An assembly is provided for an aircraft propulsion system. This assembly includes a first acoustic panel and a second acoustic panel. The first acoustic panel includes a first perforated skin, a first non-perforated skin and a first cellular core arranged between and connected to the first perforated skin and the first non-perforated skin. The first perforated skin is configured with a plurality of first perforations. A first of the first perforations has a first width. A second acoustic panel includes a second perforated skin, a second non-perforated skin and a second cellular core arranged between and connected to the second perforated skin and the second non-perforated skin. The second perforated skin is configured with a plurality of second perforations. A first of the second perforations has a second width that is smaller than the first width.