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Localization of harmonics generated in nonlinear shallow water waves.
Géraldine L Grataloup1, Chiang C Mei
1Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
Summary
Nonlinear shallow water waves interacting with random seabeds experience multiple scattering, affecting harmonic generation. This study models these effects, revealing how seabed disorder influences wave energy and propagation.
Area of Science:
- Fluid dynamics
- Oceanography
- Wave propagation
Background:
- Shallow water waves exhibit complex behaviors, especially nonlinear ones.
- Random seabed topography introduces significant challenges in wave propagation modeling.
- Understanding wave-seabed interactions is crucial for coastal and marine engineering.
Purpose of the Study:
- To investigate the impact of random seabed bathymetry on nonlinear shallow water wave propagation.
- To derive and analyze the nonlinear evolution equations governing harmonic amplitudes under multiple scattering conditions.
- To study the effects of spatial attenuation and localization on harmonic generation.
Main Methods:
- Application of the multiple scales method to derive nonlinear evolution equations.
- Incorporation of multiple scattering effects through linear damping terms with complex coefficients.
- Derivation of an equation for total wave energy.
- Numerical solutions of amplitude equations to analyze spatial attenuation.
Main Results:
- Multiple scattering from a random seabed significantly influences resonant interactions and harmonic generation.
- Seabed disorder introduces linear damping terms with complex coefficients into the wave evolution equations.
- The study quantifies the effects of spatial attenuation (localization) on harmonic generation through numerical simulations.
- An equation governing the total wave energy for a finite number of harmonics was derived.
Conclusions:
- Random seabed bathymetry leads to complex wave scattering phenomena.
- The derived models accurately represent the influence of seabed disorder on nonlinear wave dynamics.
- Numerical analysis provides insights into wave energy dissipation and localization in random environments.