## 2 The Summation Method  
The energy attenuation coefficient is calculated in accordance with ISO 9613-1. The calculation method in the standard is valid for pure tones, and it is assumed to be the most accurate basis that is available. The air attenuation can be expressed either as the sound attenuation coefficient, \(\alpha\) in decibels per meter or as the energy attenuation coefficient, \(m\) in reciprocal meters. In the following, the time \(t\) is relative to the arrival time of the direct sound at the receiver. The attenuation due to air absorption of the sound pressure level of a pure tone with frequency \(f\) after propagating the distance \(r(t)\) is given by:  
\[\delta L(f,t)=\alpha(f)r(t)=Km(f)r(t),\text{(dB)} \tag{2}\]  
where  
\(\delta L(f,t)\) is the attenuation of the sound pressure level due to air absorption, in decibels \(\alpha(f)\) is the pure-tone sound attenuation coefficient, in decibels per meter \(m(f)\) is the energy attenuation coefficient, in reciprocal meters \(K=10\lg(e)\approx 4.343\) dB is a constant \(r(t)\) is the distance that the sound wave has travelled at the time \(t\), in meters, calculated as  
\[r(t)=ct+d,(\text{m}) \tag{3}\]  
where  
\(c\) is the speed of sound at the relevant temperature, meters per second  
\(t\) is the time relative to the start of the arrival of the direct sound, in seconds  
\(d\) is the distance from the sound source to the microphone, in meters. The speed of sound in air \(c\) is mainly a function of the atmospheric temperature. Although the speed of sound also changes a little with the humidity, up to 0.3% compared to dry air ([PERSON], 1971) ([PERSON], 1971), this effect is here considered to be negligible ([PERSON], 1971) Paragraph A.6 and ([PERSON], 1971) Figure 2). Thus, the speed of sound is calculated from ([PERSON], 1971) Equation (A.5):  
\[c=c_{0}\sqrt{\frac{273.15+T}{293.15}},(\text{m}/\text{s}) \tag{4}\]  
where  
\(c_{0}=343.2\) m/s is the speed of sound at the reference temperature (20 \({}^{\circ}\)C)  
\(T\) is the ambient atmospheric temperature, in \({}^{\circ}\)C  
In room acoustic measurements, the impulse response is a function of the time \(t\) starting at the arrival of the direct sound. The distance \(d\) is not inherent in the measured impulse response, and this information must be provided by the user. Especially at frequencies well above 1 kHz, the air attenuation from source to receiver can be significant, and errors of several dBs may occur if the distance is not included. If this distance is not known, a default value of \(d=10\) m may apply. The octave-band average energy attenuation coefficient changes according to how long the sound wave has travelled. The attenuation increases with frequency, and, thus, it varies within the bandwidth of an octave. This means that the higher frequencies within the octave are attenuated more than the lower frequencies, and during the sound propagation, the energy balance is shifted towards lower frequencies in the octave band. For this reason, the attenuation of sound in an octave band is not linear. In the following, it is assumed that the frequency spectrum of the sound emitted from the source is flat (like pink noise) within the octave band. The octave-band average energy attenuation coefficient can be calculated from several frequencies equally distributed in a logarithmic scale within the octave band using Equation (5). \[m_{\text{B}}(t)=-\frac{1}{r(t)}\text{ln}\left(\frac{1}{N}\sum_{i=1}^{N}\exp(- r(t)m(f_{i}))\right),\left(\text{m}^{-1}\right) \tag{5}\]  
where  
\(m_{\text{B}}(t)\) is the energy attenuation coefficient of the octave band, in reciprocal meters  
\(n\) is the number of frequencies used for the averaging within the octave band  
\(m(f_{i})\) is the energy attenuation coefficient at the frequency \(f_{i}\), calculated in accordance with ISO 9613-1, in reciprocal meters  
The calculation with Equation (5) can be made by dividing the octave band into several frequencies, each carrying the same energy at the time \(t=0\).