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We investigate the spin wave spectra in a magnonic waveguide using the plane wave expansion method. The structure under the investigation has the form of thin strips of permalloy and cobalt, alternately arranged as a planar waveguide. The structure is assumed to be infinite in length and finite in thickness and width. Spin wave propagation is assumed along the length of the stripe, parallel to the external applied field, in a backward volume configuration. We derive both static and dynamic fields in the magnonic waveguide using the plane wave method, after reducing the linearized Landau-Lifshitz equation to an eigenfrequency problem. The eigenfrequencies corresponding to a wave vector are then numerically calculated and plotted, with the eigenmodes yielding the spatial variation in spin wave amplitudes. The demagnetizing fields, along the length and thickness, were derived from the magnetostatic potential and shows both bulk and edge mode characteristics. In a nonuniform demagnetizing field, low frequency spin waves concentrate their amplitude in a region of low internal magnetic field. These appear as standing wave excitations in the permalloy resulting in zero group velocity, or a flat band structure in the ω(k) dispersion diagram. Finally the dependence of the frequency band gap on the angle between spin wave vector and the applied field is also investigated.