The thin tube theory for a kink wave in a stratified flux tube is determined and explored in the case when the tube is unstratified. Perturbations are also considered for this case. Using a multiple scales approach, the wave equation is derived for the kink mode of a thin magnetic flux tube in an unstratified atmosphere, demonstrating the importance of the kink speed. The theory is illustrated for standing waves in a uniform loop and also extended to structured loops with non-uniform density along the structure. Two density profiles are considered in detail. Period ratios for standing waves under coronal conditions are explored. The role of a non-uniform magnetic field is explored, and leads to a wave equation with non-uniform kink speed. Dispersive corrections in a uniform tube are examined and compared with earlier results. Gravity effects are also examined.

The thin tube theory for a sausage wave is developed from first principles and shown to lead to the Klein-Gordon equation. The equations that hold when gravity is negligible are explored and the dispersion relation obtained and compared with earlier results. The effects of stratification are explored in detail, with contributing terms to the cutoff frequency explored for various cases ranging from a rigid and straight tube to a diverging elastic tube of the shape expected for a thin flux tube. The cutoff frequency is illustrated for a range of conditions likely to arise in the solar atmosphere. The role of cutoff for coronal loops is explored.