## Magneto-hydrodynamic waveguides

### Magnetic Flux Tube

The 3-dimensional standing mode solutions of the linearized MHD equations in a cylindrical geometry for sausage, kink and fluting modes for upto \(n=6\), can be seen to the right. The visualizations are based on the solutions obtained by Edwin and Roberts (1983).

Here, \(n\) is the azimuthal wave number. For all cases the following parameters are used:

\( c_0 = 1, \: v_A = 2, \: v_{Ae} = 5, \: \frac{\rho_e}{\rho_0} = 0.21, \: \frac{p_e}{p_0} = 0.05\)
\(, \: \frac{B_e}{B_0} = 1.1456, \: \beta_0 = 0.3, \: \beta_e = 0.0114 \).

The parameters in a magnetic flux tube waveguide are defined as:

\(r_a\) |
- Flux tube radius. |

\(v_{ph}\) |
- Phase speed, |

\(c_0, c_e\) |
- Sound speed inside/outside the flux tube, |

\(v_A,v_{Ae}\) |
- Alfvén speed inside/outside the flux tube, |

\(c_k\) |
- Kink speed, |

\(c_{T},c_{Te}\) |
- Internal/external tube speed, |

\(\rho_{0},\rho_{e}\) |
- Plasma density inside/outside the flux tube, |

\(p_0,p_e\) |
- Gas pressure inside/outside the flux tube, |

\(B_{0},B_{e}\) |
- Magnetic field strength inside/outside the tube, |

\(\beta_{0},\beta_{e}\) |
- Plasma beta inside/outside the tube. |