Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number 5 N = O ( ϵ− 1) of ϵ ‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order O ( ϵ ) . The density of the junction is of order O ( ϵ− α) on the rods from the second class and O ( 1 ) outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ϵ  → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ϵ  → 0, namely the case of ‘light’ concentrated ( α  ∈ (0,1)), ‘middle’ concentrated ( α  = 1), and ‘heavy’ concentrated masses ( α  ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated ( α  ∈ (1,2)), ‘intermediate heavy’ concentrated ( α  = 2), and ‘very heavy’ concentrated masses ( α  > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α  ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.