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The scaled boundary finite element method (SBFEM) is developed for the solution of waveguide eigenvalue problems, which combines the advantages of the finite element method and the boundary element method. A new variational principle formulation to derive the SBFEM equations for waveguide is developed. An equation of the dynamic stiffness matrix for waveguide representing the relationship between the `flux` and the longitudinal field components at the discretised boundary is established. A continued fraction solution in terms of eigenvalue is obtained. By using the continued fraction solution and introducing auxiliary variables, the flux`longitudinal field relationship is formulated as a system of linear equations in eigenvalue then a generalised eigenvalue equation is obtained. The eigenvalues of rectangular, L-shaped, vaned rectangular and quadruple corner-cut ridged square waveguides are calculated and compared with analytical solution and other numerical methods. The results show that the present method yields excellent results, high precision and less computational time and rapid convergence is observed.