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Can We Hear the Shape of a Noise Source\ulcorner

소음원의 모양을 들어서 상상할 수 있을까\ulcorner

  • Kim, Yang-Hann (Dept. of Mechanical Engineering, Korea Advanced Institute of Science and Technology)
  • Published : 2004.07.01

Abstract

One of the subtle problems that make noise control difficult for engineers is “the invisibility of noise or sound.” The visual image of noise often helps to determine an appropriate means for noise control. There have been many attempts to fulfill this rather challenging objective. Theoretical or numerical means to visualize the sound field have been attempted and as a result, a great deal of progress has been accomplished, for example in the field of visualization of turbulent noise. However, most of the numerical methods are not quite ready to be applied practically to noise control issues. In the meantime, fast progress has made it possible instrumentally by using multiple microphones and fast signal processing systems, although these systems are not perfect but are useful. The state of the art system is recently available but still has many problematic issues : for example, how we can implement the visualized noise field. The constructed noise or sound picture always consists of bias and random errors, and consequently it is often difficult to determine the origin of the noise and the spatial shape of noise, as highlighted in the title. The first part of this paper introduces a brief history, which is associated with “sound visualization,” from Leonardo da Vinci's famous drawing on vortex street (Fig. 1) to modern acoustic holography and what has been accomplished by a line or surface array. The second part introduces the difficulties and the recent studies. These include de-Dopplerization and do-reverberation methods. The former is essential for visualizing a moving noise source, such as cars or trains. The latter relates to what produces noise in a room or closed space. Another mar issue associated this sound/noise visualization is whether or not Ivecan distinguish mutual dependence of noise in space : for example, we are asked to answer the question, “Can we see two birds singing or one bird with two beaks?"

“소리의 비 가시성(非 可視性)”은 소음을 제어하고자 하는 공학자에게 주어진 근원적인 어려움 중의 하나이다. 따라서 이러한 비 가시성을 완화할 수 있는. 다시 말하면 소음을 볼 수 있는 방법에 대한 탐구는 많은 관심과 시도가 있어왔다. 이들 방법 중 대표적인 것으로 이론적 또는 수치적인 방법을 이용하여 소음을 가시화 하려는 많은 시도를 들 수 있다. 수치적인 방법의 경우에는 난류 유동의 가시화 분야에서 괄목할 만한 성과를 이루어왔으나 난류 소음의 가시화에는 부족한 상태이다 실험적인 방법의 경우는 마이크로폰 제작 기술의 발달로 인한 가격 인하로 많은 수의 마이크로본을 손쉽게 확보할 수 있게 되었고 매우 빠른 신호처리 기술의 발달이 확보됨으로써 많은 진전이 이루어졌으나, 가시화된 음장 정보의 정확한 해석 등이 여전히 어려운 분야로 남아있다. 또한 음장 가시화 결과에는 항상 유한한 측정 자료로 인한 오차가 포함되어 있으며, 이로 인하여 논문의 제목에서 강조한 바와 같이 소음인의 탐지 및 소음원의 공간적인 분포를 결정하는 데 어려움이 있다. 이 논문은 음장 가시화와 관련된 간략한 역사를 레오나르도 다빈치의 유명한 와류 유동 그림에서 출발하여 현대적 개념의 선형 혹은 평면형 마이크로폰 배열에 의해 수행되는 음향 홀로그래피 분야를 전반부에 설명하고, 음향 홀로그래피 구현에 있어서 발생하는 어려운 문제와 최근의 연구 동향을 후반부에 소개하는 방식으로 구성되어 있다. 최근의 문제 중에서는 자동차나 기차와 같이 이동하는 소음원을 가시화하기 위한 이동 후레임(moving frame)을 이용한 홀로그램 구성 방법과 닫힌 공간에서의 소음원 탐지를 위해 필요한 소리의 반사 효과 제거에 대하여도 기술하고 있다. 또한, 최근에 연구되는 또 다른 중요한 분야로서 “우리는 두 마리의 새가 지저귀는 것인지 아니면 한 마리의 새가 두 개의 목소리를 가지고 지저귀는 것인지 알 수 있을까?" 라는 질문으로 표현할 수 있는 공간상에 분포하는 소음원의 상호 의존관계를 구분하는 분야에 대한 문제 제기와 현실적인 접근 방법을 논하고 있다.

Keywords

References

  1. Am. Math. Mon. v.73 Can One Hear the Shape of the Drum Kac, M. https://doi.org/10.2307/2313748
  2. Basic Methods of Tomography and Inverse Problems Herman, G. T.;Tuy, H. K.;Langenberg, K. J.
  3. Inverse Problems in the Mechanics of Materials: an Introduction Bui, H. D.
  4. Inverse Thermal Problems Kurpisz, K.;Nowak, A. J.
  5. An Introduction to the Mathematical Theory of Inverse Problems Kirsch, A.
  6. Introduction to Inverse Problems in Imaging Bertero, M.;Boccacci, P.
  7. Inverse Problems for Partial Differential Equations Isakov, V.
  8. Inverse Problems and Inverse Scattering of Plane Waves Ghosh Roy, D. N.;Couchman, L. S.
  9. Lectures on Cauchy's Problem in Linear Partial Differential Equations Hardamard, J.
  10. Am. J. Math. v.73 An Iteration Formula for Fredholm Integral Equations of the First Kind Landweber, L. https://doi.org/10.2307/2372313
  11. J. App. Phys. v.34 Representation of a Function by its Line Intergrals, with Some Radiological applications Cormack, A. M. https://doi.org/10.1063/1.1729798
  12. J. App. Phys. v.35 Representation of a Function by its Line Intergrals, with Some Radiological Applications Ⅱ Cormack, A. M. https://doi.org/10.1063/1.1713127
  13. Solutions of Ⅲ-posed Problems Tikhonov, A. N.;Arsenin, V. Y.
  14. The random Transform and Some of Its Applications Deans, S. R.
  15. Proc. IEEE v.71 Mathematical Problems of Computerized Tomography Louis, A. K. https://doi.org/10.1109/PROC.1983.12596
  16. SIAM Review v.29 Can one Hear the Shape of a Drum? Revisited Protter, M. H. https://doi.org/10.1137/1029041
  17. Inverse Problems in Quantum Scattering Theory(2nd ed.) Chadan, K.;Sabatier, P. C.
  18. Inverse Problems v.8 Medical Imaging: State of the Art and Future Development Louis, A. K. https://doi.org/10.1088/0266-5611/8/5/003
  19. Inverse Acoustic and Electromagnetic Scattering Theory(2nd ed.) Colton, D.;Kress, R.
  20. Nature v.161 no.777 A New Microscopic Principle Gabor, D.
  21. J. Opt. Soc. Am. v.52 Reconstructed Wavefronts and Communication Theory Leith, E. N.;Upatnieks, J. https://doi.org/10.1364/JOSA.52.001123
  22. Proc. IEEE v.54 Hologram Television Kock, W. E.
  23. J. Opt. Soc. Am. v.57 Reconstructions of Visible Images from Reduced-scale Replicas of Microwave Holograms Tricoles, G.;Rope, E. L. https://doi.org/10.1364/JOSA.57.000097
  24. J. Opt. Soc. Am. v.57 Reconstructed Wave Forms with Large Diffraction Angles Sherman, G. C. https://doi.org/10.1364/JOSA.57.001160
  25. Appl. Phys. Lett. v.11 Digital Image Formation from Electronically Detected Holograms Goodman, J. W.;Lawrence, R. W. https://doi.org/10.1063/1.1755043
  26. Appl. Opt. v.6 Microwave Holograms and Optical Reconstruction Aoki, Y. https://doi.org/10.1364/AO.6.001943
  27. J. Opt. Soc. Am. v.60 Diffraction-limited, Scalar Image Formation with Holograms of Arbitrary Shape Porter, R. P. https://doi.org/10.1364/JOSA.60.001051
  28. J. Opt. Soc. Am. v.60 Determination of the Amplitude and the Phase of Scattered Fields by Holography Wolf, E. https://doi.org/10.1364/JOSA.60.000018
  29. J. Opt. Soc. Am. v.60 Computational Reconstruction of Scattering Objects from Holograms Carter, W. H. https://doi.org/10.1364/JOSA.60.000306
  30. J. Acoust. Soc. Am. v.68 Sound Source Reconstruction Using a Microphone Array Williams, E. G.;Maynard, J. D.;Skudrzyk, E. https://doi.org/10.1121/1.384602
  31. Phys. Rev. Lett. v.45 Holographic Imaging Without the Wavelength Resolution Limit Williams, E. G.;Maynard, J. D. https://doi.org/10.1103/PhysRevLett.45.554
  32. Nature v.237 Super-resolution Aperture Scanning Microscope Ash, E. A.;Nichols, G. https://doi.org/10.1038/237510a0
  33. Array Signal Processing Pillai, S. U.
  34. Array Signal Processing Concepts and Techniques Johnson, D. H.;Dudgeon, D. E.
  35. IEEE Trans. Acoust. Speech Signal Process v.34 The Statistical Performance of the MUSIC and the Minimum-Norm Algorithms in Resolving Plane Waves in Noise Kaveh, M.;Barabell, A. J. https://doi.org/10.1109/TASSP.1986.1164815
  36. J. Acoust. Soc. Am. v.61 Review of Undersea Acoustics to 1950 Lasky, M. https://doi.org/10.1121/1.381321
  37. J. Sound Vib. v.118 Wheel/rail Noise Generated by a High-speed Train Investigated with a Line Array of Microphones Barskow, B.;King, W. F.;Pfizenmaier, E. https://doi.org/10.1016/0022-460X(87)90257-4
  38. Proc. Inter-Noise 2002 A Class of Optimal Broadband Phased Array Geometries Designed for Easy Construction Hald, J.;Christensen, J. J.
  39. Proc. Inter-Noise 2003 Development of Visualization System for High-speed Noise Sources with a Microphone Array and a Visual Sensor Takano, Y.
  40. Proc. Inter-Noise 95 Source Localization with a Two-dimensional Focused Array: Optimal Signal Processing for a Cross-shaped Array Elias, G.
  41. Proc. Inter-Noise 2001 Wheel/rail Noise Separation with Microphone Array Measurements Nordborg, A.;Martens, A.;Wedemann, J.;Willenbrink, L.
  42. Proc. Inter-Noise 2000 Optimum Array Microphone Configuration Nordborg, A.;Wedemann, J.;Willenbrink, L.
  43. J. Acoust. Soc. Am. v.71 Forward and Backward Prediction of Acoustic Fields Using FFT Methods Stepanishen, P. R.;Benjamin, K. C. https://doi.org/10.1121/1.387606
  44. J. Acoust. Soc. Am. v.72 Numerical Evaluation of the Rayleigh Intergral for Planar Radiators Using the FFT Williams, E. G.;Maynard, J. D. https://doi.org/10.1121/1.388633
  45. J. Acoust. Soc. Am. v.78 Nearfield Acoustic Holography (NAH) : Ⅰ. Theory of Generalized Holography and the Development of NAH Maynard, J. D.;Williams, E. G.;Lee, Y. https://doi.org/10.1121/1.392911
  46. J. Acoust. Soc. Am. v.81 Nearfield Acoustic Holography (NAH) Ⅱ. Holographic Reconstruction Algorithms and Computer Implementation Veronesi, W. A.;Maynard, J. D. https://doi.org/10.1121/1.394536
  47. ASME J. Vib. Acoust. Stress Reliab. Des. v.110 Aperture Effects in Planar Nearfield Acoustical Imaging Hayek, S. I.;Luce, T. W. https://doi.org/10.1115/1.3269486
  48. J. Acoust. Soc. Am. v.93 Reconstruction of the Acoustic Field over a Limited Surface Area on a Vibrating Cylinder Sarkissian, A.;Gaumond, C. F.;Williams, E. G.;Houston, B. H. https://doi.org/10.1121/1.405630
  49. Proc. Inter-Noise 94 Reduction of Spatial Windowing Effects in Acoustical Holography Hald, J.
  50. J. Acoust. Soc. Am. v.98 Minimization of Bias Error Due to Windows in Planar Acoustic Holography Using a Minimum Error Window Kwon, H. S.;Kim, Y. H. https://doi.org/10.1121/1.413327
  51. J. Acoust. Soc. Am. v.110 Reduction Methods of the Reconstruction Error for Large-scale Implementation of Near-field Acoustical Holography Saijou, K.;Yishikawa, S. https://doi.org/10.1121/1.1405417
  52. J. Acoust. Soc. Am. v.106 Errors Due to Sensor and Position Mismatch in Planar Acoustic Holography Nam, K. U.;Kim, Y. H. https://doi.org/10.1121/1.427915
  53. J. Acoust. Soc. Am. v.68 Methods for Measuring Acoustic Radiation Fields Weinreich, G.;Arnold, E. B. https://doi.org/10.1121/1.384751
  54. J. Acoust. Soc. Am. v.78 Nearfield Acoustical Holography Using Underwater, Automated Scanner Williams, E. G.;Dardy, H. D.;Fink, R. G. https://doi.org/10.1121/1.392449
  55. J. Acoust. Soc. Am. v.82 Radial Extrapolation of Wave Fields From Synthetic Measurements of the Nearfield Blacodon, D.;Candel, S. M.;Elias, G. https://doi.org/10.1121/1.395381
  56. B&K Technical Review No. 1 STSF-A Unique Technique for Scan-based Near-field Acoustic Holography Without Restrictions on Coherence Hald, J.
  57. B&K Technical Review No. 2 STSF-Practical Instrumentation and Applications Ginn, K. B.;Hald, J.
  58. J. Sound Vib. v.233 A Method for the Efficient Construction of Acoustic Pressure Cross-spectral Matrices Yoon, S. H.;Nelson, P. A. https://doi.org/10.1006/jsvi.1999.2888
  59. J. Acoust. Soc. Am. v.113 Compensation for Source Nonstationarity in Multireference, Scan-based Near-field Acoustical Holography Kwon, H. S.;Kim, Y. J.;Bolton, J. S. https://doi.org/10.1121/1.1529669
  60. Proc. Inter-Noise Low Coherence Acoustic Holography Nam, K. U.;Kim, Y. H.
  61. J. Acoust. Soc. Am. Moving Frame Technique for Planar Acoustic Holography Kwon, H. S.;Kim, Y. H.
  62. J. Acoust. Soc. Am. v.104 An improved Moving Frame Acoustic Holography for Coherent Band-limited Noise Park, S. H.;Kim, Y. H. https://doi.org/10.1121/1.423958
  63. J. Acoust. Soc. Am. v.108 Effects of the Speed of Moving Noise Sources on the Sound Visualization by Means of Moving Frame Acoustic Holography Park, S. H.;Kim, Y. H. https://doi.org/10.1121/1.1322568
  64. J. Acoust. Soc. Am. v.110 Visualization of pass-by noise by means of moving frame acoustic holography Park, S. H.;Kim, Y. H. https://doi.org/10.1121/1.1404976
  65. J. Acoust. Soc. Am. v.76 Radial Extrapolation of Wave Fields by Spectral Methods Candel, S. M.;Chassaignon, C. https://doi.org/10.1121/1.391482
  66. J. Acoust. Soc. Am. v.81 Generalized Nearfield Acoustical Holography for Cylindrical Geometry: Theory and Experiment Williams, E. G.;Dardy, H. D.;Washburn, K. B. https://doi.org/10.1121/1.394904
  67. ASME J. Vib. Acoust. Stress Reliab. Des. v.110 A Noise Source Identification Technique Using an Inverse Helmholtz Intergral Equation Method Gardner, B. K.;Bernhard, R. J. https://doi.org/10.1115/1.3269485
  68. J. Acoust. Soc. Am. v.86 Broadband Nearfield Acoustical Holography for Vibrating Cylinders Williams, E. G.;Houston, B. H.;Bucaro, J. A. https://doi.org/10.1121/1.398245
  69. J. Acoust. Soc. Am. v.86 A Method for Computing Acoustic Fields Based on the Principle of Wave Superposition Koopmann, G. H.;Song, L.;Fahnline, J. B. https://doi.org/10.1121/1.398450
  70. J. Acoust. Soc. Am. v.88 Near-field Acoustic Holography for an Axisymmetric Geometry: A New Formulation Sarkissan, A. https://doi.org/10.1121/1.399746
  71. J. Acoust. Soc. Am. v.88 Spatial Fourier Transform Method of Measuring Reflection Coefficients at Oblique Incidence. Ⅰ: Theory and Numerical Examples Tamura, M. https://doi.org/10.1121/1.400068
  72. J. Acoust. Soc. Am. v.89 Numerical Errors Associated with the Method of Superposition for Computing Acoustic Fields Song, L.;Koopmann, G. H.;Fahnline, J. B. https://doi.org/10.1121/1.400701
  73. J. Acoust. Soc. Am. v.91 Phonoscopy: An Acoustical Holography Technique for Plane Structures Radiating in Enclosed Spaces Villot, M.;Chaveriat, G.;Ronald, J. https://doi.org/10.1121/1.402766
  74. Proc. Inter-Noise v.93 Multi-reference Nearfield Acoustical Holography in Reflective Environments Hallman, D. L.;Bolton, J. S.
  75. J. Acoust. Soc. Am. v.97 Wave-number Domain Separation of the Incident and Scattered Sound Field in Cartesian and Cylindrical Coordinates Cheng, M. T.;Mann Ⅲ. J. A.;Pate, A. https://doi.org/10.1121/1.411954
  76. J. Acoust. Soc. Am. v.97 Spatial Fourier-transform Method for Measuring Reflection Coefficients at Oblique Incidence, Ⅱ. Experimental Results Tamura, M.;Allard, J. F.;Lafarge, D. https://doi.org/10.1121/1.412940
  77. J. Acoust. Soc. Am. v.102 Helmholtz Equation-Least-squares Method for Reconstructing the Acoustic Pressure Field Wang, Z.;Wu, S. F. https://doi.org/10.1121/1.419691
  78. J. Acoust. Soc. Am. v.104 Reconstructing interior Acoustic Pressure Fields via Helmholtz Equation Least-squares Method Wu, S. F.;Yu, J. https://doi.org/10.1121/1.423719
  79. J. Acoust. Soc. Am. v.107 The Use of Partially Measured Source Data in Near-field Acoustical Holography Based on the BEM Kang, S. C.;Ih, J. G. https://doi.org/10.1121/1.428634
  80. J. Acoust. Soc. Am. v.107 On reconstruction of acoustic pressure fields using the Helmholtz equation least squares method Wu, S. F. https://doi.org/10.1121/1.428639
  81. J. Acoust. Soc. Am. v.107 Experimental validation of the HELS method for reconstructing acoustic radiation from a complex vibrating structure Rayess, N.;Wu, S. F. https://doi.org/10.1121/1.429325
  82. J. Acoust. Soc. Am. v.108 A computational acoustic field reconstruction process based on an indirect boundary element formulation Zhang, Z.;Vlahopoulos, N.;Raveendra, S. T.;Allen, T.;Zhang, K. Y. https://doi.org/10.1121/1.1317554
  83. J. Sound Vib. v.233 On the Accuracy of Nearfield Pressure Predicted by the Acoustic Boundary Element Method Kang, S. C.;Ih, J. G. https://doi.org/10.1006/jsvi.1999.2807
  84. J. Acoust. Soc. Am. v.109 Use of Non-singular Boundary Integral Formulation for Reducing Errors Due to Near-field Measurements in the Boundary element Method Based Near-field Acoustic Holography Kang, S. C.;Ih, J. G. https://doi.org/10.1121/1.1350401
  85. J. Acoust. Soc. Am. v.109 Visualization of Acoustic Radiation from a Vibrating Bowling Ball Wu, S. F.;Rayess, N.;Zhao, X. https://doi.org/10.1121/1.1361059
  86. Proc. Inter-Noise A new technique combining eigenfunction expansions and boundary elements to solve acoustic radiation problems Maynard, J. D.
  87. Fourier acoustics: sound radiation and nearfield acoustical holography Williams, E. G.
  88. Proc. Symp. Biomed. Eng. v.1 Ultrasound Holography and Visual Reconstruction Thurstone, F. L.
  89. Proc. Second International Symposium on Acoustical Holography Computer Reconstruction of Images from Ultrasonic Holograms Boyer, A. L.;Jordan, J. A.;Van Rooy, Jr.;D. L.;Hirsch, P. M.;Lesem, L. B.
  90. J. Acoust. Soc. Am. v.42 Introduction to Acoustical Holography Metilerell, A. F.;El-Sum, H. M. A.;Dreher, J. J.;Larmore, L. https://doi.org/10.1121/1.1910644
  91. Proc. Second International Symposium on Acoustical Holography Long-wavelength Holography and Visual Reproduction Methods Cram, L. A.;Rossiter, K. O.
  92. Proc. Second International Symposium on Acoustical Holography A New Method for Studying Acoustic Radiation Using Long-wavelength Acoustical Holography Graham, T. S.
  93. J. Acoust. Soc. Am. v.54 Detection of Acoustic Sources Using Long-wavelength Acoustical Holography Watson, E. E. https://doi.org/10.1121/1.1913649
  94. Acustica v.60 Restoring an Acoustic Source from Pressure Data Using Wiener Filtering Fleischer, H.;Axelrad, V.
  95. J. Acoust. Soc. Am. v.97 Supersonic Acoustic Intensity Williams, E. G. https://doi.org/10.1121/1.412991
  96. J. Acoust. Soc. Am. v.97 Acoustical Source Characterization by Using Recursive Wiener Filtering Bai, M. R. https://doi.org/10.1121/1.411897
  97. Appl. Acoust. v.48 Spherical Acoustical Holography of Low-frequency Noise Sources Lee, J. C. https://doi.org/10.1016/0003-682X(95)00068-K
  98. J. Acoust. Soc. Am. v.105 The Effect of Sensor Placement Errors on Cylindrical Near-field Acoustic Holography Carroll, G. P. https://doi.org/10.1121/1.426833
  99. J. Acoust. Soc. Am. v.85 Digital Holographic Reconstruction of Sources with Arbitrarily Shaped Surfaces Veronesi W. A.;Maynard, J. D. https://doi.org/10.1121/1.397583
  100. J. Acoust. Soc. Am. v.88 Conformal Generalized Near-field Acoustic Holography for Axisymmetric Geometries Borgiotti, G. V.;Sarkissan, A.;Williams, E. G.;Schuetz, L. https://doi.org/10.1121/1.399941
  101. J. Acoust. Soc. Am. v.88 The Relationship of Singular Value Decomposition to Wave-vector Filtering in Sound Radiation Problems Photiadis, D. M. https://doi.org/10.1121/1.399811
  102. J. Sound Vib. v.136 3-D Sound Source Reconstruction and Field Reprediction Using the Helmholtz Integral Equation Kim, G. T.;Lee, B. H. https://doi.org/10.1016/0022-460X(90)90854-S
  103. J. Acoust. Soc. Am. v.90 Acoustic Radiation from Finite Structures Sarkissan, A. https://doi.org/10.1121/1.401231
  104. J. Acoust. Soc. Am. v.90 A Numerical Solution for the General Radiation Problem Based on the Combined Methods of Superposition and Singular Value Decomposition Fahnline, J. B.;Koopmann, G. H. https://doi.org/10.1121/1.401878
  105. J. Acoust. Soc. Am. v.92 Application of BEM (boundary Element Method)-based Acoustic Holography to Radiation Analysis of Sound Sources with Arbitrarily Shaped Geometries Bai, M. R. https://doi.org/10.1121/1.404263
  106. J. Acoust. Soc. Am. v.92 The Determination of the Far Field of an Acoustic Radiator from Sparse Measurement Samples in the Near Field Borgiotti, G. V.;Rosen, E. M. https://doi.org/10.1121/1.403951
  107. J. Acoust. Soc. Am. v.100 On the Reconstruction of the Vibro-acoustic Field Over the Surface Enclosing an Interior Space Using the Boundary Element Method Kim, B. K.;Ih, J. G. https://doi.org/10.1121/1.417112
  108. J. Acoust. Soc. Am. v.107 Design of an Optimal Wave-vector Filter for Enhancing the Resolution of Reconstructed Source Field by Near-field Acoustical Holography (NAH) Kim, B. K.;Ih, J. G. https://doi.org/10.1121/1.429401
  109. J. Sound Vib. v.233 Estimation of Acoustic Source Strength by Inverse Methods: Part Ⅰ. Conditioning of the Inverse Problem Nelson, P. A.;Yoon, S. H.
  110. J. Sound Vib. v.233 Estimation of Acoustic Source Strength by Inverse Methods: Part Ⅱ. Experimental Investigation of Methods for Choosing Regularization Parameters Yoon, S. H.;Nelson, P. A.
  111. J. Acoust. Soc. Am. v.110 Regularization Methods for Near-field Acoustical Holography Williams, E. G. https://doi.org/10.1121/1.1404381
  112. J. Acoust. Soc. Am. v.112 Combined Helmholtz Equation - Least Squares Method for Reconstructing Acoustic Radiation from Arbitrarily Shaped Objects Wu, S. F.;Zhao, X. https://doi.org/10.1121/1.1487845
  113. J. Acoust. Soc. Am. v.113 Sound source reconstruction using inverse boundary element calculations Schuhmacher, A.;Hald, J.;Rasmussen, K. B.;Hansen, P. C. https://doi.org/10.1121/1.1529668
  114. J. Sound Vib. v.265 Spatial Resolution Limits for the Reconstruction of Acoustic Source Strength by Inverse Methods Kim, Y.;Nelson, P. A. https://doi.org/10.1016/S0022-460X(02)01452-9
  115. J. Acoust. Soc. Am. v.105 Holographic Reconstruction of Active Sources and Surface Admittance in an Enclosure Kim, Y. K.;Kim,Y. H. https://doi.org/10.1121/1.426842
  116. Methods of theoretical physics Morse, P. M.;Feshbach, H.
  117. Noise and vibration control engineering Beranek, L. L.;Ver, I. L.
  118. Random Data: Analysis and Measurement Procedures(2nd ed.) Bendat, J. S.;Piersol, A. G.
  119. Proc. Inter-Noise 92 Multi-reference Nearfield Acoustical Holography Hallman, D.;Bolton, J. S.
  120. Proc. Noise-Con 98 Separation of Leading Edge, Trailing Edge, and Sidewall Noise Sources from Rolling Tires Ruhala, R. J.;Burroughs, C. B.
  121. Proc. Noise-Con 98 Partial Field Decomposition in Nearfield Acoustical Holography by the Use of Singular Value Decomposition and Partial Coherence Procedures Kwon, H. S.;Bolton, J. S.
  122. Appl. Acoust. v.57 Partial Source Discrimination in Near Field Acoustic Holography Tomlinson, M. A. https://doi.org/10.1016/S0003-682X(98)00058-9
  123. J. Acoust. Soc. Am. v.109 Visualization of Multiple Incoherent Sources by the Backward Prediction of Near-field Acoustic Holography Nam, K. U.;Kim, Y. H. https://doi.org/10.1121/1.1358888
  124. Proc. Inter-Noise Visualization of Speaker, Vortex Shedding, Engine, and Wind Noise of a Car by Partial Field Decomposition Nam, K. U.;Kim, Y. H.;Choi, Y. C.;Kim, D. W.;Kwon, O. J.;Kang, K. T.;Jung, S. G.
  125. An introduction to acoustical holography B. P. Hilderbrand;B. B. Brenden