New Methods and Analysis for Wave Propagation Problems

我们对波动现象的了解支撑着我们的社会所依赖的许多技术，例如雷达、声纳、移动电话、超声波、光纤和结构中的裂缝检测。许多波动现象可以用偏微分方程组(PDE)来数学描述；通过研究这些数学模型，可以获得关于物理过程的信息。数学家有一套技术工具包来研究偏微分方程并提取有用的信息。这个项目寻求使研究人员在开发过程中发挥关键作用的两种工具更加锋利，并将它们不仅相互结合，而且与来自不同数学领域的其他尖端技术相结合。这种结合将增加这些技术的力量，并允许数学家在新的情况下应用它们，进一步增加我们对波的知识。这个项目将研究的一些问题的例子包括：-从尖角和尖边的障碍物中散射声波和电磁波。这些问题在许多工程应用中具有重要意义，因此多年来得到了广泛的研究。然而，目前的数学工具仍然不足以解决许多重要的实际问题。波在所谓的超材料中的传播，人造材料被设计来产生自然界中找不到的特性，以及周期性介质的应用，这些材料在光通信中使用光子晶体。材料表面的裂缝的检测；这显然对测试许多工程结构的完整性很重要，特别是核反应堆和化学反应堆。

Our understanding of wave phenomena underpins many technologies upon which our society depends, for example, radar, sonar, mobile phones, ultrasound, optical fibres, and crack-detection in structures.Many wave phenomena can be described mathematically by Partial Differential Equations'' (PDEs); and information about the physical processes can be obtained by studying these mathematical models. Mathematicians have a toolkit of techniques to study PDEs and extract useful information. This project seeks both to sharpen'' two tools, which the investigator has played a key role in developing, and to combine them, not only with each other, but also with other cutting-edge techniques from different areas of mathematics. This combination will increase the power of these techniques and allow mathematicians to apply them in new situations, further increasing our knowledge of waves.Some examples of problems this project will investigate are:- The scattering of sound and electromagnetic waves from obstacles with sharp corners and edges. These problems are of fundamental importance to many engineering applications, and hence have been extensively studied for many years. However, the current mathematical tools are still not powerful enough to solve many important practical problems.- The propagation of waves through so-called meta-materials'', artificial materials engineered to produce properties not found in nature, and periodic media'', which have applications in photonic crystals used in optical communication.- The detection of cracks in the surface of materials; this is obviously important for testing the integrity of many engineering structures, but in particular nuclear and chemical reactors.