Spark gaps are often used to commute energy in the discharge of a capacitive storage to a load. In some applications, a unipolar pulse is not feasible, and an oscillatory (underdamped sinusoidal) regime must be realized for the discharge of the capacitor bank. Spark gaps, which were developed for unipolar discharge, cannot be directly employed in an under-damped (oscillatory) regime since at the transition of the current through zero, the spark channel could stop motion and ignite in the following half period. This work has two main objectives: (i) To develop and test a simulation model of spark channel motion in linear rail geometry, which must be valid for both the oscillatory and unipolar regimes of capacitor bank discharge; and (ii) to investigate arc motion and electrode heating, depending on the current and charge transfer, over a wide range of operation. A self-consistent treatment of plasma motion and electrode heating (taking into account the radiation of a plasma channel) is applied in the present paper, and it is shown that radiation can significantly impact on the temperature of the electrodes. Electrode ablation and the temperature dependence of the main thermal parameters are also taken into account. Stainless steel (Cr/Ni/Ti 18/10/0.6÷0.8), copper (Cu), chromium (Cr), tungsten (W), and molybdenum (Mo) are used here as electrode materials since these materials are widely used for the manufacture of electrodes. The results of numerical calculations are compared with experimental results, and conditions are defined for reduced electrode erosion.