A structured panel is provided that includes a first skin with a plurality of perforations, a second skin and a core. The core forms a plurality of cavities vertically between the first skin and the second skin. Each of the cavities is respectively fluidly coupled with one or more of the perforations. The cavities include a first cavity. An element of the core is configured with a multi-layered structure. The multi-layered structure includes a first layer and a second layer attached to the first layer. The first layer is configured with a first rib projecting into the first cavity.