TG~waves~in~$\theta $-asymmetric magnetized plasma column.*

An unusual~$m_{\theta } =0$ TG wave~amplitude modulation due to the interaction with a~$m_{\theta } =2$ diocotron wave~is observed for the first time~in a~pure electron plasma column.~Trivelpiece-Gould (TG) modes describe longitudinal electrostatic plasma oscillation, modified by the cylindrical boundary, on an axially magnetized plasma column~[1]. Here we perform the first experiments on TG waves propagating in an azimuthally perturbed density (and potential) distribution~in~a pure electron plasma confined in a Penning-Malmberg trap. We~apply~a small $m_{\theta } =2$ diocotron~mode~to an~originally~axisymmetric equilibrium~density distribution $n_{0} (r)$, and~we~measure~the~amplitude modulation~of regular $m_{\theta } =0,\;k_{z} =1,\;m_{r} =1,2,3...$ TG modes:~in the Fourier space, this appears~as~a~wave triplets~consisting of a~central branch (continuation of the original mode) as well as of the upper~and~lower~sideband~shifted in frequency~by~$\Delta f_{SB} $.~The~splitting $\Delta f_{SB} /f_{TG} $ depends on the asymmetry strength $q_{2} $~(the quadrupole moment of the $n(r,\theta ))$ approximately as~$\sqrt {q_{2} } $.~Diagnostics also show two~triplets signals near the~TG wave, with frequency$_{\mathrm{~}}f_{TG} \pm f_{E\times B} $,$_{\mathrm{~}}$each with~$\Delta f_{SB}_{\mathrm{~}}$also related to~$\sqrt {q_{2} } $.~These features in the interaction of two orthogonal waves (diocotron and TG)~are not~yet~explained by nonlinear waves coupling theory.