# Effects of the Top Edge Impedance on Sound Barrier Diffraction
## 1 Introduction  
Sound barriers have been widely used for the alleviation of noise nuisance in outdoor engineering projects, such as highways [1], railways [2], and high-rise buildings [3]. Although acoustic performance is important for noise barriers, other factors, such as noise source visibility and barriers aesthetics, also need to be considered when designing noise barriers, as they affect the subjective perception of the barrier effectiveness in complex ways [4; 5; 6; 7; 8]. Therefore, although the acoustic performance of barriers can be simply improved by increasing the barrier height, this cannot be done in practical applications due to various objective and subjective factors such as cost and visual effects. To improve the acoustic performance of a noise barrier without increasing its height, various types of barrier have been investigated. One type of the new barriers involves modifying the shape of the barrier, and barriers with new profiles, such as T-shape barriers, Y-shape barriers, circular barriers [9], multiple edge barriers [10], and wave-trapping barriers [11] have been designed. Another type involves suppressing the sound pressure at the barrier's edge by installing an acoustically soft [12] or absorbent material [13]. These two types can be considered simultaneously when designing the barriers. [PERSON] and [PERSON] tested the acoustic performance of noise barriers with various shapes and surface conditions using a two-dimensional Boundary Element Method (BEM), and they found that the absorbing and soft edges improved the efficiency of the barrier significantly and the T-shape design was the most effective in the absorbing and the soft cases [9]. In further improving the acoustic performance, barriers with complex shapes of top devices have attracted a lot of attention. [PERSON] and [PERSON] investigated the acoustic performance of noise barriers with quadratic residue diffuser (QRD) tops using a two-dimensional BEM and found that insertion loss produced by utilizing QRD on different barrier profiles varied significantly with frequency and was strongly dependent on the design frequency of the QRD [14]. Later, [PERSON] et al. found that the diffuser performance could be improved by treating the diffuser with perforated sheets either on the top surface or inside the wells [15]. [PERSON] and [PERSON] investigated a noise barrier with a Waterwheel cylinder installed on the top edge to improve the sound shielding efficiency of a noise barrier, and the effective range was extended by assigning additional channels corresponding to the ineffective range [16; 17]. However, due to the lack of research on the effects of the top edge impedance on the sound barrier diffraction, the barrier performance can only be improved through optimization algorithms [18; 19; 20]. Besides, some phenomena in prior studies cannot be properly explained, such as findings that barriers with \"acoustically soft surface\" edge perform better than barriers with \"theoretical soft surface\" edge in certain frequency bands [21]. To further investigate the effects of top edge impedance on sound barrier diffraction, it is important to accurately calculate the sound field around barriers with arbitrary top edge impedance. The diffraction of waves by a thin half plane has been studied since the 18 th century [22]. [PERSON] [23] formulated a mathematically rigorous solution to solve generalized diffraction problems by planes due to cylindrical and spherical incident waves, and [PERSON] and [PERSON] proposed an analytical model to predict the sound pressure diffracted by a wedge [24]. Besides these two conventional analytical solutions, the geometrical theory of diffraction (GTD) developed by [PERSON] is another commonly used method, which shows that the source of the diffracted rays is determined by the incident rays toward the barrier's top and an appropriate diffraction coefficient [25].