## 2 Theoretical Background and Mathematical Formulation
### Theoretical Background of Vibration and Acoustic Analysis  
The interlayer noise inside apartment houses is created by floor impact noise transmitted downstairs and caused by impact sources such, as children running and adults walking upstairs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Center Frequency (Hz)**} & \multicolumn{3}{c}{**KS Level (dB)**} & \multirow{2}{*}{**Proposed (dB)**} & \multirow{2}{*}{**Improved (dB)**} \\ \cline{2-2} \cline{4-5}  & **Low** & & & **Middle** & **Upper** \\ \hline
31.5 & 38.0 & 39.0 & 40.0 & 38.376 & 38.364 \\
63.0 & 29.5 & 31.0 & 32.5 & 30.861 & 30.845 \\
125.0 & 21.5 & 23.0 & 24.5 & 22.820 & 22.580 \\
250.0 & 15.0 & 17.0 & 19.0 & 16.473 & 16.180 \\
500.0 & 10.5 & 12.5 & 14.5 & **8.330**1  & 12.530 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Impact exposure level for impact ball’s numerical model. Figure 7: Improved analytical impact force model for the impact ball: (**a**) Time series data; and (**b**) computed impact exposure level. In this case, the medium through which sound is transmitted is the room air downstairs, and the noise source that causes the noise is vibrations transmitted through the walls and ceiling (slab). If slabs or walls vibrate, air particles that come in contact with the slabs or walls vibrate, which causes noise. Therefore, the velocity of the air particles caused by the transmission vibration, which is a boundary condition of the acoustic analysis, should be obtained to analyze the noise arising from the floor impact sound. Through a structural vibration analysis, the exact vibration velocity of structures, such as slabs or walls, should be calculated, and an accurate acoustic analysis can be performed from the calculated vibration velocity. As shown in Figure 8, vibration and acoustic analysis procedures can be divided into a structural vibration analysis dealing with the vibration that occurs inside the structure due to an external impact, and an acoustic analysis that deals with sound radiation arising from the vibration of the structure [24; 25]. Generally, for the structural vibration analysis, vibrations caused by the external force (\(F\)) are obtained using an equation of motion as in Equation (7):  
\[m\ddot{x}+c\dot{x}+kx=F \tag{7}\]  
where \(x\) is displacement, \(\dot{x}\) is velocity, \(\ddot{x}\) means acceleration, \(m\), \(c\), \(k\) mean the mass, damping, and stiffness values of the structure, respectively, and \(F\) represents the external force applied to the structure. Based on the location, member elements, connection relationship, and geometric information of the target structure, as well as the modulus of elasticity, density, Poisson's ratio, and material properties, a numerical model for the structural analysis was produced. This structural analysis was performed using a bang machine or impact ball to calculate the vibration response of the target structure. An understanding on the generation of noise due to the vibration of the structure is needed to perform the noise analysis using the calculated vibration response of the target structure. The structural vibration that occurs in walls or slabs in apartment houses vibrates the indoor air particles that come in contact with the walls or slabs, causing noise. This indoor air vibration allows the sound source that causes a sound wave to change the pressure in a medium, move the particles of the medium, and in turn change the density of the medium. Then, the change in the density leads to a change in the pressure, and thereby causes sound. As shown in Figure 9, on the basis of this relationship, an acoustic wave equation can be induced by utilizing an equation of state that represents a relationship between the density and the pressure, an equation of motion representing a relationship between the pressure and the displacement, and an equation of continuity that shows a relationship between the displacement and the density.