## 2 The Summation Method
This summation method is like that used by [PERSON] and [PERSON] ([PERSON] and [PERSON], 1971), and it was used as a reference method by [PERSON] et al. ([PERSON] et al., 1983). Using \(N=9\) is found to be sufficient for an accurate calculation. Tests were conducted with \(N=15\) without any noticeable change in the results. The actual frequencies for \(N=9\) are calculated by Equation (6). \[f_{i}=10^{(n+\frac{5}{3})/10},(\text{Hz}) \tag{6}\]  
where\(n\) is the number of the octave band center frequency according to ANSI S1.6 [19], (e.g., \(n=30\) for the 1000 Hz octave band),  
\(i\) is a number [1; 9]. The octave-band energy attenuation coefficient \(m_{\text{B}}\) is a function of temperature, relative humidity, and time of sound propagation. Two examples for the 4 kHz octave band are shown in Figure 1 for temperatures of 0 \({}^{\circ}\)C and 20 \({}^{\circ}\)C, respectively. The attenuation is calculated for different times of sound propagation, 0.25 s, 1 s, 4 s and 8 s. The initial octave-band energy attenuation coefficient is seen to be close to the pure-tone energy attenuation coefficient of the center frequency of the octave band. However, for longer propagation times, the octave-band energy attenuation coefficient approaches asymptotically the pure-tone energy attenuation coefficient one-third octave below the center frequency of the octave band, i.e., the nominal frequency 3150 Hz in this case. The behavior at 4 kHz seen in Figure 1 is similar in other octave bands and at other air temperatures at least from \(-10\)\({}^{\circ}\)C to 40 \({}^{\circ}\)C.  
The time dependency of the octave-band energy attenuation coefficient \(m_{\text{B}}\) is more clearly seen when displayed as an attenuation of the sound pressure level in decibels as a function of time. Like Equation (2) for pure tones, the time-dependent octave-band attenuation of the sound pressure level (SPL) due to air absorption is given by Equation (7). \[\delta L_{\text{B}}(t)=Km_{\text{B}}r(t),\ (\text{dB}) \tag{7}\]  
Examples of this relationship are calculated using Equation (5) and displayed in Figure 2 for the four octave bands from 1 kHz to 8 kHz for the reference case of 20 \({}^{\circ}\)C and 50% RH. It is seen that the octave-band attenuation is not linear. For comparison, the linear pure-tone attenuation functions of the center frequency of the octave band and at the frequency one-third octave below the octave band center frequency are also shown. In the 8 kHz octave band, the energy attenuation coefficient reaches and goes below the pure-tone energy attenuation coefficient at 6.3 kHz nominal frequency after more than 2 s of propagation. However, the air attenuation at 8 kHz is more than 60 dB after 2 s of propagation, so the air attenuation at 6.3 kHz can be considered the asymptotic approximation in practice. Figure 1: Air absorption in terms of energy absorption coefficient as function of the relative humidity. Full lines are for the 4 kHz octave band at various times of propagation up to 8 s. Dotted line is pure tone attenuation at 5 kHz, dash-dotted line is pure tone attenuation at 4 kHz, dashed line is pure-tone attenuation at 3150 Hz. (**a**) Temperature 0 \({}^{\circ}\)C. (**b**) Temperature 20 \({}^{\circ}\)C.  
The results shown in Figure 2 show that the pure-tone attenuation of the octave center frequency is a valid approximation for short times of propagation or, more precisely, up to attenuations around 15 dB. This confirms the information on the upper limit for the approximation given in ISO 9613-1 [1]. It is also seen that the pure-tone attenuation at a frequency one-third octave below the octave center frequency is not a good approximation. However, this is the approximation used in the European norm for calculation of sound absorption in closed spaces, EN 12354-6 [11]. It must be concluded that this method is not accurate. Another suggested approximation for the octave-band attenuation is given as a nonlinear formula in Annex E of ANSI S1.26 [9].