Inverse construction of exact atmospheric models and the 3.7--4.4\,mHz acoustic resonances
Abstract
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(τ). 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.
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BibTeX
@misc{kaptsov2026inverse,
title = {{Inverse construction of exact atmospheric models and the 3.7--4.4\,mHz acoustic resonances}},
author = {O. V. Kaptsov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07073},
eprint = {2610.07073},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07073},
note = {CC0},
}
RIS
TY - UNPB
TI - Inverse construction of exact atmospheric models and the 3.7--4.4\,mHz acoustic resonances
AU - O. V. Kaptsov
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07073
UR - https://arxiv.org/abs/2610.07073
AB - 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(τ). 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.
ER -