Dynamical Models for the Interaction of Shocks with Dispersive Waves

点击翻译按钮获取中文摘要

In this work, the awardee will study models for the propagation of shock waves through a dispersive environment. These kinds of nonlinear, multi-mode interactions occur in a wide range of natural settings. One example arises when acoustic shocks from an underwater explosion propagate through the ocean and interact with internal dispersive waves or surface gravity waves. Another example arises in medical applications where acoustic waves and shocks are used in the destruction of undesirable objects such as kidney stones. A third arises when shock waves due to supernova explosions propagate through the interstellar plasma media. Despite the fact that these three physical settings are characterized by vastly different spatial scales and disparate physical mechanisms, there are important underlying mathematical similarities that unite them. Each is characterized by a multi-component amplitude equation governing a fast-moving hyperbolic wave (shock), interacting nonlinearly with a slowly-moving dispersive wave. This wide difference of timescales, characterized by the ratio of dispersive group velocity to shock speed, forms a small parameter which allows one to perform a multi-scale asymptotic analysis of the interaction process. The physical problem that the awardee will focus on is an example of a shock-dispersive wave system that arises when a supersonic object travels over the ocean surface. The shock wave associated with a sonic boom leaves a pressure `footprint' on the water's surface. The interest is in modelling the interaction of this shock with the nonlinear ocean surface waves in order to understand the penetration pressure below the ocean surface. It is suspected that the finite amplitude surface waves can act as focusing `acoustic lenses', which in principle can increase the noise level below the surface and create `hot spots' of more intense acoustic energy. Such problems are of interest to oceanographers and marine biologists who are concerned with the amount of noise propagating below the ocean surface and its influence on marine life. The model equations that will be studied are interesting in their own right and capture important physical mechanisms requiring new mathematical tools to analyse and understand them.