Analytical Modeling To The Oscillation Of Water Waves In The Entire Wave Range Within A Rectangular Harbor Of Variable Depth

The occurrence of harbor oscillation comes from the interaction of water waveswithin and outside the harbor, once the frequency of incident waves from the outer seaequals to the natural frequency of the water waves within the harbor, the resonanceoccurs and the harbor may be agitated into oscillated state.Though the qualitative explanation of harbor oscillation is easy, and there are alsoseveral of numerical and experimental solutions, analytical solutions for the purpose ofquantitative analysis are not many. Almost all the previous analytical solutions arerestricted to infnite water depth [23], fnite constant water depth [28] or piecewiseconstant water depth [20,25] within the harbor. Recently, two analytical solutionsof harbor oscillation for variable seabed are obtained [29,32], but both of them arerestricted to the long wave range.This paper studies the oscillation within a narrow rectangular harbor with idealizedor quasi-idealized depth in the direction perpendicular to the coastline (denoted by x-axis direction) and with constant depth in another direction (y-axis direction). Thewater depth in the outer open sea is assumed to be constant, thus it is easy to construct afar-feld solution. The key step is to seek an analytical solution to the modifed mild-slopeequation (MMSE)[2] within the harbor. By using the technique of variable separation,the2D implicit MMSE is transformed into two ordinary diferential equations (ODE)in both x-and y-directions. The equation in the y-direction is easy to solve due to theconstant depth. But in the x-direction, the related ODE is implicit due to the variablewater depth. By employing the technique of variable transform [18], the implicit ODE inthe x-direction is transformed into an explicit equation which permits a series solutionto be constructed. Finally, by using the matching conditions at the harbor mouth forboth inner and outer solutions, the analytical formula of the wave amplifcation is given.The comparison among various solutions for a rectangular harbor with constant depthshows that the correctness of our solution. Based on the solution, the infuence of theharbor seabed and incident waves on longitudinal oscillation is analyzed.