# Effects of the Top Edge Impedance on Sound Barrier Diffraction
## 2 Theory  
### Theoretical Equation  
A two-dimensional sound field was assumed throughout this paper for less computational resources and calculation time. It has already been shown by scale-model experiments that the two-dimensional analyses can be applied to the three-dimensional situation when a point source and a receiver are in the vertical plane which is perpendicular to the barrier [17]. Figure 1 shows the barrier in the free field to be investigated and the coordinate system for solving the problem. The barrier was assumed to be located at \(-L/2<x<L/2\), \(y<0\), where \(L\) is the thickness of the barrier. The normal admittance of the top edge B\({}_{2}\) was supposed to be \(\beta\) in the following text, where \(\beta=0\) denotes the rigid boundary while \(\beta=\infty\) denotes the soft boundary. Since the purpose of this research was to study the effects of top edge impedance on sound barrier diffraction, barrier front and back surfaces B\({}_{1}\) and B\({}_{3}\) were assumed to be rigid. A point source \(S\) was located at \(\mathbf{r_{s}}=(x_{s},y_{s})\) on the left side of the barrier. The transmission loss of the barrier was supposed to be large enough to ensure that the sound field behind the barrier was only the diffraction wave. The whole space was divided into three subspaces as shown in Figure 1. Space I consisted of the space \(x<-L/2\), \(y<0\), barrier B\({}_{1}\)virtual boundary \(\Sigma_{1}\), and quadrant C\({}_{1}\), where point S indicates the primary source. Space II consisted of the space \(y>0\), barrier B\({}_{2}\), virtual boundary \(\Sigma_{1}\), \(\Sigma_{2}\), and quadrant C\({}_{2}\). Space III consisted of the space \(x>L/2\), \(y<0\), barrier B\({}_{3}\), virtual boundary \(\Sigma_{2}\), and quadrant C\({}_{3}\). The diameter of C\({}_{1}\), C\({}_{2}\), and C\({}_{3}\) is infinite. Following the Kirchhoff-Helmholtz equation, the sound field at location \(\mathbf{r}\) in each of the three subspaces can be described as [31]:  
\[p^{(\mathrm{I})}(\mathbf{r})\ =\ p_{\mathrm{I}}(\mathbf{r})+\int_{\mathrm{B}_{ 1}+\Sigma_{1}+\mathrm{C}_{1}}[G_{1}(\mathbf{r}|\mathbf{r})\frac{\partial p^{( \mathrm{I})}(\mathbf{r})}{\partial n\mathbf{r}}-p^{(\mathrm{I})}(\mathbf{r}) \frac{\partial G_{1}(\mathbf{r}|\mathbf{r})}{\partial n\mathbf{r}}]\mathrm{ d}S\mathbf{,} \tag{1}\]  
\[p^{(\mathrm{II})}(\mathbf{r})\ =\ \int_{\Sigma_{1}+\mathrm{B}_{2}+\Sigma_{2}+ \mathrm{C}_{2}}[G_{2}(\mathbf{r}|\mathbf{r})\frac{\partial p^{(\mathrm{II})}( \mathbf{r})}{\partial n\mathbf{r}}-p^{(\mathrm{II})}(\mathbf{r})\frac{ \partial G_{2}(\mathbf{r}|\mathbf{r})}{\partial n\mathbf{r}}]\mathrm{d}S \mathbf{,} \tag{2}\]  
\[p^{(\mathrm{III})}(\mathbf{r})\ =\ \int_{\mathrm{B}_{3}+\Sigma_{2}+ \mathrm{C}_{3}}[G_{3}(\mathbf{r}|\mathbf{r})\frac{\partial p^{(\mathrm{III})}( \mathbf{r})}{\partial n\mathbf{r}}-p^{(\mathrm{III})}(\mathbf{r})\frac{ \partial G_{3}(\mathbf{r}|\mathbf{r})}{\partial n\mathbf{r}}]\mathrm{d}S \mathbf{,} \tag{3}\]  
where \(\mathbf{r}^{\prime}\) is the position vector on the boundaries and \(p_{\mathrm{I}}(\mathbf{r})\) is the sound field generated by the point source, which can be obtained by \(p_{\mathrm{I}}(\mathbf{r})=\mathrm{j}\omega\rho_{0}q_{\mathrm{s}}G_{1}( \mathbf{r}|\mathbf{r}_{\mathrm{s}})\). \(\omega\) is the angular frequency, \(\rho_{0}\) is the air density, \(q_{\mathrm{s}}\) is the source strength of the point source, and \(G_{1}\), \(G_{2}\), \(G_{3}\) are the Green functions of the three subspaces, respectively.