Effect of density stratification and the magnetic field variation on the period ratio P₁/P₂ and on the eigen functions of both the fundamental mode and the first overtone. Panel a: period ratio varying with L/πH. Panel b: similar to panel a, but with L/πD. Panel c: amplitude profiles of the fundamental mode for different values of L/πH, denoted by different line styles. Panel d: similar to panel c, but for different values of L/πD. Panel e: amplitude profiles of the first overtone for different values of L/πH. Panel f: similar to panel e, but for different values of L/πD. Panel g: Alfvén speed distribution for different L/πH. Panel h: similar to panel g, but for different L/πD. In panels a,c,e, we fixed L/πD = 0 to ensure that eigenmodes and eigenvalues are only affected by the density variation, and in panels b,d,f, we fixed L/πH = 0.