Abstract: This study consists of three parts: Part Ⅰ. Calculations; Part Ⅱ. Global realdata analysis and Part Ⅲ. Global spectral model. In Part L, the present paper, it is presented to review and discuss the accuracy of calculating Bough function and frequency-the eigensolutions and eigenvalues of Iinearized shallow water model over a sphere, and a physical interpretation of Û, and is also given. The results of calculation will be applied to Part Ⅱ. and Part Ⅲ, which will be written as separate papers.