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Abstract The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearityof the Korteweg–de Vries equation and the full linear dispersion relationof unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux functionand a general linear dispersion relation. Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of generalandestablishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability.
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Investigating Spectral Distortion of Local Volcano Infrasound by Nonlinear Propagation at Sakurajima Volcano, Japan
Abstract Sound waves generated by erupting volcanoes can be used to infer important source dynamics, yet acoustic source‐time functions may be distorted during propagation, even at local recording distances (15 km). The resulting uncertainty in source estimates can be reduced by improving constraints on propagation effects. We aim to quantify potential distortions caused by wave steepening during nonlinear propagation, with the aim of improving the accuracy of volcano‐acoustic source predictions. We hypothesize that wave steepening causes spectral energy transfer away from the dominant source frequency. To test this, we apply a previously developed single‐point, frequency domain, quadspectral density‐based nonlinearity indicator to 30 acoustic signals from Vulcanian explosion events at Sakurajima Volcano, Japan, in an 8‐day data set collected by five infrasound stations in 2013 with 2.3‐ to 6.2‐km range. We model these results with a 2‐D axisymmetric finite‐difference method that includes rigid topography, wind, and nonlinear propagation. Simulation results with flat ground indicate that wave steepening causes up to2 dB (1% of source level) of cumulative upward spectral energy transfer for Sakurajima amplitudes. Correction for nonlinear propagation may therefore provide a valuable second‐order improvement in accuracy for source parameter estimates. However, simulations with wind and topography introduce variations in the indicator spectra on order of a few decibels. Nonrandom phase relationships generated during propagation or at the source may be misinterpreted as nonlinear spectral energy transfer. The nonlinearity indicator is therefore best suited to small source‐receiver distances (e.g.,2 km) and volcanoes with simple sources (e.g., gas‐rich strombolian explosions) and topography.
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- PAR ID:
- 10449120
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Journal of Geophysical Research: Solid Earth
- Volume:
- 125
- Issue:
- 3
- ISSN:
- 2169-9313
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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