In the context of the extensively studied problem for finite amplitude radial motion of incompressible isotropic hyperelastic cylinders, we consider the effect of material property discontinuity on the response of finite thickness multilayer tubes. This enables explicit characterization of the effect of layer differences (density mismatch, constitutive parameter mismatch (e.g. shear moduli), and interface location). Example demonstration is given in terms of the phase-plane treatment for free oscillation of a two-layer Mooney-Rivlin tube. Equivalent results are then obtained by a fully independent energy treatment which proceeds with no mention of the concept of stress. The second order equation of motion emerges both by a Lagrangian treatment and also by Hamilton's principle.