# Half-Space Sound Field Reconstruction Based on the Combination of the Helmholtz Equation Least-Squares Method and Equivalent Source Method

## TL;DR

This paper introduces two new techniques for sound field reconstruction in half-space environments using the HELS method and equivalent sources, improving accuracy in the presence of reflective surfaces.

## Contribution

The paper proposes two novel half-space NAH techniques combining HELS and equivalent sources to enhance sound field reconstruction accuracy.

## Key findings

- Using spherical waves as equivalent sources improves reconstruction accuracy in half-space sound fields.
- Monopole-based equivalent sources also enhance accuracy without requiring knowledge of the reflecting surface's impedance.
- Both techniques outperform the conventional HELS method in half-space environments.

## Abstract

In practical conditions, near-field acoustic holography (NAH) requires the measurement environment to be a free sound field. If vibrating objects are located above the reflective ground, the sound field becomes non-free in the presence of a reflecting surface, and conventional NAH may not identify the sound source. In this work, two types of half-space NAH techniques based on the Helmholtz equation least-squares (HELS) method are developed to reconstruct the sound field above a reflecting plane. The techniques are devised by introducing the concept of equivalent source in HELS-method-based NAH. Two equivalent sources are tested. In one technique, spherical waves are used as the equivalent source, and the sound reflected from the reflecting surface is regarded as a linear superposition of orthogonal spherical wave functions of different orders located below the reflecting surface. In the other technique, some monopoles are considered equivalent sources, and the reflected sound is considered a series of sounds generated by simple sources distributed under the reflecting surface. The sound field is reconstructed by matching the pressure measured on the holographic surface with the orthogonal spherical wave source in the vibrating object and replacing the reflected sound with an equivalent source. Therefore, neither technique is related to the surface impedance of the reflected plane. Compared with the HELS method, both methods show higher reconstruction accuracy for a half-space sound field and are expected to broaden the application range of HELS-method-based NAH techniques.