.