For Josephson junctions in sandwich geometry and in thin-film geometry we formulate the equations of nonlocal electrodynamics which take account of the influence of normal electrons in superconductors. On the basis of these equations we examine the spectrum and decay of generalized Swihart waves. We find that the decay of these waves by normal electrons becomes especially prominent in the short-wavelength region. Comparison of various dissipation mechanisms has revealed the conditions for efficient emission of electromagnetic waves from Josephson structures with finite dimensions.