  We develop a framework shedding light on common features of coupled
variational formulations arising in electromagnetic scattering and acoustics.
We show that spurious resonances haunting coupled domain-boundary formulations
based on direct boundary integral equations of the first kind originate from
the formal structure of their Calderon identities. Using this observation, the
kernel of the coupled problem is characterized explicitly and we show that it
completely vanishes under the exterior representation formula.
