## 3 Accuracy Assessment with 2D Long Duct Problem
### Results and Discussion  
To facilitate intuitive understanding in the agreement between the theory and FEM, Figure 3a,b presents the attenuation levels, \(L^{th}_{\text{att.}}\) and \(L^{fem}_{\text{att.}}\), at two receivers \(x=100\) m and \(1000\) m, computed using the ISO 9613-1 model (theory) and the lossy Helmholtz equation approach (FEM) for condition (a). Apparently, \(L^{fem}_{\text{att.}}\) agrees well with \(L^{th}_{\text{att.}}\) at the two receivers and at frequencies up to 20 kHz. The result is also true for other receivers and conditions (b) and (c). Regarding the error level associated with the lossy Helmholtz equation approach, Figure 4 shows \(e_{\text{att.}}\) at the most distant receiver \(\mathbf{r}=(1000,0)\) for atmospheric conditions (a)-(c). The relative errors become larger for higher frequencies. The error magnitudes are almost equal, irrespective of the atmospheric conditions. For conditions (a)-(c), the maximum values of the relative errors caused at 20 kHz are, respectively, 0.84%, 0.77%, and 0.72%. The results demonstrated that the FEM based on the lossy Helmholtz equation approach can include the air absorption effect accurately at audible frequencies up to 20 kHz using the pure-tone sound attenuation coefficient described in the ISO 9613-1 model.  
The error increase over frequencies is attributable to the dispersion error--an inherent error deriving from spatial discretization of the space with FEM. The dispersion error becomes large at higher frequencies for a given FE mesh. A simple strategy to reduce the dispersion error is to use a mesh with higher spatial resolution. Here, we show further how the error magnitude increases or decreases using a mesh with a coarser or higher spatial resolution. Figure 5 shows the \(e_{\text{att.}}\) at the most distant receiver \(\mathbf{r}=(1000,0)\) for the  
Figure 1: Plane wave propagation problem in 2D long duct.  
Figure 2: Pure-tone sound attenuation coefficient for three atmospheric conditions (a)–(c).  
atmospheric conditions (a) when using FE meshes with a spatial resolution of 3.4 (Mesh1) and 13.6 (Mesh3) elements per wavelength at 20 kHz. The two FE meshes have a half spatial resolution of 400,002 DOF and a two times higher spatial resolution of 2,400,003 DOF. In the figure, Mesh2 represents the baseline mesh with a spatial resolution of 6.8 elements per wavelength. Since the used FEM has fourth-order accuracy, the relative errors of Mesh1 increase to about one-sixteenth compared to those of the baseline Mesh2. The error magnitude exceeds 2% at frequencies above about 12 kHz. However, it is important that the error magnitude becomes less than 0.84% at frequencies below 10 kHz, where the spatial resolution of 6.8 elements per wavelength is satisfied. On the other hand, the relative errors of Mesh3 decrease to about one-sixteenth compared to those of Mesh2. The maximum error at 20 kHz is 0.05% for the two-times-higher resolution mesh. Note that we limit our discussion to the idealized case with a mesh discretized by rectangular elements. However, the proposed problem is also helpful in estimating the error characteristics for finite elements with different shape functions and a mesh discretized with irregularly shaped elements.  
Figure 4: Relative errors in attenuation levels for three atmospheric conditions (a)–(c) when using the FE mesh with a spatial resolution of 6.8 elements per wavelength.  
Figure 3: Attenuation levels at two receivers for atmospheric conditions (**a**) computed using the ISO 9613-1 model (theory) and the Helmholtz equation approach (FEM): (**a**) \(x=100\) m, and (**b**) \(x=1000\) m.