Schematic diagram showing the flow around a working surface of a jet (in this case the leading working surface, but the diagram also applies for an internal working surface). The jet is the horizontal, red rectangle at the bottom of the graph, with the source located at z = 0. The working surface in yellow is located at a distance z_j from the source and travels at a velocity v_j. It ejects material away from the axis at an initial velocity v₀. The jet is surrounded by a “disk wind”, which travels along the outflow axis at a velocity v_w. The shape of the thin-shell bow shock (thick cyan line) is given by z_b as a function of r and ends at the edge of the bow wing (cyan point; z_{b f}, r_{b f}). The bow shock leaves behind a “cavity” (black region), which is partially refilled by the disk wind (brown region). The boundaries of the initial swept up cavity (in black dashed line) and of the refilled region (cyan dash-dotted line) are given by z_f an z_c (respectively) as a function of cylindrical radius r (see Eqs. (14) and (15)).