## 2 Theoretical Approach for Barrier Materials  
In vibration isolation applications, maintaining slope stability in soft ground conditions presents a significant engineering challenge. Deep excavations require soil support along the inner surfaces of the open trench to ensure stability. It is important to note that the configuration of the wave barrier, encased by a thin solid wall, can alter the transmission of incident surface waves from the dynamic source to the affected structure, impacting the damping mechanism of radiated vibration energy. The reflection and transmission of Rayleigh waves at different material interfaces rely on the impedance ratio between the surrounding soil and the outer wall material of the isolation barrier. Thus, the material stiffness of the wave barrier relative to the underlying soil greatly influences vibration isolation performance. An alternative solution to traditional open trench applications for ground-borne vibration isolation was developed using a vibration-damping box wave barrier model constructed from cardboard and balsa wood, which have low impedance. Evaluating the vibrational energy transmitted to the protected area, designated as \"E\({}_{n}\)\" in the literature, relied on the barrier's impedance [2]. Impedance is calculated using shear wave velocity and material density (Equation (1)). Energy transmission, denoted as E\({}_{n}\), is calculated based on the impedance of the barrier and the impedance of the soil (Equation (2)).  
\[Z_{n}=c_{n}\times\rho_{n} \tag{1}\]  
\[E_{n}=\frac{4Z_{1}Z_{2}}{\left(Z_{1}+Z_{2}\right)^{2}} \tag{2}\]  
Theoretically, high insulation performance can be achieved with barrier models exhibiting either very high or very low-impedance values; however, a rigid wall can alter the mechanics of the vibration wave. The design calculations for the vibration wave barrier, referred to as BCCB Trench (Balsa and Corrugated Cardboard Barrier) in this study, determined the approximate unit volume weight to be in the range of 0.1-0.3 t/m\({}^{3}\). Using the formula given in Equation (1), it was concluded that the impedance value of the vibration barrier BCCB Trench is in the range of \(Z_{1}\) = 25-45 t/m\({}^{2}\)s. The very low unit volume weight of corrugated cardboard (0.09 t/m\({}^{3}\)) significantly contributed to the low impedance (\(Z_{1}\)of the barrier. This low impedance ensures that the behavior of vibration waves when they encounter the barrier is similar to their behavior when entering an open trench. The aim is to create a vibration barrier with an impedance close to that of a cavity, \(Z_{0}=0\), achieving isolation comparable to the vibration performance of an open trench without altering the refraction mechanism of ground waves. Additionally, the high modulus of elasticity of the balsa wood used for the main frame, along with the orthogonal design of the internal reinforcement system, allows the barrier to bear the lateral soil pressure. Thus, the BCCB Trench, with its high strength capable of bearing lateral soil pressure, will prevent lateral stability issues that open trenches can cause in soft soil deposits lacking self-supporting properties. Simultaneously, its low impedance will enable it to achieve insulation performance similar to that of an open trench. Based on this approach, instead of focusing only on high-impedance rigid wave barrier models, this study proposes the development of a wave barrier model based on low impedance, environmentally friendly, and recyclable materials as an alternative solution.