Exact solutions for standing kink modes of the longitudinally stratified   coronal loops
  The influence of longitudinal structuring on the fast kink modes of coronal loops is investigated. Analytical dispersion relations and mode profiles are derived for the second-order ordinary differential equation governing the z- component of the perturbation in the magnetic field, $\delta B_z$. All other components are given in terms of $\delta B_z$. Deviations from the frequencies and mode profiles of homogenous loops are given as functions of the density scale height. The effects of the fixed and variable column masses, negative scale heights, and density contrasts inside and outside of the loops are studied. The frequency ratios, mode profiles, and the antinode shifts from those of the sine profiles of the homogenous loops, are tools to estimate solar photospheric parameters. To this end, we have expanded the relevant factors up to the second order in the stratification parameter. In particular, we verify that the first overtone antinode shifts are in the Mm range and are within the reach of the resolutions of the present day observations. 
N. Dadashi, H. Safari, S. Nasiri, and Y. Sobouti