Nonlinear resonant mode conversion and the Kramers–Kronig relations

Understanding Acoustics: An Experimentalist’s View of Sound and Vibration by Steven L. Garrett is extraordinary for the spectrum of topics covered: both theoretical and experimental. As an example I will discuss Steve’s work on resonant mode conversion in liquid helium; a nonlinear process whereby a propagating temperature wave can create a first sound or pressure wave. This research established the theory and demonstrated its measurement. Steve loved to quote Izzy Rudnick’s overarching insight that the experiment should work because “Helium is more nonlinear than water.” Resonant mode conversion also exists for a sound wave interacting with a spectrum of sound; a topic pioneered by Landau which surprisingly is not in his monograph on Fluid Mechanics. Another topic in Steve’s book that is surprisingly missing from Landau’s hydrodynamics is the Kramers–Kronig relations. Using the clear exposition in Steve’s book as the jumping-off point I will discuss difficulties that appear when one tries to unify resonant mode conversion with the Kramers–Kronig relations.