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. ***
