Eigenfrequencies of a rectangular atomic force microscope cantilever in a medium

We calculate the eigenfrequencies of a rectangular cantilever of an atomic force microscope immersed in a fluid or a gas. To do so, the problem of combined elastomechanical and hydrodynamical equations is solved approximatively. The results are compared with experimentally obtained frequencies. For water the difference between experiment and theory is less than 4% if the ratio of the height of the cantilever to the width is less than $\text{1/20}$ and if the corresponding eigenmode has at least four nodes.