A Prepositive Regularization Based On Wave Superposition Of Ray Wave Function For Near-field Acoustic Holography

Near-field acoustical holography(NAH)is a very effective technique for source identification,location and field visualization.By measuring the complex sound pressure,sound intensity or particle vibration velocity on the near-field holographic surface of the sound source,the sound pressure on the surface of the sound source,the normal vibration velocity and even the sound pressure,particle velocity and sound intensity at any point in the whole three-dimensional sound field can be inversely performed by using the sound field reconstruction algorithm,which has a very high application value for noise source identification and noise suppression.After decades of development,the research of nearfield acoustical holography algorithm has made great progress,including near-field acoustical holography algorithm based on spatial Fourier transform(SFT),near-field acoustical holography algorithm based on boundary element method(BEM),near-field acoustical holography algorithm based on wave superposition approach(WSA)and so on.Among them,wave superposition method has been widely used in the study of near-field acoustic holography of any sound source because of its advantages in numerical calculation and adaptability.The sound field reconstruction by the WSA represents an ill-conditioned inverse problem.However,even a small measurement error can cause reconstruction result distortion.The traditional method to obtain a stable approximation solutions is to use the subsequent regularization method after getting the linear equations.The related research has shown that when linear equations are severely ill-conditioned due to the ill-posed model,the subsequent regularization cannot provide a satisfactory approximate solution.In order to improve the ill-posed nature of the WSA mathematical model,this paper proposes a prepositive regularization method(PRM),in which the traditional monopole equivalent source spherical wave functions are replaced with strong directional wave functions that satisfy the sound field definite conditions,so the ill-conditioned problem can be improved when building the integral equations of a near-field acoustic holography model.The directional wave functions are constructed by the zeroth-order of completed solutions of the Helmholtz equation in a spherical coordinate system.Using directional wave functions as an integral kernel function can make the transfer matrix be a main-diagonally-dominant matrix.The PRM correctness in the sound field reconstruction is verified by numerical simulations,and the reconstruction stability with measurement noise is also implemented.Furthermore,the selection method of the directional wave function with different model parameters is introduced.The results show that the PRM can reconstruct the sound field correctly,especially when the transfer matrix of the traditional equivalent source method is ill-conditioned,and it is shown that stable calculation results cannot be obtained without subsequent regularization methods.Besides,the PRM can obtain satisfactory results directly and also reduce the condition number of the transfer matrix greatly and make it be in good form.The main research work and chapters are as follows:(1)The research status and development trend of acoustical holography are reviewed,and several main acoustical holography calculation methods are described.Aiming at the advantages and disadvantages of the existing acoustical holography algorithms and the problems still existing,the main research direction and content of this paper are determined.(2)Based on the physical meaning of the transfer matrix of wave superposition method,this paper analyzes in detail the cause of the ill condition of the transfer matrix caused by the traditional spherical wave function,and puts forward the core method of this paper,namely the prepositive regularization method(PRM),which uses the ray wave function with strong directivity to replace the traditional spherical wave function to improve the reconstruction stability.(3)Four construction methods of ray wave function are put forward: 1.Ray wave function of derivative type of Green function constructed by the way of finding direction derivative of Green function of free field;2.Ray wave function of spherical harmonic type constructed by the 0-th subset of complete solution of Helmholtz equation in spherical coordinate system.3.On the basis of a single n-order spherical harmonic function,the 0-order spherical harmonic function is added as the ray wave function of double spherical harmonic function.4.Using the Dirac δ function with strong directivity to constrain the solution set of Helmholtz equation to the order of 0,the δ function constrained ray wave function is obtained.In this paper,the derivation process of these four ray wave functions is given in detail,and their effectiveness and stability in sound field reconstruction are verified by a large number of numerical examples.(4)The main research work of this paper is summarized,and the future research direction is put forward.