Research Article An Analytical Solution for Acoustic Emission Source Location for Known P Wave Velocity System Longjun Dong, 1 Xibing Li, 1 and Gongnan Xie 2 1 School of Resources and Safety Engineering, Central South University, Changsha 410083, China 2 School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710129, China Correspondence should be addressed to Longjun Dong; rydong001@csu.edu.cn Received 31 December 2013; Revised 24 February 2014; Accepted 25 February 2014; Published 27 March 2014 Academic Editor: Massimo Scalia Copyright © 2014 Longjun Dong et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. his paper presents a three-dimensional analytical solution for acoustic emission source location using time diference of arrival (TDOA) measurements from N receivers, ⩾5. he nonlinear location equations for TDOA are simpliied to linear equations, and the direct analytical solution is obtained by solving the linear equations. here are not calculations of square roots in solution equations. he method solved the problems of the existence and multiplicity of solutions induced by the calculations of square roots in existed close-form methods. Simulations are included to study the algorithms’ performance and compare with the existing technique. 1. Introduction he solution of the problem of locating a signal source using time diference of arrival (TDOA) measurements has numerous applications in aerospace, surveillance, structural health, nondestructive testing, navigation, industrial process, speaker location, machine condition, the monitoring of nuclear explosions, and mining induced areal seismology [1–17]. Many authors have discussed and faced numerous problems connected with the location of acoustic emission. he time diference of arrival TDOA method, based on estimates of time delay for a correlated signal as detected by spatially distributed sensor elements in an array, remains a commonly used technique for source location [5]. he TDOAs are proportional to the diferences in sensor- source range, called range diferences (RDs). Conventionally, the source location is estimated from the intersection of a set of hyperboloids deined by the RD measurements and the known sensor locations [18]. he inverse problem for TDOA source location is usually solved by an iterative technique such as nonlinear least squares, minimum error, or an optimization method in recognizing that the equations are nonlinear with respect to source location [19–23]. Kalman iltering has also been used to iteratively solve the TDOA source location problem for microphone speaker location [24, 25]. Although these iterative algorithms are resilient to varying extents to errors in arrival time data, they may be computationally expensive. his is a key consideration in some real-time applications [21, 22]. Closed-form solutions are usually less computationally burdensome than iterative, nonlinear minimization, or the ML method and achieve good accuracy [18]. Several closed- form analytical solutions to the TDOA source location problem have been developed [4, 21, 26, 27]. Closed-form solutions have been found in terms of intersecting spheres of distance from each sensor to an arbitrarily located source spherical intersection method, for a monitoring array of four or more sensors, in some cases admitting dual source location solutions [28, 29]. A spherical interpolation (SI) method based on linear least-squares equation error minimization with respect to source range has also been developed [30]. he two-dimensional (2D) TDOA source location problem has also been solved [21, 31]. A technique has been successfully developed by Kundu et al. [32] for localizing acoustic source in anisotropic plates that avoids the need to solve a system of nonlinear equations. he advantage of the technique is that the knowledge of the wave velocity in isotropic or anisotropic plate is not required in two-dimensional structure, but the Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 290686, 6 pages http://dx.doi.org/10.1155/2014/290686