This chapter addresses the eigenvalue problem (EVP) with a focus on its application to describing standing waves or stationary quantum systems. A numerical method known as the shooting method is introduced to solve the EVP. Using a cylindrical waveguide (e.g., optical fibre) as a system model, the normal modes of a scalar wave are explored. The wave equation is examined, with considerations made for axial symmetry, boundary conditions, and the influence of refraction coefficients. A significant part of the study is devoted to finding the normal modes and associated wave numbers in an optical fibre. The latter part of the chapter presents the shooting method, a recursive technique to ascertain the eigenvalue in numerical calculations. The applicability of this method is further examined in the context of a quantum well. This chapter offers a thorough exploration of the EVP, highlighting its relevance to real-world research and introducing a robust numerical method for its resolution.