Asymptotic behavior of the eigenvalues of the Steklov problem on a junction of domains of different limiting dimensions

The asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem on a junction of rectangles: a thin rectangle with a width of $\varepsilon>0$ and a rectangle with unit dimensions, is studied. In addition to asymptotic formulas for the main series of eigenvalues (in the low-frequency region), other series with stable characteristics are found in the medium-frequency region and explicit formulas for the correction terms are derived. In the framework of the linear theory of surface waves, the results of this work describe the effect of wave localization in shallow water.