When barriers are placed in parallel on opposite sides of a source, their performance deteriorates markedly. However, barriers made from materials of inhomogeneous impedance eliminate this drawback by altering the behavior of sound as it undergoes multiple reflections between the barriers. In this paper, a theoretical approach is carried out to estimate the performance of the proposed barriers. By combining the ray-tracing method and sound diffraction theory, the existence of different ray paths between the proposed barriers is revealed. Compared to conventional rigid-walled barriers, barriers having inhomogeneous surfaces may have the potential to be widely used in environmental noise control.

## 1. Introduction

Sound barriers are common structures extensively used to protect against noise, both in transport and industry. The acoustic performance of a barrier was investigated theoretically, experimentally, and numerically.^{1–3} Of particular interest, a degradation in performance was reported when barriers were placed on opposite sides of a source; this resulted from the multiple sound reflections between the barriers.^{4,5} The reduction in performance can be avoided by applying an absorption layer on the inner surfaces of the parallel barriers.^{5} However, using available absorptive materials may cause environmental problems such as the accumulation of dust and bacteria as well as the irritation of the human respiratory system due to fibers from porous materials. Therefore barriers from materials of inhomogeneous impedance were proposed^{6} that offered a significant improvement over the performance of those rigid-walled ones by affecting the way sound undergoes multiple reflections.

Following the previous work,^{6} in the present study, we provide a theoretical approach by combining the ray tracing method^{7} and the sound diffraction theory.^{1} In Sec. 2 of this paper, the behavior of sound interacting with an inhomogeneous impedance surface is discussed in brief. In Sec. 3, the theoretical model is given to investigate the performance of the inhomogeneous-impedance barriers (IIBs). Finally, in Sec. 4, numerical simulations based on the finite element method (FEM) are carried out to verify the accuracy of the theoretical prediction.

## 2. Manipulation of a wavefront using a surface of inhomogeneous impedance

The wavefront manipulation due to the surface inhomogeneity has been described in detail in previous works.^{6,8} Accordingly, we present the basic concept of this subject, which is illustrated in Fig. 1(a), before launching into an analysis of the IIBs. For a specular reflection, it is known that the incident angle equals to the reflected one. However, consider now that the impedance of a boundary is no longer homogeneous but shifts over the scale of a wavelength along the reflection path. If the phase shift gradient along the boundary is constant, the direction of the reflected ray is governed by the generalized law of reflection^{8}

where *k* is the wave number of sound in air, $\theta i$ and $\theta r$ denote the angles of incidence and reflection, respectively, $\varphi $ is the phase shift due to sound reflection at the surface, which varies with the position *x* along the boundary. Therefore $d\varphi /dx$ represents the phase shift gradient along the boundary. As illustrated in Ref. 6, the inhomogeneous surface theoretically proposed can be realized by an array of slender tubes with their lengths appropriately tuned. For a surface structured by a subwavelength-spaced tubes array with tuned tube lengths, its normal specific impedance can be written as^{6,9}

where *l*(*x*) is the length of the tube at the position *x*, as illustrated by the dashed-line in Fig. 1(a).

Consequently, such a boundary with inhomogeneous impedance provides great flexibility in engineering the direction sound propagates. The core concept of the present paper is that it is possible to make use of inhomogeneous barrier surfaces to control wave reflections between parallel barriers and hence trap sound energy in a zone between the barriers and the intervening ground (called the semi-bounded domain hereafter).

## 3. Theoretical analysis on barriers having inhomogeneous impedance surfaces

In this section, a theoretical model is presented to investigate the performance of the IIBs. Figure 1(b) shows the cross sections (in the *x-y* plane) of a pair of barriers placed on opposite sides of a line source on the ground. This investigated model is equivalent to a traffic road surrounded by two parallel barriers with constant cross-section facades in three dimensions. The inner surfaces of both barriers were assumed to be inhomogeneous. For simplicity, the source **S** and receiver **R** were placed on the ground, and **S** was set in the middle between the two barriers. However, one can straightforwardly extend the current work to models with different geometries, including different source and receiver positions.

When a source is bounded by the parallel barriers, as shown in Fig. 1(b) with the given geometrical configurations, the receiver **R** is shielded from the source **S** by the near-side barrier **B1**. In this paper, the insertion loss (IL) was used to access the performance of the parallel barriers, which is defined as

IL is the decibel value of the ratio of sound pressure at **R** without ($Pw/o$) and with ($Pw$) the presence of the barriers. Moreover, the sound pressure at **R** with the presence of the barriers can be given as $pw=pdiffract$, where the term, $pdiffract$, represents the overall contributions from the diffracted rays, which leave the source, undergo reflections with different orders before diffracted at the edge **E** and then reaching **R**.

In this paper, the phase gradient of the barrier surfaces was set as $\beta =\u22122/2$, where $\beta $ has been given in Eq. (1). For the given inhomogeneous surface, the relations between the incident and reflected angles were shown in Fig. 2 of Ref. 6. One feature of such a surface is that if the sound impinges obliquely in the positive direction along the boundary surface with incident angle of 45°, the reflected angle is 0°, meaning that the sound will reflect normally from the surface.

As demonstrated by Li *et al.*,^{7} for the parallel barriers with rigid walls, countless image sources were formed by using the ray tracing method. In other words, due to the multiple reflections between the barriers, there were countless ray paths that can reach **E** and then **R**. In contrast, for the proposed IIBs with negative phase gradient, the wave behavior in the semi-bounded domain is considerably different: the behavior of the sound is altered when it reflects at the inhomogeneous barrier surfaces. Accordingly, in the process of continual reflections between the IIBs, the reflected angle of a ray gradually decreases, and even become negative, indicating that reflection occurs on the same side of the normal as the incident sound. As a result, there are only a few rays that can indeed reach **R**. In addition, for the investigated model, no ray path from **S** to **R** includes any reflection from the ground for rigid-walled barriers. In contrast, for the IIBs, it is possible that a ray may be reflected back to the ground by the inhomogeneous surface in its propagation process before reaching **R**. Furthermore, the reflections of a ray at the walls of IIBs are no longer specular but governed by Eq. (1), so the conventional image source method^{7} cannot be adopted straightforwardly. All these factors add to the complexity on theoretically investigating the performance of the IIBs.

In general, the possible ray paths, which leave **S**, undergo continual reflections before diffracted at **E**, and then eventually reach **R**, can be classified into several categories. For the investigated model, it is easy to find the first path $P0,0$, which propagates directly from **S** to **E**, as shown by Fig. 2(a). Herein the subscript before the comma represents the category of the path (group 0 for this path); while that after the comma denotes number of reflection from the barrier walls the ray undergoes before reaching **E** (or its image **E′**), which is also regarded as the order of the path.

Paths of group 1 are illustrated in Figs. 2(b) and 2(c). As shown by Fig. 2(b), the path $P1,1$ leaves **S**, reflects on **B2**, then reaches **E**. It can be expected that there are similar paths with higher orders, indicating that the rays undergo multiple reflections by **B1** and **B2**. These paths are denoted as $P1,N$ (*N* = 1,2,…). Figure 2(c) shows a path with odd reflections, i.e., $P1,(2n+1)$; for that with even reflections, the trajectory is similar except that the ray first hits the near-side barrier **B1**. For a path $P1,N$, the incident angle of the *i*th (1 ≤ *I* ≤ *N*) reflection is denoted as $\u2220i$. Consequently, by using the simple geometrical relationship and the generalized law of reflection, i.e., Eq. (1), the path $P1,N$ can be determined by the following equations:

where *D* is the distance between **S** and **B1**/**B2**. Notice that, sound rays of group 1 always propagate upward before reaching **E**. Thus the incident and reflected sound of each reflection should be on different sides of the normal of the surface. In other words, both the incident and reflected angles should be positive (the detailed description on the relation between the incident and reflected angles can be referred to Ref. 6). Hence for a truly allowable path $P1,N$, there should be $\u2220i>0i=1,2,...,N.$ However, for the IIBs with a certain negative phase gradient, the reflected angle is smaller than the incident one, and thus, there is $\u2220(i+1)<\u2220i$. Therefore only those paths below a certain threshold order are physically possible. A truly allowable path $P1,N$ should satisfy the following restriction:

where $\u2220(N+1)$ represents the incident angle to the edge **E**, as illustrated by Fig. 2(c), which equals to the reflected angle of the *N*th reflection.

A ray path of group 2 involves reflections from the ground: the ray from **S**, in turn, hits one of the barriers, reflects back to strike the ground, reflects to hit the other barrier, reflects again to strike the ground; such procedure repeats several times before the ray reaches **E**. Notice that, although a reflection from the IIB wall is complicated, a reflection by the ground is specular. Consequently, the ray tracing method^{7} can be used to track the trajectory when a ray is reflected by the ground, as shown by Figs. 2(d)–2(i). Figure 2(d) shows the lowest-order ray in group 2, which is denoted as $P2,1$. The ray hits **B2**, reflects back to strike the ground, and then reflects to hit the edge image **E′**. Figure 2(e) shows a higher-order ray that undergoes odd reflections (2*N* + 1) by the IIBs before reaching **E′**. A path of group 2 that undergoes even reflections is denoted as $P2,2N$, where $P2,2$ is shown by Fig. 2(f), and Fig. 2(g) shows the higher-order one. For paths of group 2, by using the simple geometrical relation and the generalized law of reflection, it is found that all the incident angles to **B2** are the same, denoted as $\u22201$; so do all the incident angles to **B1**, denoted as $\u22202$. Therefore a path $P2,M$ can be solved by the following equations, as

where *M* represents the total reflections by the barrier walls the ray undergoes before reaching **E,** or **E′**. Similarly, for the IIBs with a given negative phase gradient, only those paths below a certain threshold order are physically possible. The algorithm needs to address the situation where a ray path will, in reality, not strike the ground when it propagates across the barrier canyon to add to the total diffracted sound level at **R**. These unallowable paths are illustrated in Figs. 2(e) and 2(g). Consequently, a truly allowable path $P2,M$ should satisfy the following restriction:

Notice that for a ray path $Pi,N$, the subscript after the comma denotes the number of reflections from the walls. For rays in group 2, each propagation across the barrier canyon includes a reflection by the ground; thus a path $P2,M$ represents that the ray undergoes *M* reflections from the walls and *M* reflections from the ground.

The propagation of a ray in group 3, denoted as $P3,N+M$, is more complicated, which can be regarded as the combination of rays in groups 1 and 2. The ray $P3,N+M$ leaves **S**, first propagates upward by a process of continual reflections (of order *N*) from barrier walls with its reflected angle continuously decreasing (similar to the behavior of rays in group 1); then the reflected angle becomes negative, and hence the ray begins to propagate downward to the ground. This ray then undergoes continual reflections (of order *M*) from the walls and the ground before reaching **E** (similar to the behavior of rays in group 2). For clarity, only the lowest-order $P3,1+1$ are shown in Fig. 2(h). Ray paths in the last group (group 4) can be regarded as the combination, in turn, of groups 1, 2 and then 1 with different orders. In general, these rays can be represented as $P4,N+M+O$. For clarity, only the lowest-order one, denoted as $P4,1+1+1$, is illustrated in Fig. 2(i). By using the similar approaches for groups 1 and 2, these rays in groups 3 and 4 can be straightforwardly determined.

When considering the IIBs with a given negative phase gradient, for each category, there is a certain threshold order, thus only those paths below the threshold order are truly allowable. In other words, paths of higher orders cannot meet the corresponding constraint equations. In the present study, the negative phase gradient of the IIBs was set as $\beta =\u22122/2$. By using the proposed approach in the preceding text, one can find out all ray paths that indeed reach **E,** or **E′**. For the investigated model, the physically possible paths are $P0,0$, $P1,1$, $P3,1+1$, and $P4,1+1+1$. It is known that, for the rigid-wall parallel barriers, there are countless rays.^{7} In contrast, only four rays are formed for the investigated IIBs. Therefore, an improvement on the shielding effect by the IIBs can be expected.

In general, for the IIBs, a ray-based algorithm combined with the theory of sound diffraction^{1} can be used to estimate the contributions of these rays to the overall diffracted sound at the receiver **R**. This procedure is quite similar to that adopted in Ref. 7 for rigid-walled barriers. The major difference is that for the rigid-walled barriers, all ray paths can be determined straightforwardly by using the simple ray tracing method; while for the IIBs, it is much more difficult to determine the truly allowable paths. One should use the constraint equations, i.e., Eqs. (6) and (9), to verify the paths of different categories. Roughly, this ray-based algorithm requires the following parameters to determine the contribution of the diffracted sound from each ray path: the respective distances from the source **S** (*r*_{s}) and receiver **R** (*r*_{n}) to the diffraction point **E**, as well as the respective angles, $\theta S$ and $\theta n$. These parameters can be straightforwardly determined by using the corresponding governing equations, i.e., Eqs. (4)–(5), and (7)–(8), for each truly allowable ray path. Once all allowable paths have been found with these parameters determined, one can use this ray-based algorithm, which was described in detail in Refs. 1 and 7, to calculate the overall diffracted sound $pdiffract$ at **R**.

## 4. Numerical simulations based on the finite element method

In this section, the IIBs are analyzed by using the two-dimensional finite element method (FEM) with the commercial software comsol Multiphysics^{®}.^{10} Figure 1(b) shows the calculation domain (in grey) of the investigated model. The inner surfaces of the IIBs were set to be inhomogeneous with the normal specific impedance $Z(y)=\u2212j\rho c\u2009cot(\u22122ky/4)$, which was derived by substituting $\beta =\u22122/2$ into Eq. (2); while other surfaces, including the ground and the outer surfaces of both barriers, were rigid. As shown in Fig. 1(b), the calculation domain was bounded by perfectly matched layers (PMLs), which are artificial absorbing layers allowing waves to propagate out from the domain without reflection.^{11} The frequency range of interest was 200–4000 Hz. To ensure numerical accuracy, a fine mesh was used to divide the model into more than 130 000 triangular elements whose dimensions were kept below 0.02 m.

Figure 3(a) shows the comparison of the IL curve predicted by the theoretical model with the FEM simulation result. The general trend of the IL curve such as the position of the peaks and dips predicted by the theoretical approach coincides well with the FEM result, especially at frequencies higher than 630 Hz. Although there are some discrepancies for the predicted magnitudes between these two methods, these differences will be less significant if the results are averaged over a frequency range band. The comparison of IL curves in 1/3 octave band is shown in Fig. 3(b), which also exhibits good agreement between these two methods at frequencies higher than 630 Hz. At lower frequencies, noticeable deviation between these two methods is observed, indicating that the theoretical model, which is an energy based method, cannot provide acceptable accuracy in this range. For comparison, the IL curve of the parallel rigid-walled barriers from the previous work^{6} is also given in Fig. 3(b). It can be observed that the IIBs have a significant improvement on the performance than those with rigid walls. The shielding effect introduced by the IIBs can be observed over the frequency range of interest, which reaches around 12.3 dB at 4000 Hz by the theoretical prediction.

## 5. Conclusions

The proposed IIBs work by altering the way sound propagates as it undergoes multiple reflections between two opposite barrier walls. As long as a suitable constant-phase gradient along the surface is provided, the IIBs offer a drastic improvement in noise shielding compared to rigid-walled barriers. In this paper, a theoretical model was carried out to investigate the performance of the IIBs. Due to the surface inhomogeneity, the ray paths between the IIBs differed significantly from those between rigid-walled barriers. These ray paths were classified into several categories and were described by the corresponding governing equations. And those truly allowable rays were justified by the corresponding constraint conditions. Although in this paper, only the case with $\beta =\u22122/2$ was illustrated, it is straightforward to use the proposed algorithm to calculate the performance of IIBs with different phase gradients. Numerical simulations using FEM were also conducted that agreed well with the theoretical results. Physically, implementing the inhomogeneous boundaries causes sound energy redistribution,^{6} which leads to more energy trapped inside the semi-bounded domain with less energy escaped and hence results in a better shielding effect in the whole shadow zone. As discussed in the previous work,^{6} the theoretically proposed inhomogeneous surface can be realized by the subwavelength-structured tubes array with tuned tube lengths. Although no flow is considered in our barrier model, the wind-induced tones may not occur because the openings of these tubes^{6} are small enough compared to the thickness of the boundary layer.^{12} Furthermore, such barriers having inhomogeneous surfaces, although have different underlying working mechanism, have similar profiles with those having multiple wedges^{13} and quadratic residue diffusers.^{14} In practical applications, one can attach the tube-structured surface to a base panel^{13} to ensure the reliability of the proposed barrier.

## Acknowledgments

This work was financially supported in part by the National Natural Science Foundation of China under Grant No. 11304229 and National Key Scientific Instrument and Equipment Development Project of China under Grant No. 2012YQ15021306 as well as the Fundamental Research Funds for the Central Universities.