An application of additional harmonic cavities operating at multiplies of the main RF system frequency of 400 MHz is currently under discussionin the framework of the High Luminosity LHC upgrade program [1, 2] . A structure consisting of two 800 MHz single cell superconducting cavities with grooved beam pipes coupled by drift tubes has been suggested for implementation. However, it is desirable to increase the number of single cells installed in one cryomodule in order to decrease the number of

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... transitions between "warm" and "cold" parts of the collider vacuum chamber. Unfortunately, it can lead to the appearance of higher order modes (HOM) trapped between the cavities. In order to solve this problem the methods of HOM damping with rectangular waveguides connected to the drift tubes were investigated and compared. We describe the results obtained for arrays of 2, 4 and 8 cavitiesin this paper.