# Numerical Modeling of Bowel Sound Propagation: Impact of Abdominal Tissue Properties
## 2 Theory and Model
### Transmission of Bowel Sounds  
#### 2.2.1 Principle of Sound Propagation  
This study adopts the assumption that the bowel sound source is not significantly influenced by complex fluid dynamics in the abdominal cavity, such as the movement or interaction of gases, liquids, or other substances. By modeling the IWC as an ideal point source, the approach bypasses the need to incorporate these fluid behaviors, allowing the focus to remain on fundamental acoustic phenomena. This assumption provides a practical framework for simulating intestinal sound propagation while maintaining both computational efficiency and analytical accuracy. Additionally, other components of the abdominal cavity, such as gaseous or liquid substances present in the peritoneal cavity, do not significantly impact the propagation or attenuation of sound waves generated by the IWC, and their effect on sound wave propagation is considered negligible. This study primarily focuses on the acoustic properties of the various tissue layers in the abdominal cavity, such as density \(\rho_{0}\) and sound speed \(c\)[20]. These tissue properties determine the propagation, reflection, refraction, and attenuation of sound as it passes through the complex anatomical structures of the abdomen, enabling a more accurate simulation of intestinal sound within the human body. By concentrating on these acoustic properties, the model provides a simplified but effective representation of sound wave propagation. Figure 1: An IWC with its bounding envelope. To simulate the propagation of bowel sounds within the complex environment of the abdominal cavity, this study employs the pressure acoustics-elasticity model available in COMSOL, a widely recognized software platform for solving complex physics-based simulations. Specifically, the study focuses on time-domain simulations, which facilitate the analysis of sound wave behavior over time as it propagates through various tissue layers within the abdominal cavity [21]. The simulation is based on the classical wave equation, a fundamental principle in acoustics that describes the propagation of sound waves as they travel through different media. This wave equation captures the dynamic relationship between sound pressure and the physical properties of the medium [22]. The equation is mathematically represented as  
\[\frac{1}{\rho_{0}c^{2}}\frac{\partial^{2}p}{\partial t^{2}}+\
abla\cdot\left( -\frac{1}{\rho_{0}}\
abla p\right)=0 \tag{3}\]  
where \(p\) is the sound pressure, \(\rho_{0}\) is the density of the medium, \(c\) is the speed of sound, and \(\
abla\) represents the gradient operator. The first term \(\frac{1}{\rho_{0}c^{2}}\frac{\partial^{2}p}{\partial t^{2}}\) in the equation describes the temporal variation of sound pressure. It provides inputs into how sound waves evolve over time as they travel through the medium. This term captures the relationship between sound pressure and wave acceleration, influencing the velocity of sound wave propagation over time. Essentially, it models the inertial effects of the medium, reflecting the time-dependent behavior of sound waves. The second term, \(\
abla\cdot\left(-\frac{1}{\rho_{0}}\
abla p\right)\), describes the propagation and diffusion of sound waves through space. Specifically, the gradient operator \(\
abla p\) quantifies the variation of sound pressure at each spatial location, while the divergence operator \(\
abla\) describes the spatial spreading or focusing of the sound waves as they propagate through the medium. The negative sign in this term reflects the physical principle that pressure gradients drive the movement of the sound waves. As a result, this term captures how sound waves expand or contract as they travel through different tissue layers, potentially experiencing attenuation or changes in direction, depending on the properties of the surrounding medium. To solve the wave equation numerically, COMSOL discretizes both space and time, transforming the continuous wave equation into a set of algebraic equations that can be solved iteratively.