# Boussinesq Equation

Numerical Model of Breaking Regular Wave Runup Propagation by Using 1D Boussinesq Equation

 Yuliarko Sukardi B.E. Ocean Engineering Institute Technology of Bandung M. Eng. Coastal Engineering University of Tokyo

Numerical Model of Breaking Regular Wave Runup Propagation by Using 1D Boussinesq Equation
This study is concerned with the development of a high-order numerical model to solve incompressible water wave motion based on improved nonlinear dispersive Boussinesq equations. A third-order Adams-Bashforth and a fourth-order Adams-Mouton predictor-corrector scheme was selected in an attempt to eliminate the truncation error terms that would be of the same form as the dispersive terms in the Boussinesq equations with second order schemes as in many other studies.
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### Simulation of Non-breaking Regular Wave Runup Propagation on a Sloping Beach

As a first check of the moving boundary model, a monochromatic wave train is let to runup and rundown a plane beach. The model is compared with the analytic solution derived by Carrier and Greenspan (1958) for monochromatic long wave runup on a constant slope. Their derivation makes use of the non-linear shallow water equations, and thus for consistency the dispersive terms will be ignored in the numerical simulations for this comparison.

Simulation of non-breaking regular wave runup propagation on a sloping beach is shown below

Figure below shows a comparison between analytical (Carrier and Greenspan, 1958) and numerical solution of vertical shoreline movement. The analytic solution is indicated by a red line while the numerical solution is indicated by a blue line. It is concluded that non-breaking regular wave runup is accurately predicted by the proposed model, yielding a validation of the moving boundary technique.

Simulation of Non-breaking Regular Wave Propagation on a Constant Depth

Simulation of Non-breaking Regular Wave Propagation on a Sloping Beach up to Breaking Location

Simulation of Breaking Regular Wave Propagation on a Sloping Beach up to Very Shallow Water Depth

Simulation of Breaking Regular Wave Runup Propagation on a Sloping Beach

BOUSSINESQ EQUATION's CONTENT

Joseph Valentin Boussinesq
He (born March 13, 1842 in Saint-André-de-Sangonis (Hérault département), died February 19, 1929 in Paris) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.

Master Research
This study was supported by the Asian Development Bank-Japan Scholarship Program.
Numerical Model of Breaking Regular Wave Runup Propagation by Using 1D Boussinesq Equation

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