# Using Mechanical Metamaterials in Guitar Top Plates: A Numerical Study
## 2 Materials and Methods  
The investigation was carried out via the finite element method (FEM), the standard method for the simulation of musical instruments in the literature [22; 23; 24; 25; 26; 27; 28; 29]. The reference 3D model of the guitar body was based on an 1884 instrument by Antonio de Torres, as described in [30], and it was realised with Autodesk Fusion 360%. For the computation of the sound pressure level of the instrument, we coupled the FEM and boundary element method (BEM), as explained in [31]. This model includes a traditional bracing design, with 7 fan braces glued on the underside of the lower bout of the soundboard, all 3 mm thick (see Figure 1a). A simplified model of the bridge, used in the static load simulations, is also included. The soundboard was then altered by cutting a pattern of elliptical holes in the underside, to a depth of 2 mm. We note that, unlike in the studies in [17], the plate here is not cut all the way through. Instead of holes, the metamaterial is composed of periodic cavities, of 2 mm in depth. This leaves a layer of solid material 0.5 mm thick on the outside of the guitar. Moreover, the material of the soundboard here is not homogeneous over its entire extent, as the cavities are cut only in the lower bout of the instrument. The hole patterns are generated by the juxtaposition of several copies of the same unitary cell (see Figure 1). The cell is rectangular, with a size of 26 \(\times\) 12 mm for all simulations, but we distinguish between two kinds of holes based on the orientation: here and throughout the rest of this paper, when the long side is parallel to the longitudinal axis of the soundboard's wood, we talk about _longitudinal_ holes, while when it is aligned with the radial axis, we talk about _radial_ holes. The holes are centred with respect to the cell, and their size is varied while keeping the aspect ratio constant, by changing a dimensionless scaling parameter \(p\leq 1\), which multiplies the sizes of the major and minor axes of the ellipses. With \(p=1\), the major semiaxis of the elliptical holes is 10 mm long, while the minor semiaxis is 2.5 mm long. All the materials were modelled as linear elastic and orthotropic, taking the mechanical properties for each particular wood (12% moisture content) from [32]. As detailed in Figure 1, three different woods were used: Engelmann spruce for the soundboard and the soundboard bracing (\(E_{L}^{\text{top}}=9.8\) (GPa), \(E_{R}^{\text{top}}=1.3\) (GPa), \(\rho^{\text{top}}=350\) (kg/m\({}^{3}\))), Honduran mahogany for the sides, back, and back braces (\(E_{L}^{\text{back}}=10.8\) (GPa), \(E_{R}^{\text{back}}=10.1\) (GPa), \(\rho^{\text{back}}=450\) (kg/m\({}^{3}\))), and rosewood for the bridge (\(E_{L}^{\text{bridge}}=15.8\) (GPa), \(E_{R}^{\text{bridge}}=1.64\) (GPa), \(\rho^{\text{bridge}}=609\) (kg/m\({}^{3}\))). Appropriate care was taken to correctly define the material axes of each component of the model. The simulations were performed using the COMSOL Multiphysics(r) simulation software, in particular the Solid Mechanics interface from the Structural Mechanics module. For the solid-air coupling, there are then 3 Multiphysics modules that deal with the boundaries between the domains: Solid Mechanics-Pressure Acoustic FEM, Solid Mechanics-Pressure Acoustic BEM, and Pressure Acoustic FEM-Pressure Acoustic BEM; this last module works on the sound-hole virtual surface where the portion of air inside the cavity meets the outer air. The numerical model here differs from the one without air in one way: whereas earlier, we ignored the effect of structural damping, here it is accounted for in the form of proportional, or Rayleigh damping. In particular, following [22],  
Figure 1: (**a**) Diagrams of the soundboards in three different configurations: solid, with longitudinal holes (\(p=1\)), and with radial holes (\(p=1\)). The figure shows the bracing and the hole patterns, and each surface is coloured according to its height with respect to the lower surface of the plate.