We hereby develop a theoretical framework for analyzing Fluid Structure Interaction (FSI) waves propagation occurring in liquid filled pipes to manage a large family set of boundary conditions (e.g. junctions coupling effects). A self-adjoint operator theory framework leads to the analytical derivation of a transcendental equations for operator’s spectrum. The latter provides the system’s natural resonant frequencies as well as permit to find the discrete mode orthogonal basis decomposition. This theoretical framework also permits to demonstrate that the spectrum is uniquely composed into simple eigenvalues enabling explicit time-domain solutions from inverse-Laplace transform. The analysis is directly conducted in the time-domain but the obtained spectrum also applies to Fourier transformed frequency analysis. The obtained analytical solutions are successfully confronted with numerical simulation obtained using the Method of characteristic (MOC) for the same four equations (FSI) model on the very same configurations. The spectrum sensitivity matrix is also explicitly evaluated.