Mathematical Sciences: Inverse Nodal Problems and Perturbation Theory in Higher Dimensions

[9410700 . McLaughlin]这个项目涉及的问题是确定弹性膜的哪些性质可以从膜的固有频率和节点线的知识中推断出来。当膜以固有频率被激发时，结线是膜上没有激发的地方。这个项目有两个主要部分。首先是得到固有频率和振型的摄动结果。由于频率在高维问题中不是很好地间隔，所以必须解决小因子问题。该项目的第二个主要部分与这样一个事实有关：节点域（由节点线定义的膜中的连接域）可以是长而薄的条带，其直径与膜的直径一样大。我们需要的是定义近似的节点域，当特征值的阶数趋于无穷时，它的直径趋于零。本研究的目的是建立从间接测量中发现振动系统特性的方法，特别是对固有频率和节点线的测量。固有频率可以通过脉冲响应数据的频谱分析来确定。当膜以固有频率激发时，可以通过将激光对准振动表面来测量结线。后向散射中多普勒频移最小的线是节点线。根据这些数据，可以根据这些量的显式公式确定膜上外力的振幅和膜的（非恒定）密度的表达式。这些公式对于获得有效的数值算法来确定所讨论的量是非常有用的。＊＊＊

9410700 McLaughlin This project is concerned the problem of determining which properties of an elastic membrane can be deduced from knowledge of the natural frequencies and the nodal lines of the membrane. The nodal lines are places on the membrane where there is no excitation when the membrane is excited at a natural frequency. There are two major parts to the project. The first is to obtain perturbation results for the natural frequencies and mode shapes. Since the frequencies are not well spaced in higher dimensional problems, small divisor problems must be solved. The second major part of the project is related to the fact that nodal domains (connected domains in the membrane defined by the nodal lines) can be long, thin strips whose diameter is as large as the diameter of the membrane. What is required then is to define approximate nodal domains whose diameter goes to zero as the order of the eigenvalue goes to infinity. The goal of this research is to establish methods for finding properties of vibrating systems from indirect measurements, in particular, measurements of natural frequencies and nodal lines. The natural frequencies can be determined by spectral analysis of impulse response data. The nodal lines can be measured by directing a laser at the vibrating surface when the membrane is excited at a natural frequency. The lines where the Doppler shift in the backscatter is minimized are the nodal lines. From this data the amplitudes of external forces on the membrane and an expression for a (nonconstant) density of the membrane may be determined from explicit formulas for these quantities. These formulas are expected to be very useful for obtaining efficient numerical algorithms to identify the quantities in question. ***