# Effects of the Top Edge Impedance on Sound Barrier Diffraction
## 3 The Effects of Top Edge Impedance on Sound Barrier Diffraction
### Mechanism Analysis  
The good (\(\beta=27.3\mathrm{j}\beta_{0}\)) and bad (\(\beta=11.2\mathrm{j}\beta_{0}\)) impedances of 500 Hz were used as examples to explain the mechanisms. The normal sound intensity, sound pressure amplitude, normal particle velocity amplitude, and the phase difference between the sound pressure and normal particle velocity at each point on the boundary between the sound shadow zone and the sound bright zone under different boundary conditions are shown in Figure 5. For the rigid case, the sound energy in the sound shadow area mainly flowed from the area within 0.1 m from the top edge because the normal sound intensity at \(x=0.1\) m was only 1% near the top edge. The effects of different top edge impedance on the diffraction sound field were mainly reflected in the change of the normal sound intensity in this area. The sound pressure in the range of \(x<0.1\) m for the soft boundary was much less than that for the rigid boundary even though their normal particle velocities were similar, indicating that the control mechanism of the soft boundary was to reduce the sound pressure near the top edge of the barrier. For the bad top edge impedance, both the sound pressure and normal particle velocity were large with a small phase difference, resulting in the largest normal sound intensity. For the good top edge impedance, the normal particle velocity increased but was out of phase (phase difference is larger than \(\pi\)/2) with the sound pressure in the range of \(x<0.005\), as evidenced by the negative normal sound intensity in this range. The sound pressure in the range of \(x>0.005\) m was much less than that of the soft case, so normal sound intensity was the minimum.  
Figure 4: The good and bad top edge impedance: (**a**) The SPL curve at R\({}_{1}\) as a function of the imaginary part of top edge admittance; (**b**) the SPL curve at R\({}_{2}\) as a function of the imaginary part of top edge admittance; (**c**) the good and bad top edge impedance regarding the sound power flowing from \(\Sigma_{2}\) to Space III; (**d**) the SWL flowing into the sound shadow zone under different boundary conditions.  
The SPL and sound intensity distribution near the top edge of the barrier under the four boundary conditions are shown in Figure 6. In Figure 6a,b, when the bad top edge impedance was used, the pressure increased significantly at the top edge and the direction of the sound intensity was nearly parallel to the top edge, which can increase the diffracted sound according to the GTD theory [25]. In Figure 6c, the incident sound wave interfered with the secondary sound wave generated at the top edge, forming a dipole-like radiation above the barrier, so the sound energy radiating into the sound shadow zone was reduced. Compared with the soft boundary sound field shown in Figure 6d, the size of the low sound pressure range at the top of the barrier with the good top edge impedance was larger.  
Figure 5: The physical quantity on the boundary between the sound shadow zone and the sound bright zone: (**a**) The normal sound intensity; (**b**) the sound pressure amplitude; (**c**) the normal particle velocity amplitude; (**d**) the phase difference between the sound pressure and normal particle velocity, where bad impedance is for \(\beta=11.2\mathrm{j}\beta_{0}\), rigid boundary is for \(\beta=10^{-10}\), good impedance is for \(\beta=27.3\mathrm{j}\beta_{0}\) and soft boundary is for \(\beta=10^{10}\).