# Calculation of the Insertion Loss of Barriers on Rigid Ground in the Time Domain
## 1 Introduction  
Sound barriers have been widely used for noise reduction in daily life, such as roads [1] and buildings [2]. Sound barrier research essentially can boil down to the diffraction problem. Hence, to better improve the acoustic performance of noise barriers, it is important to accurately calculate the sound field diffracted by barriers.  
Since the 19 th century, many studies have been carried out on sound diffraction by barriers. In 1896, [PERSON] [3] first presented a mathematically rigorous solution for the diffraction problem of a plane wave and a cylindrical wave incident on a thin rigid half plane. [PERSON] [4] and [PERSON] [5] extended [PERSON]'s method to the spherical incident case. [PERSON] and [PERSON] [6] developed an analytical model for wedge diffraction based on Green's function and used the Laguerre technique for numerical integration. [PERSON] et al. [7] proposed an analytical method for calculating a sound field based on the integral equation method. Then, [PERSON] et al. [8] extended this method to investigating the effects of top edge impedance on sound barrier diffraction.  
Besides the frequency-domain analytical solutions above, time-domain solutions for diffraction have also been developed. [PERSON] and [PERSON] [9] provided a closed-form time-domain solution for the impulse response of an infinite rigid wedge. [PERSON] et al. [10] proposed a discrete Huygens interpretation to model diffraction using a finite wedge by assuming the existence of secondary edge sources located along the apex of the wedge. [PERSON] et al. [11, 12] extended this model by deriving analytical directivity functions for the secondary edge sources.  
In addition to the solutions above, the geometrical theory of diffraction (GTD) [13], the uniform geometrical theory of diffraction (UTD) [14], and the physical theory of diffraction (PTD) [15] are widely used diffraction theories. Moreover, the classic boundary element method (BEM) [16, 17, 18] and the finite element method (FEM) [19, 20, 21] have found application in the calculation of barrier diffraction.  
It is worth noting that most of the studies above neglected the effect of ground reflections. [PERSON] et al. [22] combined the Hadden-Pierce solution with the imaging method to study the insertion loss of absorbent barriers on the ground. [PERSON] [23] utilized a semi-empirical model to investigate the dipole sound field diffracted by a barrier above an impedance ground. [PERSON] et al. [24] developed an integral equation method for calculatingdiffraction using a rigid barrier on the ground. However, most of these methods focused on the frequency domain and rarely adopted time-domain solutions to address ground reflections. It is convenient to perform a method in the time domain as it handles the whole frequency range at the same time due to the time-domain nature, while frequency-domain methods have to repeat the process point by point in the range of interest. Moreover, this is a basis for future extension of this method to a moving sound source.  
This paper combines the time-domain Svensson method [11; 12] of wedge diffraction with the image method to investigate the insertion loss of a rigid barrier on a rigid ground. The validity of the proposed method was verified by comparing the theoretical results with those derived from the MacDonald method, which has been verified by experiments and other methods [7; 24]. A valid approximate substitution for the infinite barrier was discussed.