This study investigates the higher order hydrodynamic phenomena of a breaking wave impact on a vertical impermeable wall. During the collision, a small amount of air is entrapped between the water and the wall. Here, we idealize the problem under consideration and we assume that the hydrodynamic pressure in the air-pocket is zero and that the brief collision does not change significantly the volume of the entrapped air. In addition, no compression effects are taken into account. We use a perturbation method along with the small-time approximation to cope with the kinematic and the dynamic boundary conditions of the free-surface. This approach allows us to satisfy the free-surface boundary conditions on the still water level and to avoid satisfying them on the a-priori unknown free-surface elevation induced by the collision. The breaking wave is idealized in a logical manner, allowing the consideration of the problem in an analytical context. In this way, we treated two separate mixed boundary value problems (BVP). Clearly, the solution of the higher order problem dictates the solution of the leading order problem. The higher order problem leads to a substantially complicated Sturm–Liouville problem, with mixed conditions and consequently, requires to be treated in a sophisticated manner. The analysis is performed in the two dimensions.

这项研究调查了破碎波冲击垂直不透水墙的高阶流体动力学现象。在碰撞过程中，少量空气滞留在水和壁之间。在这里，我们将所考虑的问题理想化，并假设气袋中的流体动压为零，并且短暂碰撞不会显着改变截留空气的体积。此外，没有考虑压缩效果。我们使用微扰方法和小时间近似来处理自由表面的运动学和动态边界条件。这种方法允许我们在静止水位上满足自由表面边界条件，并避免在碰撞引起的先验未知自由表面高程上满足它们。破碎波以合乎逻辑的方式被理想化，允许在分析环境中考虑问题。这样，我们处理了两个单独的混合边值问题（BVP）。显然，高阶问题的解决方案决定了前导问题的解决方案。高阶问题导致了相当复杂的 Sturm-Liouville 问题，具有混合条件，因此需要以复杂的方式处理。分析是在两个维度上进行的。高阶问题导致了相当复杂的 Sturm-Liouville 问题，具有混合条件，因此需要以复杂的方式处理。分析是在两个维度上进行的。高阶问题导致了相当复杂的 Sturm-Liouville 问题，具有混合条件，因此需要以复杂的方式处理。分析是在两个维度上进行的。