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A new computationally efficient 2-D model of the beam-wave interaction in coupled-cavity traveling-wave tubes (CC-TWTs) has been developed. The model provides self-consistent time-dependent solutions of Maxwell's equations together with a fully relativistic solution of the electron equations of motion. The model is based on different treatments of the RF fields in the region occupied by an electron beam and in the region of the coupled-cavity structure. The RF fields inside the beam tunnel are represented as a sum of eigenmodes of the local cross section of the beam tunnel. The fields outside the beam tunnel are represented as a superposition of modes of an equivalent circuit with lumped capacitors, inductors, and resistors. The model has been implemented in the TESLA-CC code. The results of the code predictions agree well with measured data for a wideband CC-TWT operating in the Ka-band. The code also shows good agreement with predictions of the 1-D code CHRISTINE-CC in regimes in which a 1-D approximation is applicable. A numerical study of CC-TWT operation shows that, in the small-signal regime, the code is able to predict a gain enhancement due to transverse motion at focusing magnetic fields comparable with Brillouin equilibrium values, which is the major 2-D effect. In the large-signal regime, the code is also capable of treating cases in which the transverse displacement of electrons becomes large and of determining the dependence of the spent beam energy distribution on radial position.