The aim of this work is to study the fundamental compressional (L(0,1)) Pochhammer-Chree mode interaction with non-axisymmetric damages in cylinders. To this end, experimental and numerical investigations of non-axisymmetric vertical cracks are considered. A non-contact magnetostrictive device is used for experimental investigations. Magnetostrictive transducers are used to generate and receive compressional guided waves. These are enabled by using an axisymmetric and longitudinal magnetic polarising field. Both, the incident and the reflected signals are acquired by the same receiver which allows a direct calculation of the reflected power flow. Different vertical cracks with various depths milled in steel cylinders are considered. The power flows are compared with those obtained by a three dimensional numerical method. This numerical method is based on a hybrid three dimensional (3D) approach combining the classical finite element (FE) method with the semi-analytical finite element (SAFE) technique. The near field surrounding the damage is analysed with the 3D FE method whereas transparent conditions are applied to the wave guide sections for the far field analysis. These transparent conditions are based on modal expansions on cross-sections. The SAFE technique is used to compute the eigenmodes. Eigenforces and modal power flows are post-processed on a straightforward way. First, the hybrid method is validated with published results in the literature obtained for a free-end cylinder. Finally, numerical and experimental results are compared with success.