The modes of oscillation of a magnetic flux tube are explored, working from the fundamental differential equations obtained in Chapter 3. Sausage modes and kink modes (as in a magnetic slab) are investigated and their dispersion relations understood. Fluting modes also occur. Dispersion relations and diagrams, each similar to those arising in a slab, are derived and displayed, for both photospheric and coronal conditions. Leaky waves are explored. Resonant absorption in a flux tube is examined, with the decay rate obtained for a $\beta = 0$ tube. Two profiles of density across a thin layer on the boundary of the tube are explored, the linear profile and the sinusoidal profile, with decay rates obtained for both.

The differential equations in Cartesian geometry are solved for the magnetoacoustic waves in a magnetic slab. The case of a field-free environment is also investigated as is the $\beta = 0$ plasma. Sausage and kink waves arise and their properties are described. The notion of surface waves and body waves is introduced. Dispersion diagrams are displayed under two sets of conditions, the photospheric medium and the coronal medium. Impulsive waves are examined. Also, waves in smoothly varying profiles are explored, especially the Epstein profile. Cutoff frequencies are obtained for a range of profiles.

Surface waves are introduced, and the surface wave dispersion relation derived. Some general properties of this relation are investigated. Surface waves in certain special cases, including when one interface is field-free or when both sides of the interface are $\beta = 0$ plasmas are discussed in detail.

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.

The twisted flux tube is explored. Its dispersion relation is obtained for an incompressible plasma and examined in the special case of a thin tube. The case of a compressible medium is also discussed for small twist. The special case of a twisted annular region is also explored.

Here the fundamental problem of MHD waves in a uniform medium is discussed in detail, principally from the viewpoint of partial differential equations. The tube speed is introduced. Dispersion relations are obtained and their properties determined, as well as the properties of the perturbations. Two special cases are also discussed: the incompressible medium, and the $\beta = 0$ plasma.