For us to have a non-trivial solution, the determinant of the matrix must be 0. This leads us to the following expression: cos ⁡ ⁡ ( k a ) = cos ⁡ ⁡ ( β β b ) cos ⁡ ⁡ [ α α ( a − − b ) ] − − α α 2 + β β 2 2 α α β β sin ⁡ ⁡ ( β β b ) sin ⁡ ⁡ [ α α ( a − − b ) ] .