Reflection and transmission at an attachment half-space

The reflection and transmission at the connection of an attachment half-space to a core half-space that contains a source of mechanical energy and one or more scatterers is considered. Both the attachment and core half-spaces have property measures that are invariant with distance from the connection plane, but may vary with position in planes parallel to the connection plane. At issue is the construction of a reflection operator that both applies at the connection plane and is expressed in terms of operators that separately describe the responses of the half-spaces at the connection plane, i.e., of impedance operators that can be written for the half-spaces in isolation. A successful construction is achieved for a connection that spans the entire connection plane. For this scenario, the construction achieved is easily identified as a generalization of a ‘‘nonreflecting boundary condition,’’ in that the reflection operator obtains no reflection for any incident field, if the impedance operators for the two half-spaces are the same. The achieved construction is next demonstrated to not apply for a connection that spans only a finite portion of the connection plane, i.e., an aperture. For this second scenario, pressure release conditions apply for both half-spaces at points on the connection plane that are outside the aperture. The reason for the failure is the impedance operators that apply in the context of a full-plane connection are not uniquely described for a connection that applies to a finite aperture. That is, mobility operators that can be identified with the DtN map obtained on factoring a two-way description of the fields in the half-spaces into separated one-way descriptions, are uniquely defined regardless of the connection region. However, the inverse of this mobility operator requires connection across the entire plane. The implication of this demonstration for more complex experimental scenarios in both propagation modeling and structural acoustics is discussed.