Abstract The scaled boundary finite-element method is a semi-analytical fundamental-solution-less boundary-element method based solely on finite elements. Using the simplest wave propagation problem and discretizing the boundary with a two-node line finite element, which preserves all essential features, two derivations of the scaled boundary finite-element equations in displacement and dynamic stiffness are presented. In the first, the scaled-boundary-transformation-based derivation, the new local coordinate system consists of the distance measured from the so-called scaling centre and the circumferential directions defined on the surface finite element. The governing partial differential equations are transformed to ordinary differential equations by applying the weighted-residual technique. The boundary conditions are conveniently formulated in the local coordinates. In the second, the mechanically based derivation, a similar fictitious boundary is introduced. A finite-element cell is constructed between the two boundaries. Standard finite-element assemblage and similarity lead to the scaled boundary finite-element equations after performing the limit of the cell width towards zero analytically.