Determination of Thickness of an Inaccessible Thin Film under a Multilayered System from Natural Frequencies

We investigate the relationship between natural frequencies of a multilayered system of different elastic materials and the thickness of the undermost thin film. The natural frequencies are numerically calculated from the reflection
coefficient of a sample system of ``steel--epoxy resin--aluminium--thin polymer' with normal incidence. Strain energy ratio is defined and calculated to give the physics explanation why some frequencies are sensitive to thickness of the thin film in certain range. Experiments of three specimens indicate that the measured natural frequencies agree well with the theoretical ones. It is found in our experiments that the ratio of the lowest film thickness to wavelength is about 1/5. The average relative errors for the inverted polymer film thicknesses are found to be 11.8%, -4.8% and -1.3%, respectively.