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

Simulation of breaking regular wave propagation on a sloping beach up to very shallow water is computed in order to test wave breaking model. Figure below shows numerical experiment setup where initial wave properties and bathymetry condition are described. At the left boundary, time history of wave elevation and horizontal water particle velocity based on 5th order Stokes wave are applied.

In simulation below wave transformation increasing in their wave heights during propagate to shore are well simulated. With further decrease in water depth an asymmetrical wave profile are also clearly observed and once the wave exceed breaking location, their height is decreasing due to energy dissipation.

Although there is a reflected wave due to introduction of discontinuity of dissipation breaking term, comparison between numerical and analytical solution show fairly good agreement. The analytic solution is indicated by a red line while the numerical solution is indicated by a blue dotted-line.

Secondary effects like wave setdown seaward of the breaker and wave setup landward of the breaker are also clearly indicated. The minimum mean sea water level is about 5 % of wave height at the breaking location which is also inline with nonlinear properties derived from conservation equation.

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 Non-breaking Regular Wave Runup Propagation on a Sloping Beach

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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