## 3 Approximation Using Linearization  
It is necessary to realize that the air attenuation in octave bands is not a linear function of distance/time of propagation. Therefore, a linearization method is searched for. It is suggested to approximate the octave-band air attenuation by using the pure-tone attenuation of two different frequencies, one frequency\(f_{1}\) valid below a certain distance \(r_{\rm k}\) and another frequency\(f_{2}\) valid at longer distances. The approximation has been tested for a wide range of temperatures and humidities, and the best results are obtained if \(f_{1}\) is the exact center frequency of the octave band and \(f_{2}\) is the center frequency of the lower one-third octave within the octave band, see Table 1 and Figure 4. The two pure-tone attenuation lines cross in a knee point, and the distance \(r_{\rm k}\) to the knee point is as in Equation (3) with \(t=t_{\rm k}\). From the information in ISO 9613-1 [1] (8.1.3), it could be expected that the knee point is where the air attenuation is approximately 15 dB. A closer study has led to the criterion \(\delta L_{\rm k}=14\) dB in the octave bands 1 kHz and above, and \(\delta L_{\rm k}=6\) dB in the 500 Hz octave band. Below 500 Hz, a knee point is not relevant because the air attenuation is very small and nearly linear. The distance of sound propagation to the knee point \(r_{\rm k}\) is calculated by Equation (8a):  
\[r_{\rm k}=\frac{\partial L_{k}}{4.343\ m(f_{1})},\ (\rm m) \tag{8a}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Nominal Center Frequency, \(f_{\rm c}\), Hz** & **63** & **125** & **250** & **500** & **1000** & **2000** & **4000** & **8000** \\ \hline Band number \(n\) & 18 & 21 & 24 & 27 & 30 & 33 & 36 & 39 \\ \(f_{1}\), Hz & 63 & 126 & 251 & 501 & 1000 & 1995 & 3981 & 7943 \\ \(f_{2}\), Hz & NA & NA & NA & 398 & 794 & 1585 & 3162 & 6310 \\ \(\delta L_{\rm k}\), dB & NA & NA & NA & 6 & 14 & 14 & 14 & 14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Octave band numbers, equivalent frequencies\(f_{1}\) and \(f_{2}\), and the attenuation \(\delta L_{\rm k}\) at the knee point. Figure 4: Air attenuation in an octave band and approximation with the air attenuation of two specific frequencies. The 4 kHz octave and (20 \({}^{\circ}\)C, 50% RH) is used as example. Full line: Calculated for octave band with the summation method. Dashed lines: Pure-tone attenuation of octave center frequency. Dot-dashed line: Pure-tone attenuation at center frequency of lower one-third octave band within the octave, shifted to go through the knee point at \(t_{\rm k}\). The corresponding time \(t_{\text{k}}\) of the knee point in the impulse response depends on the speed of sound \(c\) and the distance \(d\) from source to receiver. It is calculated by Equation (8b):  
\[t_{\text{k}}=\frac{\partial L_{k}}{4.343\;c\;m(f_{1})}-\frac{d}{c},\;(\text{s}) \tag{8b}\]  
The approximation for the attenuation of the sound pressure level in an octave band due to the absorption in the air is calculated in the distance domain by Equation (9):  
\[\delta L_{\text{B}}(r)\cong \begin{cases}Km(f_{1})r(t),\;(\text{dB})&\text{for $r<r_{ \text{k}}$}\\ K\{m(f_{1})r_{\text{k}}+m(f_{2})(r(t)-r_{\text{k}})\},\;(\text{dB})&\text{for $r \geq r_{\text{k}}$}\end{cases}  \tag{9}\]  
where  
\(\delta L_{\text{B}}(r)\) is the octave-band attenuation of the sound pressure level due to air absorption as function of the distance \(r\), in decibels  
\(m(f_{1})\) is the energy attenuation coefficient at the frequency \(f_{1}\), in reciprocal meters  
\(m(f_{2})\) is the energy attenuation coefficient at the frequency \(f_{2}\), in reciprocal meters. An example of the attenuation of the sound pressure level due to the absorption in the air is shown in Figure 4 for the reference condition of 20 \({}^{\circ}\)C and 50% RH. The change of slope is seen when the attenuation exceeds 14 dB. The example in Figure 4 could suggest a knee point lower than 14 dB to minimize the difference between the approximation and the summation method.