Numerical Simulation of Wave Resonance in Harbour with Bottom Friction using MacCormack Method

Abstract

This study investigates harbour wave resonance through numerical simulations considering three distinct topographic shapes: rectangular, triangular, and semi-parabolic. The simulations examine the effects of both linear and nonlinear bottom friction on wave dynamics. The numerical model is based on the one-dimensional shallow water equations, solved using the MacCormack method, a second-order finite difference numerical approach known for its second order accuracy. The analytical solution, derived from prior studies using the separation of variables method, provides the fundamental resonance period of the wave and serves as a validation benchmark for the numerical scheme, ensuring the reliability of the results. The findings reveal that harbour configurations incorporating nonlinear bottom friction are significantly more effective in attenuating wave resonance compared to those with linear bottom friction. An increase in bottom friction values leads to a notable reduction in the wave amplitude growth rate, highlighting the critical role of friction in resonance mitigation. Among the topographic shapes analysed, the rectangular harbour exhibits the highest amplitude growth rate, followed by the semi-parabolic shape, while the triangular harbour demonstrates the lowest amplitude growth rate. The rectangular topographic harbour is most effective at dampening wave resonance because it exhibits the largest change in the rate of increase with each increment in bottom friction. These results underscore the importance of both topographic shape and bottom friction in influencing wave resonance behaviour within harbours.

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