# Prediction of Vibration-Mode-Induced Noise of Structure-Acoustic Coupled Systems
## 2 Induction Characteristics of Acoustic Modes
### Prediction of Vibration-Mode-Induced Noise in the Acoustic Domain  
For a given vibration mode, the nearby acoustic mode can be obtained through the Laplace transformation of Equation (8). \[q(s)=\frac{\rho_{a}g_{a}}{F_{a}(s)}\ddot{\eta}(s) \tag{11}\]  
In other words, the response of the single degree-of-freedom system to the vibration mode (acceleration) is the acoustic mode. The vibration mode or acoustic mode exhibit a wide spectrum at frequencies approaching the natural frequency. The sum of the power spectra near the corresponding frequency represents the vibration or acoustic magnitude. Therefore, the dispersion of the vibration mode and the acoustic mode induced from it can be used to estimate the acceleration level or sound pressure level, as follows:  
\[ \begin{split} E\big{[}\ddot{\eta}(s)\ddot{\eta}(\bar{s})\big{]}= \int\big{|}\ddot{\eta}(\omega)\big{|}^{2}d\omega\\ E[q(s)q(\bar{s})=\rho_{a}^{2}g_{a}^{2}\int\bigg{|}\frac{\ddot{ \eta}(\omega)}{F_{a}(\omega)}\bigg{|}^{2}d\omega\end{split}  \tag{12}\]  
Here, \(E[]\) is a symbol for equalization; specifically, \(s=i\omega\), where \(\omega\) denotes the angular velocity. Equations (12) can be substituted into Equation (9), respectively, to calculate the dispersion of the vibration and acoustic modes. If the external load in Equation (9) is assumed to be an impact load, the leading term is a quartic equation for \(s\), and thus, the dispersion can be calculated with a closed form [40]. When the dispersion of the vibration and acoustic modes in Equation (12) is expressed in a closed form, it consists of parameters including \(g_{s}\) and \(g_{a}\), representing the natural frequencies, damping ratios, and coupling effect. The dispersion of the vibration and acoustic modes is determined using these parameters, and then, the acceleration and sound pressure levels are calculated. Through this process, the correlation between the two modes can be identified in the frequency domain. If the planar density of the structure in the acoustic domain is considerably greater than the density of the acoustic domain medium, the damping effect diminishes. Extremely large masses, such as slabs in apartment buildings, may exhibit a diminished coupling effect owing to the minimal influence by the sound pressure of the surrounding air. If the coupling between the structure and acoustic domain is negligible and only the structural vibration induces a sound field in the acoustic domain, the dispersion of the vibration and acoustic modes can be determined in a facile manner. Specifically, if the effect of the sound field on the structure is ignored, \(g_{s}=0\) in Equation (8). Since the vibration affects the acoustic domain in a cascading manner, the vibration and acoustic modes can be expressed by maintaining \(g_{a}\) in Equation (8) as follows:  
\[ \begin{split}\ddot{\eta}(s)=\frac{s^{2}}{F_{s}(s)}X(s)\\ q(s)=\frac{\rho_{a}g_{a}s^{2}}{F_{s}(s)F_{a}(s)}X(s)\end{split}  \tag{13}\]  
If the ideal impact load on the structure is a constant (band-limited white noise) at a specific frequency domain (\(\omega_{L}<\omega<\omega_{\text{U}}\)), the following expression holds:  
\[S_{xx}(\omega)=\left|X(i\omega)\right|^{2}=S_{0}\left(constant\right) \tag{14}\]  
By substituting Equation (13) into Equation (12), respectively, and integrating them based on the assumption of a constant impact load, the dispersion of the vibration and acoustic modes can be determined in the closed form as follows (7):  
\[ \begin{split} E\big{[}\ddot{\eta}(s)\ddot{\eta}(\widehat{s}) \big{]}=S_{0}\int\left|\frac{i\omega^{2}}{F_{s}(i\omega)}\right|^{2}\!d \omega=\frac{\pi S_{0}}{2}\frac{\omega_{s}\big{(}1+4\xi_{s}^{2}\big{)}}{\xi_ {s}}\simeq\frac{\pi S_{0}}{2}\frac{\omega_{s}}{\xi_{s}}\\ E[q(s)q(\widehat{s})]=S_{0}\int\left|\frac{\rho_{a}g_{a}(i\omega)^ {2}}{F_{s}(i\omega)F_{a}(i\omega)}\right|^{2}\!d\omega=\pi S_{0}\rho_{a}^{2}S _{a}^{2}\frac{N}{D}\end{split}