Snapshot of the perturbed surface density after ten planetary orbits for our locally isothermal 3D model. Left: entire simulation domain. Spiral shocks are weaker near either radial end of the disk due to wave damping. Top right: a zoom-in into the orange box on the left, highlighting the region where gap opening is driven. Bottom right: angular momentum deposition function ∂F_dep/∂R and the surface density evolution rate ∂ς/∂t in arbitrary units computed according to Cordwell & Rafikov (2024). Saturated and faded colors indicate smoothed or raw data, respectively. Dashed and dotted vertical lines mark the isothermal shock distance x_sh (Eq. (15) with γ = 1) and the planet’s corotating region (Eq. (22) with γ = 1), respectively.