Non-reflective propagation of kink waves in magnetic-flux tubes in the solar atmosphere

We study the non-reflective propagation of kink waves in inhomogeneous magnetic-flux tubes. We use the thin-tube and zero-beta plasma approximations. The wave equation with the variable velocity is reduced to the Euler–Poisson–Darboux equation. This equation contains one dimensionless parameter. There are two infinite sequences of this parameter, one monotonically increasing and the other monotonically decreasing, when exact analytical solutions for the Euler–Poisson–Darboux equation can be obtained. For the monotonically increasing sequences the Euler–Poisson–Darboux equation becomes the equation describing spherically symmetric waves in multi-dimensional spaces. The general results are applied to kink-wave propagation in coronal magnetic loops. We consider a coronal magnetic loop of a half-circular shape. We find that for a fixed loop height there is a one-parametric family of dependences of the loop cross-sectional radius on the coordinate along the loop corresponding to the non-reflective kink-wave propagation.