## 1 Introduction
Extended-reaction models that solve the partial differential equations of materials are used to describe the incident-angle-dependent absorption characteristics. A simple equivalent fluid model based on the lossy Helmholtz equation and the poroelastic material model based on the Biot theory have been used [33; 34; 35]. By contrast, standard room acoustics simulation usually solves the lossless wave equation and Helmholtz equation for sound propagation in the air inside rooms. Their lossless equations do not address the sound attenuation effects of air absorption caused by viscothermal effects and relaxation processes. However, considering the current situation in the applicable range of wave-based room-acoustics simulation to high frequencies, wave-based room acoustics simulation including air absorption might be an aspect for increasing their predictive reliability. Some approaches can be used to include the air absorption effect. A recent report of the literature [36] proposes a low pass filter-based post-processing approach that uses a cascade of three time-varying IIR filters, approximating the air absorption model described in ISO 9613-1 [37]. The air absorption effect is added by filtering impulse responses without air absorption with time-varying IIR filters. For one earlier study [38], FIR filter-based approximation of air absorption was used to study the audibility of the dispersion error in a room acoustic FDTD solver. As a physically based approach for time-domain analysis, an acoustic FDTD solver discretized on Stokes' equation has been used to include the air absorption effect caused by relaxation absorption from oxygen molecules, which has a dominant role in sound attenuation at high frequencies within the audible frequency range [39]. A time-domain wave-based room acoustics model, including viscothermal and relaxation effects in air, has also been proposed with corresponding FDTD discretization schemes [40]. Stokes' equation can be reduced to the lossy Helmholtz equation with a complex-valued wave number assuming time-harmonic sound pressure [41]. Therefore, for frequency-domain analysis, a simple approach that solves the lossy Helmholtz equation with the pure-tone attenuation coefficient describing sound attenuation because of classical absorption and molecular relaxation absorption is available to implement air absorption effects. The study described herein specifically examines the simple lossy Helmholtz equation approach in the frequency domain. As described previously, the lossy Helmholtz equation with a complex wave number has been used to model viscothermal dissipation of sound-absorbing materials such as porous and resonator absorbers as equivalent fluids [42; 43; 44; 45; 46; 47; 48; 49]. Nevertheless, how the lossy Helmholtz equation approach models air absorption effects accurately and efficiently has not been discussed adequately in earlier reports of the relevant literature. This report of our study is intended to demonstrate, via two numerical problems, the accuracy and computational efficiency of the lossy Helmholtz equation-based FEM to include air absorption effects. One problem is long-range plane wave propagation in a long duct for an accuracy assessment up to 20 kHz. Another is a sound propagation problem in a real-scale 2D office for computational efficiency estimation. As a salient contribution of this study, we will reveal empirically that lossy Helmholtz equation-based FEM in the frequency domain has attractive features in terms of computational efficiency for room acoustics simulation at high frequencies. The structure of this paper is as follows. Section 2 presents the lossy Helmholtz equation approach, with the pure-tone attenuation coefficient based on the ISO 9613-1 model to add air absorption effects to FEM in the frequency domain. Then, Section 3 presents accuracy assessment results obtained for the lossy Helmholtz equation-based FEM via the long-range plane wave propagation problem in a long duct.