The acoustical impulse response of a line source of finite length above an absorbing plane

Martin Ochmann

doi:10.1121/10.0001314

Abstract

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.

Highlights

Acoustic formulations in time domain are more seldom than in frequency domain, but are required in many different areas of acoustics to describe the sound emitted by transient sources occurring, e.g., in outdoor sound propagation,1 room acoustics,2 medical ultrasound, etc
In a previous paper,3 it was shown that the impulse response of a point source in three-dimensional space (3-D) above an infinite plane can be expressed in a closed form if the plane is characterized by a masslike or pure absorbing behavior
A closed form expression is obtained for the impulse response of a line source of finite length above absorbing ground in the 3-D space, which is the main result of the present paper

Summary

INTRODUCTION

Acoustic formulations in time domain are more seldom than in frequency domain, but are required in many different areas of acoustics to describe the sound emitted by transient sources occurring, e.g., in outdoor sound propagation, room acoustics, medical ultrasound, etc. An analytical representation for the impulse response of a point source in two dimensions over an absorbing plane was derived, which can be equivalently considered as the sound field of an infinite line source in three dimensions (see Ref. 4). The aim of the present paper is to derive a closed-form, integral-free solution for the impulse response of a finite line source above absorbing ground. The first signal is called the mirrored signal, since it can be considered as radiated from the image point source with respect to the infinite plane and weighted with a reflection coefficient R1. It is clear that after a sufficiently long time, only the pressure tail can be observed at the location of the receiver It can be interpreted as reverberation of the infinite plane due to the excitation with a signal of finite duration. The complete solution for the impulse response of the finite line source is discussed, evaluated numerically, and presented in the graphical form

DESCRIPTION OF THE RADIATION PROBLEM

THE FINITE LINE SOURCE IN FREE SPACE

CLOSED FORM SOLUTION FOR THE FINITE LINE SOURCE ABOVE AN ABSORBING PLANE

SOLUTION FOR FINITE LINE SOURCES WITH ARBITRARY TIME SIGNALS

NUMERICAL EVALUATION

CONCLUSIONS