Numerical versus analytical solutions for the linear loop geometry, uniform magnetic field, and stratified density profile. (a) Numerical results of Eq. (18). The blue solid line represents the solution of fundamental tone with eigenvalue ω₁ = 2.90, the orange dotted line indicates the first overtone with eigenvalue ω₂ = 5.57, and the green dashed line denotes the second overtone with eigenvalue ω₃ = 8.54, respectively. (b) Analytical results. The blue solid line represents the solution of fundamental tone with eigenvalue ω₁ = 3.37, the orange dotted line indicates the first overtone with eigenvalue ω₂ = 5.81, and the green dashed line denotes the second overtone with eigenvalue ω₃ = 8.26. Here, we choose .