Various Aspects of the Water-Wave Problem

这项提案中概述的研究在欧拉方程的框架内解决了水波问题的两个不同方面。首先，我们将研究二维和三维周期行波的频谱稳定性。结果将使用Ablowitz、Fokas和Musslimani提出的水波问题的非局部公式进行数值和解析。其次，这项提议将开发从压力测量中重建表面波的方法。在以前与合作者合作的基础上，这项建议将推广压力和表面高程之间的关系，以允许在任意深度(不仅仅是底部)进行压力测量。此外，难以物理测量先验的物理参数将在没有近似的情况下从模型中正式删除。文中还将推导出渐近模型，并与数值实验进行比较。了解产生无赖海浪或海啸所需的物理条件有助于预测这些事件可能发生的时间。这项建议试图通过研究描述地表水波的数学模型来更好地了解这些物理条件。由于这些方程的复杂性，稳定性分析等技术增加了我们对波浪在一段时间内的行为的了解。这个项目将调查行波解的稳定性，以便更好地了解可能引起无赖波或海啸的物理参数和扰动类型。仅有这些信息在实际使用中是不够的。人们还必须依赖当前海洋状况的实时测量，作为任何预测或预报系统的一部分。作为这项建议的一部分，将确定从压力数据重建当前波高的快速而准确的方法，并将其与数值和现场数据进行比较。

The research outlined in this proposal addresses two separate aspects of the water-wave problem within the framework of Euler's Equations. First, the spectral stability of both two- and three-dimensional periodic traveling waves will be examined. Results will be obtained numerically and analytically using a nonlocal formulation of the water-wave problem due to Ablowitz, Fokas, and Musslimani. Second, this proposal will develop methods for the reconstruction of surface waves from pressure measurements. Building upon previous work with collaborators, this proposal will generalize the relationship between pressure and the surface elevation to allow for pressure measurements to be taken at arbitrary depths (not just the bottom). In addition, physical parameters that are difficult to physically measure a priori will be formally eliminated from the model without approximation. Asymptotic models will also be derived and compared with numerical experiments. Understanding the physical conditions necessary to generate rogue waves or tsunamis aids in predicting when these events could occur. This proposal seeks to better understand these physical conditions by investigating the mathematical models that describe surface water waves. Due to the complicated nature of these equations, techniques such as stability analysis increase our understanding of how waves behave over some period of time. This project will investigate the stability of traveling wave solutions in order to gain better insight into the physical parameters and types of perturbations that could give rise to rogue waves or tsunamis. Alone, this information is not sufficient in practical use. One must also rely on real-time measurements the current ocean state as part of any prediction or forecasting system. As part of this proposal, fast and accurate methods of reconstructing the current wave-height from pressure data will be identified and compared with numerical and field data.