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The equations governing traveling-wave interaction between an electron beam and a slow-wave circuit are formulated in terms of amplitudes of circuit mode and slow and fast space charge modes. The resulting equations are solved to find expressions for the matrix which relates the mode amplitudes at the output of the traveling-wave coupler to the mode amplitudes at the input. The properties of this matrix are discussed and numerical values given for Kompfner Dip. Matrices for velocity jumps and drift regions are also given, and the characteristics of couplers which are preceded by or followed by a drift region and velocity jump are discussed. It is shown that necessary and sufficient conditions for the removal of beam noise from the fast space-charge wave by any lossless coupler are that, for a circuit input, there be no circuit output (M11 = 0) and no slow space-charge wave output (M21 = 0). These results are then applied to the design of fast space-charge wave couplers for longitudinal beam type parametric amplifiers.