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We study the effects of random rough surface on electromagnetic wave propagation in a parallel-plate waveguide excited by a line source. The second-order small perturbation method is applied to derive a closed-form expression for the coherent wave propagation and the power loss. The derived result is expressed in terms of a double Sommerfeld integral. The double Sommerfeld integral is evaluated by direct numerical integration and by approximate methods. The results of the two methods are shown to be in good agreement with each other. Numerical results of coherent wave propagation and power loss are illustrated as a function of roughness characteristics and waveguide thickness. The enhancement factors are compared with previous results of plane-wave scattering. The waveguide model and the plane-wave model are in good agreement for moderate to large waveguide thicknesses. For small waveguide thickness, the waveguide model shows significantly different power loss as compared to the plane-wave model. Full-wave simulations by the finite-element method are used to validate the second-order small perturbation in waveguide model.