Abstract
A wavelet Kq(t), that satisfies the q‐advanced differential equation for q > 1, is used to model N‐wave oscillations observed in tsunamis. Although q‐advanced ODEs may seem nonphysical, we present an application that model tsunamis, in particular the Japanese tsunami of March 11, 2011, by utilizing a one‐dimensional wave equation that is forced by Fq(t, x) = Kq(t)qSin(x). The profile Fq is similar to tsunami models in present use. The function is a wavelet that satisfies a q‐advanced harmonic oscillator equation. It is also shown that another wavelet, , matches a rogue‐wave profile. This is explained in terms of a resonance wherein two small amplitude forcing waves eventually lead to a large amplitude rogue. Since wavelets are used in the detection of tsunamis and rogues, the signal‐analysis performance of Kq and is examined on actual data.
Highlights
Tsunami or maremoto waves occur in response to earthquakes or landslides on the seafloor of large bodies of water, as discussed in 1–4
An objective of this paper is to demonstrate that the modeling as well as the detection and analysis of an observed wave profile can be achieved efficiently in terms of the wavelets 2.2 - 2.3, see 18–20
The paper is completed with the details of a perturbation analysis in q > 1 that is needed to establish the existence of a resonance for rogue waves
Summary
Tsunami or maremoto waves occur in response to earthquakes or landslides on the seafloor of large bodies of water, as discussed in 1–4. The consequential runup to the shore is such that the tide goes out, returns as a large surge, only to be followed by several diminishing cycles of similar events 5 An understanding of this behavior involves a consideration of the effects from the seafloor near the shore where the wave velocity decreases 6, 7. Tsunami and rogue waves are perturbations of the water-surface elevation function η t, x, y. −α u, v η Q, where H x, y is the depth of the water, w ≡ ∂η/∂t is the vertical velocity of the wave surface, Px and Py are variable external forcings, and Q is a mass source term. An objective of this paper is to demonstrate that the modeling as well as the detection and analysis of an observed wave profile can be achieved efficiently in terms of the wavelets 2.2 - 2.3 , see 18–20. The paper is completed with the details of a perturbation analysis in q > 1 that is needed to establish the existence of a resonance for rogue waves
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