## 2 Statement of the Problem in the Frequency Domain  
The physical space \(\mathbb{R}^{3}\) refers to a cartesian reference system, and the generic point of \(\mathbb{R}^{3}\) is denoted by \(\mathbf{x}=(x_{1},x_{2},x_{3})\). For any function \(f(\mathbf{x})\), the notation \(f_{;j}\) designates the partial derivative with respect to \(x_{j}\). The classical convention for summations over repeated Latin indices is used, but not over Greek indices. The vibration problem is formulated in the frequency domain. Therefore, the Fourier transform is introduced for various quantities involved. For instance, for the displacement field \((\mathbf{x},t)\mapsto\mathbf{u}(\mathbf{x},t)\), the simplified notation \((\mathbf{x},\omega)\mapsto\mathbf{u}(\mathbf{x},\omega)=\int_{-\infty}^{+ \infty}e^{-i\omega t}\,\mathbf{u}(\mathbf{x},t)\,dt\) is introduced and consists of using the same symbol for a quantity and its Fourier transform, in which the circular frequency \(\omega\) (rad/s) is real. The vibroacoustic system is assumed to be in linear vibrations around a static equilibrium state taken as a natural state at rest. **Structure**. In general, the structure of a complex vibroacoustic system is composed of a main part called the master structure that is accessible to conventional modeling including uncertainties modeling, and a secondary part called the fuzzy substructure related to the structural complexity and including for example many equipment units attached to the master structure. In the present paper, we will not consider fuzzy substructures, and concerning fuzzy structure theory, we refer the reader to Chapter 15 of [41] for a synthesis, and to [96] for an extension of the theory to the modeling of an uncertain complex vibroacoustic system with fuzzy interface. At equilibrium, the structure occupies the three-dimensional bounded domain \(\Omega_{5}\) with a boundary \(\partial\Omega_{5}\) that is made up of a part \(\Gamma_{E}\) that is the coupling interface between the structure and the external acoustic fluid, a part \(\Gamma\) that is a coupling interface between the structure and the internal acoustic fluid, and finally, a part \(\Gamma_{Z}\) that is another part of the coupling interface between the structure and the internal acoustic fluid with acoustic properties. The structure is assumed to be free (free-free structure). The outward unit normal to \(\partial\Omega_{5}\) is denoted as \(\mathbf{n}^{S}=(n_{1}^{S},n_{2}^{S},n_{3}^{S})\) (see Figure 1). In \(\Omega_{S}\), the displacement field is denoted by \(\mathbf{u}(\mathbf{x},\omega)=(u_{1}(\mathbf{x},\omega),u_{2}(\mathbf{x}, \omega),u_{3}(\mathbf{x},\omega))\). A surface force field \(\mathbf{G}(\mathbf{x},\omega)=(G_{1}(\mathbf{x},\omega),G_{2}(\mathbf{x}, \omega),\)\(G_{3}(\mathbf{x},\omega))\) is given on \(\partial\Omega_{5}\), and a body force field \(\mathbf{g}(\mathbf{x},\omega)=(g_{1}(\mathbf{x},\omega),\)\(g_{2}(\mathbf{x},\omega),g_{3}(\mathbf{x},\omega))\) is given in \(\Omega_{S}\). The structure is a dissipative medium for which the frequency-dependent constitutive equation is detailed in Appendix A. **Internal dissipative acoustic fluid**. Let \(\Omega\) be the internal bounded domain filled with a dissipative acoustic fluid (gas or liquid) that is described in Section 4. The boundary \(\partial\Omega\) of \(\Omega\) is \(\Gamma\cup\Gamma_{Z}\). The outward unit normal to \(\partial\Omega\) is denoted by \(\mathbf{n}=(n_{1},n_{2},n_{3})\), and we have \(\mathbf{n}=-\mathbf{n}^{S}\) on \(\partial\Omega\) (see Figure 1). The acoustic properties of boundary \(\Gamma_{Z}\) are modeled by a wall acoustic impedance \(Z(\mathbf{x},\omega)\) satisfying the hypotheses defined in Section 4.2. In \(\Omega\), the pressure field is denoted by \(p(\mathbf{x},\omega)\) and the velocity field by \(\mathbf{v}(\mathbf{x},\omega)\). It is assumed that there is no Dirichlet boundary condition on any part of \(\partial\Omega\). An acoustic source density \(Q(\mathbf{x},\omega)\) is given inside \(\Omega\). **External inviscid acoustic fluid**.