Instability of exact finite-amplitude acoustic-gravity waves
Abstract
Godin (\textit{J. Fluid Mech.}, vol. 767, 2015, pp. 52--64) showed that compressible fluids in a uniform gravity field support finite-amplitude incompressible wave motions, in which every parcel conserves pressure and density, and constructed exact Gerstner-type surface and edge waves, possibly carried by currents. Using the short-wavelength instability method for compressible flows, we show that these waves are linearly unstable above a critical steepness. The stability of each fluid level reduces to a Floquet problem depending on two parameters: the steepness e=eka2 and ν2=(1−e2)ν02, where ν02=cos2α N02/Ω2, N0 is the buoyancy frequency of the hydrostatic reference state, α the inclination of the plane of motion and Ω=(gkcosα)1/2 the intrinsic orbital frequency. In an isentropic fluid, a level is unstable if e>1/3, recovering Leblanc's threshold for Gerstner waves (\textit{J. Fluid Mech.}, vol. 506, 2004, pp. 245--254) and extending it to compressible surface and edge waves; weak stable stratification lowers the threshold to 1/3−32/81ν02+O(ν04). In an isothermal atmosphere, ν02=κcosα/(kH), where H is the pressure scale height. For horizontal wavelengths near 5.5 H/cosα, principal subharmonic resonance makes the critical steepness vanish. For the fast airglow wave observed by Hecht et al. (\textit{J. Geophys. Res.}, vol. 107, 2002, D20, 4444), which Godin interpreted with his solution, the lidar temperature places the layer in which this interpretation is consistent on this resonance, where the critical steepness is typically 0.1 instead of 1/3. Instability is confined to a layer beneath the free surface of surface waves and to a wedge at the shoreline of edge waves.
Related
- Interior-fold and endpoint transitions in finite-depth capillary-gravity ship wakes
- An exact finite-state representation of viscoelastic memory for accelerated sea-level modeling: SLEvolve v1.0
- Effects of waveguide tube geometry and surrounding medium conditions on acoustic emission signal attenuation
- DEMONet: Underwater acoustic target recognition based on multi-expert network and cross-temporal variational autoencoder
- Automated photogrammetric close-range imaging of small invertebrates using acoustic levitation
BibTeX
@misc{puntini2026instability,
title = {{Instability of exact finite-amplitude acoustic-gravity waves}},
author = {Christian Puntini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35210},
eprint = {2610.35210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35210},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Instability of exact finite-amplitude acoustic-gravity waves
AU - Christian Puntini
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.35210
UR - https://arxiv.org/abs/2610.35210
AB - Godin (\textit{J. Fluid Mech.}, vol. 767, 2015, pp. 52--64) showed that compressible fluids in a uniform gravity field support finite-amplitude incompressible wave motions, in which every parcel conserves pressure and density, and constructed exact Gerstner-type surface and edge waves, possibly carried by currents. Using the short-wavelength instability method for compressible flows, we show that these waves are linearly unstable above a critical steepness. The stability of each fluid level reduces to a Floquet problem depending on two parameters: the steepness e=e^ka_2 and ν^2=(1−e^2)ν_0^2, where ν_0^2=cos^2α N_0^2/Ω^2, N_0 is the buoyancy frequency of the hydrostatic reference state, α the inclination of the plane of motion and Ω=(gkcosα)^1/2 the intrinsic orbital frequency. In an isentropic fluid, a level is unstable if e>1/3, recovering Leblanc's threshold for Gerstner waves (\textit{J. Fluid Mech.}, vol. 506, 2004, pp. 245--254) and extending it to compressible surface and edge waves; weak stable stratification lowers the threshold to 1/3−32/81ν_0^2+O(ν_0^4). In an isothermal atmosphere, ν_0^2=κcosα/(kH), where H is the pressure scale height. For horizontal wavelengths near 5.5 H/cosα, principal subharmonic resonance makes the critical steepness vanish. For the fast airglow wave observed by Hecht et al. (\textit{J. Geophys. Res.}, vol. 107, 2002, D20, 4444), which Godin interpreted with his solution, the lidar temperature places the layer in which this interpretation is consistent on this resonance, where the critical steepness is typically 0.1 instead of 1/3. Instability is confined to a layer beneath the free surface of surface waves and to a wedge at the shoreline of edge waves.
ER -