q -Advanced Models for Tsunami and Rogue Waves

Abstract

A wavelet $K_q(t)$, that satisfies the q-advanced differential equation $K_q^{\prime }(t)=K_q(qt)$ 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 $F_q(t,x)=K_q{(t)}_q\text{S}\text{i}\text{n}(x)$. The profile $F_q$ is similar to tsunami models in present use. The function ${}_q{}^{\hspace{0.17em}}\mathrm{Sin}\hspace{0.17em}{(t)}^{\hspace{0.17em}}$ is a wavelet that satisfies a q-advanced harmonic oscillator equation. It is also shown that another wavelet, ${}_q{}^{\hspace{0.17em}}\mathrm{Cos}\hspace{0.17em}{(t)}^{\hspace{0.17em}}$, 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 $K_q$ and ${{}_q{}^{\hspace{0.17em}}\mathrm{Cos}\hspace{0.17em}}^{\hspace{0.17em}}$ is examined on actual data.