## 3 Accuracy Assessment with 2D Long Duct Problem  
### Outline of the Problem  
This section presents an examination of the accuracy of lossy Helmholtz equation-based FEM to include the air absorption effect. To this end, as a benchmark problem, we propose a plane wave propagation problem in a 2D long duct with a 1000 m length. Figure 1 portrays the geometry of the 2D duct. Multi-frequency analyses using the lossy Helmholtz equation-based FEM and lossless Helmholtz equation-based FEM were performed, respectively, to calculate frequency responses at ten receiving points placed at \(x=[100,1000]\) with 100 m intervals in the duct. The calculation was performed at 1 Hz to 20 kHz with 1 Hz intervals. The lossless Helmholtz equation approach solves the Helmholtz equation with real-valued wave number \(k_{0}\) to compute frequency responses without the air absorption effects. Attenuation levels \(L_{\text{att.}}^{fem}\) attributable to air absorption were computed further as the difference of frequency responses between the lossy and the lossless Helmholtz equation approaches:  
\[L_{\text{att.}}^{fem}(\mathbf{r},\omega)=\left|L_{\text{lossy}}(\mathbf{r},\omega)-L_ {\text{lossless}}(\mathbf{r},\omega)\right|\ [\text{dB}]. \tag{10}\]  
Therein, \(L_{\text{lossy}}(\mathbf{r},\omega)\) and \(L_{\text{lossless}}(\mathbf{r},\omega)\), respectively, denote the sound pressure levels computed using the lossy and lossless Helmholtz equation approaches. Then, the numerical attenuation levels \(L_{\text{att.}}^{fem}\) were compared with the theoretical ones \(L_{\text{att.}}^{theory}\) computed using the pure-tone sound attenuation coefficient in ISO 9613-1. For a quantitative evaluation of accuracy, we define the relative error \(e_{\text{att.}}\) in the attenuation levels as  
\[e_{\text{att.}}(\mathbf{r},\omega)=\frac{L_{\text{att.}}^{fem}(\mathbf{r},\omega)}{L_ {\text{att.}}^{theory}(\mathbf{r},\omega)}\times 100\ [\%]. \tag{11}\]  
This study evaluated the \(e_{\text{att.}}\) at the most distant receiver \(\mathbf{r}=(1000,0)\). Regarding the boundary conditions, the duct inlet \(\Gamma_{\text{in}}\) in Figure 1 has a vibration boundary \(\Gamma_{\text{v}}\) with vibration velocity of 1.0 m/s, radiating a plane wave. To consider only the progressive wave in the positive \(x\) direction, the duct inlet \(\Gamma_{\text{in}}\) and outlet \(\Gamma_{\text{out}}\) were assigned perfectly absorbing boundary conditions \(\Gamma_{a}\) with the characteristic impedance of the air. The side walls of the duct \(\Gamma_{\text{side}}\) were assumed as a rigid boundary \(\Gamma_{r}\). Spatial discretization of the air domain \(\Omega\) is performed using a dispersion-reduced version of Q4s [53] with the size of 0.0025 m \(\times\) 0.0025 m. The resulting mesh has 800,002 DOF with a spatial resolution of approximately 6.8 elements per wavelength at 20 kHz. The dispersion-reduced FEMs [20,47] have fourth-order accuracy. They can therefore produce more accurate solutions for the used mesh than the mesh created using standard linear elements with a spatial resolution of ten elements per wavelength.  
For the accuracy estimation, we considered three atmospheric conditions (a)-(c) at standard atmospheric pressure 101.325 kPa: (a) temperature \(T=10\)\({}^{\circ}\)C, relative humidity \(hr=15\%\), (b) \(T=20\)\({}^{\circ}\)C, \(hr=50\%\), and (c) \(T=30\)\({}^{\circ}\)C, \(hr=70\%\). Figure 2 shows the pure-tone sound attenuation coefficient [dB/m] computed according to ISO 9613-1 for atmospheric conditions (a)-(c). Overall, the degree of air absorption is greater in the order of conditions