An acoustic panel is provided that includes a perforated first skin, a second skin and a core. The core is between and connected to the perforated first skin and the second skin. The core includes a plurality of chambers, a first corrugated structure and a second corrugated structure. The chambers includes a first chamber. The first chamber extends from the perforated first skin, through the first corrugated structure and the second corrugated structure, to the second skin.