On the nodal sets of the eigenfunctions of certain homogeneous and nonhomogeneous membranes

The author finds conditions for the existence of eigenfunctions whose nodal sets consist of one line segment or two perpendicular line segments. Using nodal sets of eigenfunctions, a sufficient condition is found for the density function of a nonhomogeneous annular membrane that is a radial function. The author also shows how to use the nodal sets of eigenfunctions of some particular type and the corresponding eigenvalues to determine the density function of the annular membrane if its density function is a radial function.