## 2 Theoretical Background and Mathematical Formulation  
### Standard Heavy Weight Impact Sources  
The Ministry of Construction and Transportation (now the Ministry of Land, Infrastructure, and Transport) released a document on the heavy weight impact noise of apartment houses [20] in order to resolve the interlayer noise problem in 2015 and announced the revised notice \"Structural standards for interlayer floor impact noise insulation to prevent noise\" [21]. Figure 1 shows standard heavy weight impact sources, including the bang machine (FI-02, RION Co., Ltd., Tokyo, Japan) and impact ball (YI-01, RION Co., Ltd., Tokyo, Japan), used for field measurements of interlayer noise in apartment houses and experimental testing devices, such as the impact force measuring instrument (PF-10, RION Co., Ltd., Tokyo, Japan) and data transmission device. The standard specifications for the bang machine and impact ball are summarized in Table 1[15].  
KS presents the statistical characteristics such as the mean and standard deviation of impact force levels (\(L_{FE}\), Impact Exposure Level, dB) of the standard heavy weight impact noise based on a \(1/1\) octave band as summarized in Table 2. The impact exposure level of the standard heavy weight impact noise can be calculated using the following Equation (1):  
\[L_{FE}=10 log_{10}\left(\frac{1}{T_{0}}\int_{t_{1}}^{t_{2}}\frac{F^{2}(t)}{F_{0 }^{2}}dt\right) \tag{1}\]  
where \(F(t)\) is the impact force (N), \(F_{0}\) is the reference power (1 N), \(t_{2}\sim t_{1}\) is the time including the impact time by the impact source (s), and \(T_{0}\) is reference time (1 s).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Bang Machine** & **Impact Ball** \\ \hline Effective Mass (kg) & 7.3 & 2.5 \\ Air Pressure (Pa) & \(2.5\times 10^{5}\) & - \\ Drop Height (m) & 0.85 & 1.0 \\ Rebound Coefficient & 0.8 & 0.8 \\ Shape (mm) & - & 80 \\ Impact Time (ms) & 20 \(\pm\) 2 & 20 \(\pm\) 2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The characteristics of two impact sources in Korean industrial standards (KS).  
Figure 1: Impact sources and experimental testing devices: (**a**) bang machine; (**b**) impact ball; and (**c**) impact force source tester.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**1/1 Octave Band Center** & \multicolumn{2}{c}{**1/1 Octave Band Impact Exposure Level (dB)**} & \multicolumn{2}{c}{**Allowable Variation (dB)**} \\ \cline{2-4}
**Frequency (Hz)** & **Bang Machine** & **Impact Ball** & \\ \hline
31.5 & 47.0 & 39.0 & \(\pm\) 1.0 \\
63 & 40.0 & 31.0 & \(\pm\) 1.5 \\
125 & 22.0 & 23.0 & \(\pm\) 1.5 \\
250 & 11.5 & 17.0 & \(\pm\) 2.0 \\
500 & 5.5 & 12.5 & \(\pm\) 2.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Impact exposure level in KS.  
The impact exposure level can be calculated by applying Equation (1) with respect to each octave band center frequency, and the computing process is depicted in Figure 2. First, the entire impact force, as in Figure 2a, is divided into each 1/1 octave band center frequency, as shown in Figure 2b, and then the impact exposure level for each frequency is computed using Equation (1). KS presents a total of five 1/1 octave band center frequencies, which are 31.5 Hz, 63 Hz, 125 Hz, 250 Hz, and 500 Hz. The impact exposure level calculated by frequency can be represented as in Figure 2c. In order to develop a numerical model on the standard heavy weight impact source, the measurement was conducted 10 times, respectively, with respect to both the bang machine and impact ball.