An inverse nodal problem of a membran

We solve an inverse nodal problem for a rectangular membrane using two different algorithms. We call one algorithm the ratio method, and the other one the method of parameter identification. The algorithms use nodal positions and the corresponding natural frequencies to construct a piecewise constant approximation to the density distribution of the membrane. The ratio method approximates the value of the density at every point. The approximation is constant on subsets of each nodal domain. A nodal domain is the smallest connected domain enclosed by a zero set of the vibration mode of the membrane. Each constant in the approximation is the ratio of two eigenvalues. One is the principle eigenvalue on the nodal domain with the constant density equals to one. The other is the eigenvalue of the entire nodal domain when the membrane has the variable density. This eigenvalue divided by (2$pi)sp2$ is the square of a natural frequency for the full membrane with variable density, and is measured along with the nodal lines. The method of parameter identification approximates the density distribution also by a linear combination of piecewise constant functions. The constant values of these piecewise constant functions in this case are determined so that the principle eigenvalue on every given nodal domain is as close to the measured eigenvalue as possible.;The algorithms are numerically implemented. We use numerically simulated data in the implementation. Error in the data is assessed and error estimates given. The numerical experiments are conducted for densities that are perturbations of the constant density with constant equal to one. We consider two types of perturbations. For the first type of perturbation, we use a test function. The ratio method shows the presence of a perturbation even when the size of the subsets of the nodal domains is larger than the support of the test function. The other type of perturbation considered is a slowly varying polynomial perturbation. The ratio method gives good reconstruction of this density function. The method of parameter identification gives an approximation to this density which captures the characteristic features of the density.