A sequence of simple models is discussed to identify and quantify key theoretical issues pertaining to how such oscillators can affect sound scattering. The first such model consists of a sound wave incident obliquely on a plate with a single rib which has a continuous smear of oscillators along it. The presence of the rib alters the apparent natural frequency of the oscillators and causes the backscattered sound to be anomalously high when the incident sound has the same frequency as the apparent natural frequency. The width of the spectral line in the plot of backscattered cross section versus frequency for the fixed angle of incidence is controlled both by the radiation of flexural waves away from the rib and by the overall loss of energy through radiated sound. Other models considered are an infinitely long cylinder with a single rib, smeared oscillators along the rib, an infinitely long cylinder with a single oscillator attached to a rib, a plate with a periodic array of ribs with attached oscillators, and an infinitely long cylinder with a similar hierarchy of configurations.