  In this paper we will derive an non-local (``integral'') equation which
transforms a three-dimensional acoustic transmission problem with
\emph{variable} coefficients, non-zero absorption, and mixed boundary
conditions to a non-local equation on a ``skeleton'' of the domain
$\Omega\subset\mathbb{R}^{3}$, where ``skeleton'' stands for the union of the
interfaces and boundaries of a Lipschitz partition of $\Omega$. To that end, we
introduce and analyze abstract layer potentials as solutions of auxiliary
coercive full space variational problems and derive jump conditions across
domain interfaces. This allows us to formulate the non-local skeleton equation
as a \emph{direct method} for the unknown Cauchy data of the solution of the
original partial differential equation. We establish coercivity and continuity
of the variational form of the skeleton equation based on auxiliary full space
variational problems. Explicit expressions for Green's functions is not
required and all our estimates are \emph{explicit} in the complex wave number.
