One-dimensional numerical simulations of interacting plane-polarized Alfvén waves including magnetic diffusion and thermal energy dissipation form structures between the sources of excitation that are analogous to clumpy cool diffuse clouds surrounded by a warm low-density intercloud medium. The cloud oscillates slightly after formation but has an average state that is in both thermal and total pressure equilibrium with the intercloud medium. The cloud substructure has a realistic size-line width relation, Δv R^0.5, that is the result of a systematically decreasing correlation among random motions for increasingly large scales. The substructure also has a peak-density versus size relation ρ R^-0.18 because of self-similar hierarchical structure. The gas continuously changes its temperature in response to density fluctuations in the turbulent medium, i.e., each region picks the temperature of the nearest equilibrium phase for the current density. The gas motions are generally less than the Alfvén speed and comparable to the sound speed. This implies that supersonic turbulence in real clouds is probably the result of supersonic relative motions between very tiny clumps inside which the actual motions are nearly sonic. The magnetic field plays the important role of allowing energy from the perturbation to spread over a large distance with minimal dissipation.