According to a recently proposed vibrational interpretation of the Schrödinger equation for hydrogen-like atoms, stationary single-electron wave functions describe vibrations that are driven by phase waves associated with a bound electron moving on precessing orbits similar to orbits of the old quantum theory by Bohr and Sommerfeld. Based on this interpretation, the objective of this work is to provide insights into the structure of these wave functions by constructing similar wave functions from the corresponding electronic orbits and their associated phase waves without explicitly solving the three-dimensional Schrödinger equation. The proposed construction extends solutions of the two-dimensional Schrödinger equation into three dimensions and takes precession of the orbital angular momentum into account by a weighted average over all rotations about the quantization axis. The constructed wave functions are structurally similar to the standard wave functions of hydrogen-like atoms in spherical coordinates; thus, the proposed construction provides a new way of understanding the structure of these wave functions.