Inverse construction of exact atmospheric models and the 3.7--4.4\,mHz acoustic resonances

Large explosive eruptions and earthquakes excite atmospheric oscillations near 3.7 and 4.4,mHz that couple to the solid Earth. For vertically propagating linear acoustic–gravity waves in a hydrostatic ideal-gas atmosphere, the travel-time formulation gives a Klein–Gordon equation with a potential $Q(\tau )$. We establish three main results. First, for every potential admitting a positive zero-energy solution, the corresponding atmospheric profiles are constructed explicitly by quadrature; this extends the known reflectionless profiles of constant $Q$. Second, Darboux–Crum potentials give a smooth atmosphere that reproduces the U.S. Standard Atmosphere between 11 and 100,km to within 3,K and is exactly transparent to vertical waves above its cut-off; it supports no damped overtone near 4.4,mHz, and its ground-confined mode below the cut-off is perfectly trapped. The observed spectrum therefore requires a cut-off barrier of finite thickness connecting the lower atmosphere to the thermosphere. Third, a Rosen–Morse barrier gives an exactly solvable atmosphere with a closed-form hypergeometric resonance condition. Its resonances at 3.76 and 4.37,mHz, with leakage quality factors 173 and 20, explain the different damping of the two oscillations: the fundamental lies below the top of the mesospheric cut-off barrier and leaks by tunnelling, whereas the overtone lies above it. Full-wave computations for event-specific atmospheres reproduce the reported 3.7–3.8,mHz fundamental oscillation in all six events, of seven considered, in which such a mode was identified.