## 4 Soft and Stiff Reduction Measures in the Propagation Path  
Open and filled trenches and stiff obstacles are considered in this section. The reduction effect of an open trench can be clearly seen in the vibration mode of Figure 6. Behind the trench, there is a shadow zone where the amplitudes are quite small. For an open trench, no transfer of vibrations can occur and the soil behind the trench can only be excited by the underlying soil layers. The frequency-dependent reduction ratio of the amplitudes has been evaluated at the distances between 20 to 30 m (Figure 7). Reduction ratios down to values of 0.1 are possible with an open trench (Figure 7c). This is in agreement with previously published results in [29; 30; 31]. Note that the shortest wavelength at 80 Hz of 2.5 m is less than a third of the trench depth. Real trenches are usually stabilised by an infill material. Softer materials provide a better reduction. The relevant parameter is the compressional stiffness \(k^{\prime\prime}\) of the infill [44] which can be reduced by a softer material or by a thicker trench. Figure 5: The soil amplitudes \(v\) averaged at 20 to 30 m distance for \(\square\) a strip load of \(b\) = 3 m, \(\Circle\) a strip load of \(b\) = 5 m, and \(\triangle\) a load on a rigid strip foundation (\(b\) = 3 m), compared to the averaged reference amplitudes \(v_{R}\) of a line load (the frequency-dependent reduction effect \(v/v_{R}\)). Figure 6: _Cont._Figure 6: The vibration mode of an open trench at **(a)** 20, **(b)** 40, **(c)** 60, and **(d)** 80 Hz. Figure 7: The frequency-dependent reduction effect of a 9 m deep and 0.5 m thick trench with different infill materials: **(a)** variation in the mass density + \(\rho\) = 0, and the Poisson ratio \(\triangle\)\(v\) = 0 (\(k^{\prime\prime}\) constant), \(\Circ\) standard values \(\rho\) = 2000 kg/m\({}^{3}\), \(\
u\) = 0.33; **(b)** simultaneous variation in the thickness, \(t\), and the elasticity modulus \(E\)\(\sim\)\(t\), \(t\) = 0.5 \(\triangle\) 0.25, + 0.05 m, keeping the stiffness per area \(k^{\prime\prime}\) = 16 MN/m\({}^{3}\) constant; **(c)** variation in the stiffness per area \(k^{\prime\prime}\) = \(\Circ\) 160, \(\triangle\) 64, + 32, \(\times\) 16, \(\Diamond\) 1.6, \(\square\) 0 MN/m\({}^{3}\); **(d)** different concrete obstacles \(\Circ\) wall (9 m \(\times\) 0.5 m), \(\triangle\) plate (0.5 m \(\times\) 9 m), + block (5 m \(\times\) 5 m). The wave propagation can also be impeded by a stiff material in the soil. If the,,trench\" is filled with concrete, the reduction effect of a concrete wall is analysed in Figure 8. The deformations of the near field and the surface waves are averaged over the depth by the bending stiffness of the wall. The shorter wavelengths at higher frequencies are reduced more efficiently. The frequency-dependent reduction ratios of the wall shown in Figure 7d (circle markers) go down to high-frequency values of 0.3. Similar effects are found for a plate on top of the soil in Figure 9. The horizontal wave motion is suppressed by the longitudinal stiffness and the vertical wave motion is reduced by the bending stiffness of the plate. Wheras the impedance (the wave resistance) of the soil  
Figure 8: The vibration mode of a concrete wall at (**a**) 20, (**b**) 40, (**c**) 60, and (**d**) 80 Hz. is constant, the bending stiffness is increasing with the wave number or the frequency. The bending stiffness dominates above a certain frequency which is at 20 Hz for the present case, and the amplitudes are reduced with increasing frequency (Figure 7d triangle markers). As a compact obstacle, a concrete block of dimension 5 m \(\times\) 5 m is considered (Figure 10). The reduction effect of compact obstacles is ruled by their width compared to the wavelength [45] and it becomes stronger with increasing frequency. In a mid-frequency range around 50 Hz, the reduction of the massive block (Figure 7d plus markers) is even stronger than for a wall or plate, and almost as strong as for an open trench (Figure 7c).