A two-dimensional random-walk problem is investigated, where a vector fixed in amplitude (magnitude) is added to one that is random in phase (direction angle). This problem can arise from a study of the effect of multipath interference from an acoustic or electromagnetic-wave source, where an arrival of constant amplitude is combined with an incoherent background of randomly scattered components of the incident field. The source is considered to have harmonic time dependence. This study follows from an earlier paper, in which the required properties of the incoherent background were derived. In the present study, the joint distribution of the resultant amplitude and of the phase angle between the resultant and the fixed arrival is determined. Further, expressions are developed for the first and second moments of the resultant phase, amplitude, decibel-amplitude, and intensity. Many properties of these moments are described.