Self-excited combustion instabilities of the first tangential mode have been observed in a research combustor operated with hydrogen and oxygen in cryogenic state as propellants. A detailed analysis of this unstable mode is conducted and the experimental observations are compared with numerical simulations. A high fidelity Large-Eddy Simulation together with an acoustic eigenmode computation performed with a Helmholtz solver give complementary information on the mechanisms driving the instability. A methodology for the 3D reconstruction of the acoustic field is presented