# Historical Church Bell Tower Structural Behavior Induced by [PERSON]'s Acoustic Wave Propagation
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structural Frequency Peak** & \begin{tabular}{c} **Acoustic Frequency** \\ **Partials** \\ \end{tabular} & \
\begin{tabular}{c} **Relation Between** \\ **Simulation Output and** \\ **Computed Value** \\ \end{tabular} &
\begin{tabular}{c} **Relative Deviation** \\ **\%** \\ \end{tabular} \\ \hline
**[Hz]** & **[Hz]** & **Computed Value** & **\%** \\ \hline
1.67 & NA &
\begin{tabular}{c} Independent structural \\ vibration \\ \end{tabular} & NA \\
1.91 & 1.93 &
\begin{tabular}{c} First experimental \\ vibration mode (1.93 Hz) \\ \end{tabular} & 0.67\% \\
4.08 & 2.04 &
\begin{tabular}{c} One octave lower than the \\ second experimental \\ vibration mode (2.03 Hz) \\ \end{tabular} & \(-\)0.59\% \\
4.67 & 4.67 &
\begin{tabular}{c} Harmonic ratio with 32.670 \\ Hz (_sesquisptimum_) \\ Independent structural \\ vibration \\ \end{tabular} & 0.00\% \\
5.58 & NA &
\begin{tabular}{c} Independent structural \\ vibration \\ \end{tabular} & NA \\
5.83 & 5.85 &
\begin{tabular}{c} Harmonic ratio with 29.250 \\ Hz (_sesquiqartum_) \\ Harmonic ratio with 49.750 \\ Hz (three octaves lower) \\ \end{tabular} & \(-\)0.29\% \\
6.25 & 6.22 &
\begin{tabular}{c} Harmonic ratio with 49.750 \\ Hz (three octaves lower) \\ \end{tabular} & 0.50\% \\
23.00 & 23.04 & \
\begin{tabular}{c} Harmonic ratio with 46.080 \\ Hz (one octave lower) \\ Structural vibration \\ Harmonic ratio with bell \\ \end{tabular} &
\begin{tabular}{c} \(-\)0.17\% \\ NA \\ \end{tabular} \\
32.67 & 32.76 &
\begin{tabular}{c} Signal forcing at 229.300 Hz \\ _(sesquisptimum)_ \\ \end{tabular} & \(-\)0.27\% \\
43.33 & NA &
\begin{tabular}{c} Independent structural \\ vibration \\ \end{tabular} & NA \\
46.08 & 45.86 &
\begin{tabular}{c} Harmonic ratio with bell \\ signal forcing at 229.300 Hz \\ _(sesquiqartum)_ \\ Independent structural \\ vibration \\ \end{tabular} & NA \\
49.75 & NA &
\begin{tabular}{c} Harmonic ratio with bell \\ signal forcing at 127.7 Hz \\ (lower octave) \\ \end{tabular} & \(-\)0.03\% \\
80.75 & 80.04 &
\begin{tabular}{c} Harmonic ratio with bell \\ signal forcing at 560.300 Hz \\ _(sesquisptimum)_ \\ Harmonic ratio with bell \\ signal forcing at 229.300 Hz \\ (one octave lower) \\ \end{tabular} & 0.88\% \\
114.70 & 114.65 &
\begin{tabular}{c} Signal forcing at 229.300 Hz \\ (one octave lower) \\ \end{tabular} & 0.04\% \\
118.50 & 119.21 &
\begin{tabular}{c} Harmonic ratio with bell \\ signal forcing at 953.700 Hz \\ (three octaves lower) \\ \end{tabular} & \(-\)0.60\% \\
144.80 & 144.65 &
\begin{tabular}{c} Harmonic ratio with bell \\ signal forcing at 289.300 Hz \\ (one octave lower) \\ \end{tabular} & 0.10\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation output frequency peaks and expected acoustic frequency partials (based on the calculation of harmonic proportions), their relation and their percentage relative deviation. that a sound source with specific a frequency pattern emission (like in the case of any musical instrument, such as a bell), also known as vibroacoustic fingerprint [1], can trigger a structural response, covering a wider range of frequencies, including those lower than the original acoustic forcing source. The study of acoustic transmission through solid materials, especially in the context of structural systems, remains underexplored, despite its critical relevance to engineering and heritage conservation. This phenomenon, often referred to as solid-borne sound transmission, plays a significant role in understanding the vibroacoustic behavior of structures subjected to dynamic loads, including heritage buildings like bell towers. Surprisingly, only a limited number of indexed works in _Scopus_ explicitly address this topic, highlighting a significant gap in the literature. The foundational contribution to this area of research through the application of finite element modeling and analysis established a robust framework for analyzing fluid-structure interaction problems [24].