A theory which incorporates a surface spectrum of water waves is developed in such a way that direct computation and comparison with experiment is possible. Three theories are developed, one which applies to inviscid waves, one which applies to waves dominated by viscous conduction, and one which applies to waves dominated by viscous convection. It is found that the steady second-order streaming at the free surface increases by 20% if a sinusoidal wave is replaced by a Gaussian spectrum of standard deviation 4 radians per second of the same total energy.