It was shown in previous publications that the impulse response of an infinite line source, as well as a point source in the three-dimensional space above an absorbing plane, can be expressed in closed form. However, line sources of a finite length are much more practical. In this study, the impulse response of a finite line source above the absorbing ground is derived and presented in a closed form, which is the main result of the present work. The corresponding analytical solution is composed of three distinct signals. Those are the direct signal, the mirrored signal, and the “pressure tail.” The direct signal is the line-of-sight signal. The mirrored signal is the signal emitted from the image source. It is weighted with the reflection coefficient of the infinite line source, which is a surprising mathematical result. The direct signal and the mirrored signal are both of finite duration. However, the third signal—the pressure tail—starts at the end of the mirrored signal and continues indefinitely for a finite surface impedance. In general, this last signal is relatively weak; it can be considered as the reverberation of the source caused by the infinite plane.

中文翻译：

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吸收平面上方有限长度的线源的声脉冲响应。

在以前的出版物中已经表明，无限线源以及在吸收平面上方的三维空间中的点源的脉冲响应可以用封闭形式表示。但是，有限长度的线源更加实用。在这项研究中，得出了吸收地面以上有限线源的脉冲响应，并以封闭形式表示，这是本研究的主要结果。相应的分析解决方案由三个不同的信号组成。这些是直接信号，镜像信号和“压力尾巴”。直接信号是视线信号。镜像信号是从图像源发出的信号。它用无限线源的反射系数加权，这是令人惊讶的数学结果。直接信号和镜像信号都具有有限的持续时间。但是，第三个信号（压力尾部）从镜像信号的末端开始，并无限期地持续有限的表面阻抗。通常，最后一个信号相对较弱。它可以被认为是无限平面引起的信号源的混响。