The theory of non-linear transresonant wave phenomena and an examination of Charles Darwin's earthquake reports

Abstract

SUMMARY A non‐linear theory of transresonant wave phenomena based on consideration of perturbed wave equations is presented. In particular, the waves in a surface layer of a porous compressible viscoelastoplastic material are considered. For such layers the 3‐D equations of deformable media are reduced to 1‐D or 2‐D perturbed wave equations. A set of approximate, closed‐form, general solutions of these equations are presented, which take into account non‐linear, dissipative, dispersive, topographic and boundary effects. Then resonant, site and liquefaction effects are analysed. Resonance is considered as a global parameter. Transresonant evolution of the equations is studied. Within the resonant band, utt≈a20∇2u and the perturbed wave equations transform into non‐linear diffusion equations, either to a basic highly non‐linear ordinary differential equation or to the basic algebraic equation for travelling waves. Resonances can destroy predictability and wave reversibility. Surface topography (valleys, islands, etc.) is considered as a series of earthquake‐induced resonators. A non‐linear transresonant evolution of smooth seismic waves into shock‐, jet‐ and mushroom‐like waves and vortices is studied. The amplitude of the resonant waves may be of the order of the square or cube root of the exciting amplitude. Therefore, seismic waves with a moderate amplitude can be amplified very strongly in natural resonators, whereas strong seismic waves can be attenuated. Reports of the 1835 February 20 Chilean earthquake given by Charles Darwin are qualitatively examined using the non‐linear theory. The theory qualitatively describes the ‘shivering’ of islands and ridges, volcano spouts and generation of tsunami‐like waves and supports Darwin's opinion that these events were part of a single phenomenon. Same‐day earthquake/eruption events and catastrophic amplification of seismic waves near the edge of sediment layers are discussed. At the same time the theory can account for recent counterintuitive results of experiments with water, liquified matter and granular materials.

Numerical Study of the Propagation of One-Dimensional Surface Waves in Sedimentary Rock Layers.

“…В частности, они усиливаются на краях долин, где толщина слоя земли, по которому волна распространяется, уменьшается. В результате сейсмическая энергия аккумулируется и земля начинает сильно колебаться [6,14,15].…”

“…В работах [6,7] автор построил новую теорию распространения нелинейных поверхностных волн. Теория строится на том факте, что во время землетрясений (во время первых сейсмических колебаний) некоторые поверхностные слои осадочных пород переходят в почти жидкое состояние, т.е.…”

“…Грубо говоря, после вибраций частички почвы начинают так слабо контактировать друг с другом, что слабо работают на сдвиг, однако сжатие (давление) они передают достаточно хорошо. Используя подобные идеи, автор работ [4][5][6][7][8][9][10] построил теорию распространения поверхностных волн в слоях ожиженного материала.…”