本研究旨在探討移動聲源所引發之聲場，而內容上共分為兩個部份。第一個部份是要建立移動聲源的數學模型，以及對應的數值演算法；第二個部份則是要運用所建立的數值演算法，去探討實際的工程聲學問題。在第一個部份中，首先是推導出一個新的統馭方程式，該方程式可視為是一種修正型的Ffowcs Williams-Hawkings方程式(FW-H方程式)，而其最大的特點為能兼顧內聲場(Interior Domain)與外聲場(Exterior Domain)問題。此外，為了使新的統馭方程式能擴展至移動聲源作非等速運動的情形，也推導出新的聲學位置向量(Acoustic Position Vector)表示法。所謂的聲學位置向量，描述的是聲源點在聲波發射時，與接收點收到聲波時，兩者之間方向與距離的關係。至於統馭方程式的解，則利用格林函數(Green’s Function)將之表示成表面積方程(Surface Integral Formula, SIF)，其目的是為了以時域邊界元素法建立對應的數值演算法(Boundary Element Method in Time Domain, BEMTD)。
在第二個部份中，將以兩個實際個工程案例來展現所建立演算法之特性。第一個案例，是探討變速度移動線聲源，該案例能清楚展現本文所建立的演算法，確實能解除過去相關演算法，其僅能計算等速度移動物體的限制，另外模擬的結果也顯示出，當移動聲源以等加速或等減速運動時，其聲波頻率與振幅的變化率會比等速度運動情況下來的劇烈，而且觀測點接收到最大聲壓振福的時刻也會因此提早或提前。第二個案例則是移動音源聽覺感知的模擬，在該案例中，將利用所建立的演算法計算出耳道入口處之聲壓，並結合黃-希爾伯特轉換(Hilbert Huang Transformation, HHT)作聲壓訊號的時頻分析，以期能了解就感知移動音源之速度與方向而言，不同聽覺感知線索(Cue)之間的權重，結果顯示，雙耳位準差（Interaural Level Difference, ILD）與頻率偏移( Frequencies Shifting )比起雙耳時間差（Interaural Time Difference, ITD）是較為重要的感知線索。此外在相同條件下，移動聲源比靜止聲源對於聽者而言，會感知到較大的響度位準(Loudness Level)。上述兩個工程案例，都確實顯示出移動聲源所引發的聲場，其聲學特性相較靜止聲源所引發的有著很大的不同。而本研究所建立之分析方法，能有效模擬並分析上述類似之聲學問題（無論聲源是處於是靜止狀態或移動狀態），進而從中獲得有意義之物理現象。

Abstract
The Phenomenon of the sound field generated by a moving sound source has
been investigated in the present work with two-part subject. The first part is to establish
the mathematical model and the corresponding numerical scheme; the second part is to
employ the established numerical scheme in practical acoustic problems. In order to
establish the mathematical model, a novel governing equation is derived and it can be
viewed as a modified Ffowcs Williams-Hawkings equation (FW-H-equation). The
major characteristic of the novel governing equation is to include the interior and the
exterior domain. In addition, the acoustic position vector, the quality describes the
relations about the distance and the direction between the observer and the sound
source, is represented for applying to the condition of the sound source moving with
variable speed. As to the solution of the governing equation, it is expressed in the form
of the Surface Integral Formula (SIF) by convoluting the free-space Green’s function in
time domain. Then, this SIF is used to numerical implementation in the concept of the
Boundary Element Method in time domain (BEMTD).
After establishing the numerical scheme and verifying the correctness of the
calculating results, two acoustic problems were investigated for revealing the ability of
the numerical scheme. The first case considers that a moving line source with variable
speed. This case reveals that the restriction of the constant speed is released in the
established numerical scheme. For the simulation results, it shows that the effect of the
variable speed not only influenced the variation rate of the frequency modulation, i.e.,
Doppler effect, but also the time about the maximum acoustic pressure being observed.
In addition, the rate of the amplitude variation is shaper than that in the constant speed
case when the line source is approaching to the observer point. The second case
investigates the binaural hearing perceived by a moving sound source. For
understanding the weighting of the eventful cues about perceiving the direction and the
speed of the moving sound source, the sound pressure at the entrance of the external ear
canal was calculated by the established numerical scheme. Furthermore, the Hilbert
Huang transformation (HHT) is used to find the instantaneous frequencies of acoustic
signal. Results show that the Interaural Level Difference (ILD) and the frequencies
shifting are eventful than Interaural Time Difference (ITD). The perceived loudness
level will be larger in the motional case than that found in the stationary case. These
engineering problems are shown that the acoustic properties are different for
comparing with the sound field generated by a moving sound source and that by a
stationary sound source. As to the analytical methodology developed in the present
work, it turns out that it indeed can be used to simulate and analyze these acoustic
problems whenever the sound source moves or not. Furthermore, some meaningful
phenomenon relating to these problems then can be observed through discussing the
calculating results.

Bibliography
[1] Morgans, W.R., “The Kirchhoff Formula Extended to a Moving Surface”, Philosophical Magazine, pp. 141-161 (1930)
[2] Ffowcs Williams, J.E., and Hawkings, D.L., “Sound Generation by Turbulence and Surfaces in Arbitrary Motion”, Philosophical Transactions for the Royal Society of London, Series A, 264, pp. 321-342 (1969)
[3] Farassat, F. and Myers, M.K., “Extension of Kirchhoff's Formula to Radiation from Moving Surfaces”, Journal of Sound and Vibration, 123, pp. 451-460 (1988)
[4] Myers, M.K. and Hausmann, J.S., “On the Application of the Kirchhoff Formula for Moving Surfaces”, Journal of Sound and Vibration, 139(1), pp. 174-178 (1990)
[5] Myers, M.K. and Hausmann, J.S., “Computation of Acoustic Scattering from a Moving Rigid Surface”, Journal of the Acoustical Society of America, 91(5), pp. 2594-2605 (1992)
[6] Farassat, F. and Myers, M.K., “The Kirchhoff Formula for a Supersonically Moving Surface”, First Joint CEAS/AIAA Aeroacoustics Conference (16thAIAA Conference), Munich, Germany (1995)
[7] Farassat, F., “The Kirchhoff Formulas for Moving Surfaces in Aeroacoustics-The Subsonic and Supersonic Cases”, NASA Langley Research Center, Hampton, VA 23681-0001 (1996)
[8] Wu, X.F. and Akay, A., “Sound Radiation from Vibrating Bodies in Motion”, Journal of the Acoustical Society of America, 91(5), pp. 2544-2555 (1992)
[9] Wu, S.F. and Wang, Z., “Extended Kirchhoff Integral Formulations for Sound Radiation from Vibrating Cylinders in Motion”, Journal of Vibration and Acoustics, Transactions of the ASME, 115(3), pp. 324-331 (1993)
[10] Wu, S.F., “Nonuniqueness of solutions to extended Kirchhoff Integral Formulations”, Journal of the Acoustical Society of America, 93(2), pp. 683-695 (1993)
[11] Howe, M.S., “Theory of Vortex Sound”, Cambridge University Press (2003)
[12] Farassat, F., “The Acoustic Far-field of Rigid Bodies in arbitrary Motion”, Journal of Sound and Vibration, 32(3), pp. 387-405 (1974)
[13] Farassat, F., “Theory of Noise Generation from Moving Bodies with Application to Helicopter Rotors”, NASA Technique Report, R-451 (1975)
[14] Farassat, F., “Linear Acoustic Formulas for Calculation of Rotating Blade Noise”, AIAA Journal, 19, pp. 1122–1130 (1981)
[15] Di Francescantonio, P., “A New Boundary Integral Formulation for the Prediction of Sound Radiation”, Journal of Sound and Vibration, 202(4), pp. 491-509 (1997)
[16] Lockard, D.P., “Efficient, Two-dimensional Implementation of the Ffowcs Williams and Hawkings Equation”, Journal of Sound and Vibration, 229(4), pp. 897-911 (2000)
[17] Pilon, A.R. and Lyrintzis, A.S., “Integral Methods for Computational Aeroacoustics”, AIAA paper No. 97-0020, presented at the 35th Aerospace Science Meeting, Reno, NV (1997)
[18] Brentner, K.S. and Farassat, F., “An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulations for Moving Surfaces”, AIAA Journal, 36(8), pp.1379-1386 (1998)
[19] Lockard, D.P., “A Comparison of Ffowcs Williams-Hawkings Solvers for Airframe Noise Applications”, 8th AIAA/CEAS Aeroacoustics Conference, Breckenridge, Colorado (2002)
[20] Ianniello, S., “New Perspectives in the Use of the Ffowcs Williams-Hawkings equation for Aeroacoustic Analysis of Rotating Blades”, Journal of Fluid Mechanics, 570, pp. 79-127 (2007)
[21] Farassat, F., “Derivation of Formulations 1 and 1A of Farassat”, NASA Technique Report, 2007-214853 (2007)
[22] Lee, S.K., Brentner, K.S., Farassat, F. and Morris, P.J., “Analytic formulation and numerical implementation of an acoustic pressure gradient prediction”, Journal of Sound and Vibration, 319(3-5), pp. 1200-1221 (2009)
[23] Casalino, D., “An advanced time approach for acoustic analogy predictions”, Journal of Sound and Vibration, 261(4), pp. 583-612 (2003)
[24] Fedalaa, D., Kouidrib, S. and Rey, R., “Numerical study of time domain analogy applied to noise prediction from rotating blades”, Journal of Sound and Vibration, 321(3-5), pp. 662-679 (2008)
[26] Brebbia, C.A., “Boundary element techniques: theory and applications in engineering”, Springer-Verlag (1984)
[27] Wu, T.W., “Boundary element acoustics: fundamentals and computer codes”, Southampton, UK, Boston, WIT Press (2000)
[28] Estorff, O.V., “Boundary elements in acoustics: advances and applications”, Southampton, WIT Press (2000)
[29] Antes, H. and Baaran, J., “Noise Radiation from Moving Surfaces”, Engineering Analysis with Boundary Elements, 25(9), pp.725-740 (2001)
[30] Long, L., “The Compressible Aerodynamics of Rotating Blades Based on an Acoustic Formulation”, NASA Technique Report, 2197 (1983)
[31] Morino, L., Bernardini, G. and Gennaretti, M., “A boundary element method for the aerodynamics and aeroacoustics of bodies in arbitrary motions”, International Journal of Aeroacoustics, 2(2), pp.129–156 (2003)
[32] Gennarettia, M. and Testab, C., “A boundary integral formulation for sound scattered by elastic moving bodies”, Journal of Sound and Vibration, 314(3-5), pp. 712-737 (2008)
[33] Myers, M.K., “On the Acoustic Boundary Condition in the Presence of flow”, Journal of Sound and Vibration, 71(3), pp. 429-434 (1980)
[34] Tam , C.K.W. and Auriault, L., “Time-domain Impedance Boundary Conditions for Computational Aeroacoustics,” AIAA Journal, 34, pp. 917–923 (1996)
[35] Fung, K.Y. and Ju, H., “Time-domain Impedance Boundary Conditions for Computational Acoustics and Aeroacoustics”, International Journal of Computational Fluid Dynamics, 18, pp. 503–511 (2004)
[36] Li, X.D., Richter, C., and Thiele, F., "Time-domain impedance boundary conditions for surfaces with subsonic mean flows," Journal of the Acoustical Society of America, 119(5), pp. 2665-2676 (2006)
[37] Richter, C., Thiele, F.H., Li, X.D., and Zhuang, M., “Comparison of time-domain impedance boundary conditions for lined duct flows”, AIAA Journal, 45(6), pp. 1333-1345 (2007)
[38] Farassat, F. and Succi, G. P., “The Prediction of Helicopter Rotor Discrete Frequency Noise”, Vertica, (7), pp. 309-320 (1983)
[39] Farassat, F. and Brentner, K.S., “Supersonic quadrupole noise theory for high-speed helicopter rotors”, Journal of Sound and Vibration, 218(3), pp. 481-500 (1998)
[40] Farassat, F., “Modeling aerodynamically generated sound of helicopter rotors”, Progress in Aerospace Sciences, 39, pp. 83-120 (2003)
[41] Morino, L., Bernardini, G. and Gennaretti, M., “A boundary element method for the aerodynamic analysis of aircraft in arbitrary motions”, Computational Mechanics, 32, pp.301-311 (2003)
[42] Wang, T.Q. and Zhou, S., “Investigation on sound field model of propeller aircraft-the effect of vibrating fuselage boundary”, Journal of Sound and Vibration, 209(2), pp. 299-316 (1998)
[43] Wang, T.Q. and Zhou, S., “Investigation on sound field model of propeller aircraft-the effect of rigid fuselage boundary”, Journal of Sound and Vibration, 209(2), pp. 317-328 (1998)
[44] Seol, H., Jung, B., Suh, J.C. and Lee, S., “Prediction of non-cavitating underwater propeller noise”, Journal of Sound and Vibration, 257(1), pp. 131-156 (2002)
[45] 郭伯維,“高速鐵路噪音管制規範之探討”, 國立成功大學土木工程學系碩士班碩士論文 (2003)
[46] Howe, M.S., Iidab, M., Fukuda, T. and Maeda, T., “Theoretical and experimental investigation of the compression wave generated by a train entering a tunnel with a flared portal”, Journal of Fluid Mechanics, 425, pp.111-132 (2000)
[47] Howe, M.S., Iidab, M. and Fukuda, T., “Influence of an unvented tunnel entrance hood on the compression wave generated by a high-speed train”, Journal of Fluids and Structures, 17, pp 833-853(2003)
[48] Taeseok, Y., “Efficient prediction methods for the micro-pressure wave from a high-speed train entering a tunnel using the Kirchhoff formulation”, Journal of the Acoustical Society of America, 110(5), pp. 2379-2389 (2001)
[49] Ogata, S., Takahashi, R., Seki, S. and Hara, T., “A Practical Study of Shinkansen Tunnel Portal Noise Reduction”, The 33rd International Congress and Exposition on Noise Control Engineering, Prague, Czech Republic (2004)
[50] Lening, Y. and Bridget, M. S., “The prediction of speech intelligibility in underground stations of rectangular cross section”, Journal of the Acoustical Society of America, 109(1), pp. 266-273 (2001)
[51] Jian, K., “Modelling of train noise in underground stations”, Journal of Sound and Vibration, 195(2), pp. 241-255 (1996)
[52] Jian, K., “Method for predicting acoustic indices in long enclosures”, Applied Acoustics, 51, pp. 169-180 (1997)
[53] Jian, K., “Scale modelling of train noise propagation in an underground station”, Journal of Sound and Vibration, 202(2), pp. 298-302 (1997)
[54] Carman, R., “Prediction of Train Noise in Tunnels and Station”, The 33rd International Congress and Exposition on Noise Control Engineering, Prague, Czech Republic (2004)
[55] Nagya, A. B., Fiala, P., M□rki, F., Augusztinovicz, F., Degrande,G., Jacobs, S. and D. Brassenx, “Prediction of interior noise in buildings generated by underground rail traffic”, Journal of Sound and Vibration, 293(3-5), pp. 680-690 (2006)
[56] Lam, P. M. and Li, K. M., “A coherent model for predicting noise reduction in long enclosures with impedance discontinuities”, Journal of Sound and Vibration, 299(3), pp. 559-574 (2007)
[57] Thompson, D. J., “The influence of the contact zone on the excitation of wheel/rail noise”, Journal of Sound and Vibration, 267(3), pp. 523-535 (2003)
[58] Muto, D., Horihata, K., Makino, K., Horiuchi, M., Hashimoto, K. and Shiraishi, H., “Experimental and Computational Analysis to Reduce the Noise in High-speed Trains”, The 33rd International Congress and Exposition on Noise Control Engineering, Prague, Czech Republic (2004)
[59] Mellet, C., L□tourneaux, F., Poisson, F., Talotte, C., “High speed train noise emission: Latest investigation of the aerodynamic/rolling noise contribution”, Journal of Sound and Vibration, 293(3-5), pp. 535-546 (2006)
[60] Genesc□, M., Sol□, J., Romeu, J. and Alarc□n, G., “Pantograph noise measurements in Madrid-Sevilla high speed train (AVE)”, The 33rd International Congress and Exposition on Noise Control Engineering, Prague, Czech Republic (2004)
[61]Mitsuru I. and Takehisa T., “Evaluation Method of Low-Frequency Aeroacoustic Noise Source Structure Generated by Shinkansen Pantograph”, Quarterly Report of RTRI, 49(3), pp.184-190 (2008)
[62] Takeshi S., Mitsuru I. and Takehisa T., “Aerodynamic Noise Reduction using Porous Materials and their Application to High-speed Pantographs”, Quarterly Report of RTRI, 50(1), pp.26-31 (2009)
[63] Wu, S.F. and Zhou, Z., “Simulation of Vehicle Pass-by Noise Radiation”, Journal of Vibration and Acoustics, Transactions of the ASME, 121(2), pp. 197-203 (1999)
[64] Cheng, C. I., Wakefield, G. H., “Moving sound source synthesis for binaural electro-acoustic music using interpolated Head-Related Transfer Functions (HRTF’s)”, Computer Music Journal, 25(4), pp.57-80 (2001)
[65] Mitsuo, M., Mikio, T. and Hirofumi, Y., “A method of interpolating binaural impulse responses for moving sound images”, Acoustical Science and Technology, 24 (5), pp. 284-292 (2003)
[66] Kaczmarek, T., “Auditory perception of sound source velocity”, Journal of the Acoustical Society of America, 117 (5), pp. 3149-3156 (2005)
[67] Yukio, I., Yoiti, S., “Rendering moving sound with the doppler effect in sound space”, Applied Acoustics, 68 (8), pp. 916-922 (2007)
[68] Dowling, A., “Convective amplification of real simple sources”, Journal of Fluid Mechanics, 74(3), pp. 529-546 (1976)
[69] Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., et al. “The empirical mode decomposition and the Hilbert transform spectrum for nonlinear and non-stationary time series analysis”, Proceedings of the Royal Society of London A, 454, pp. 903–995 (1998)
[70] Qina, S. R., Zhong, Y. M., “A new envelope algorithm of Hilbert–Huang Transform”, Mechanical Systems and Signal Processing, 20, pp. 1941-1952 (2006)
[71] Danielson, D. A., “Vectors and tensors in engineering and physics”, Redwood City, Addison-Wesley (1992)
[72] Curle, N., “The Influence of Solid Boundries upon Aerodynamic Sound”, Proceedings of the Royal Society of London A, 234, pp.505-514 (1955)
[73] Wright, M. C. M., “Lecture Notes on the Mathematics of Acoustics”, Imperial College Press (2005)
[74] Aris, R.,” Vectors, tensors, and the Basic Equations of Fluid Mechanics”, Prentice-Hall (1962)
[75] Howe, M.S., “Acoustics of Fluid-Structure Interactions”, Cambridge University Press (1998)
[76] Morse, P. M. and Ungard, K. U. “Theoretical Acoustics”, McGraw-Hill, New York (1968)
[77] Edelman, A. and Murakami, H., “Polynomial Roots from Companion Matrix Eigenvalues”, Mathematics of Computation, 64(201), pp. 776 (1995)
[78] Tignol, J. P., “Galois' Theory of Algebraic Equations. Singapore”, World Scientific Publishing (2001)
[79] Hayami, K., “A projection transformation method for nearly singular surface boundary element integrals”, Springer-Verlag, Berlin (1992)
[80] Telles, J. C. F., “Self-adaptive Co-ordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integrals”, International Journal for Numerical Methods in Engineering, 24(5), pp. 959-973 (1987)
[81] Sladek, V. and Sladek, J., “Singular Integrals in Boundary Element Methods”, Computational Mechanics Publications: Southampton (1998)
[82] Chew, C. H., “Propagation of train noise in housing estates”, Applied Acoustics, 51(1), pp.1-12 (1997)
[83] Menounou, P., Busch-Vishniac, I. J. and Blackstock, D. T., “Directive line source model: A new model for sound diffraction by half planes and wedges”, ,Journal of the Acoustical Society of America, 107(6), pp. 2973-2986 (2000)
[84] Tanaka, K. and Ishii, S., “Acoustic radiation from a moving line source”, Journal of Sound and Vibration, 77(3), pp. 397-401 (1981)
[85] Wang, J. H and Lee, P. L., “Determination of the Acoustic Position Vector of a Moving Sound Source with Constant Acceleration Motion,” Engineering Analysis with Boundary Elements, 33 (8-9), pp. 1141-1144 (2009)
[86] Moller, H., Sorensen, M.F., Hammershoi, D. and Jensen, C.B., “Head-related transfer functions of human subjects”, Journal of the Audio Engineering Society, 43 (5), pp. 300-321 (1995)
[87] Strutt, J.W. (Lord Rayleigh), “On the acoustic shadow of a sphere”, Philosophical Transactions for the Royal Society of London, Series A, 203, pp.87-89 (1904)
[88] Duda, R.O. and Martens, W.L. “Range dependence of the response of a spherical head model”, Journal of the Acoustical Society of America, 104(5), pp.3048-3058 (1998)
[89] Jesteadt, W., Wier, C.C. and Green, D.M., “Intensity discrimination as a function of frequency and sensation level”, Journal of the Acoustical Society of America, 61 (1), pp.169-177 (1977)
[90] Frank, P.P. and Anthony, N.P., “Detection and identification of nonlinearities by amplitude and frequency modulation analysis”, Mechanical Systems and Signal Processing, 55(2), pp.1107-1132 (2008)
[91]Lighthill, M.J., “On the sound generated aerodynamically. Part I: General theory”, Proceeding of the Royal Society of London, Series A, 211, pp. 321-342 (1952)
[92] Kanwal, R. P., “Generalized functions: theory and applications 3rd ed”, Boston: Birkh□use (2004)
[93] Duffy, D. G., “Green’s Functions with Applications”, Chapman & Hall/CRC (2001)
[94] Farassat, F. and Brentner, K.S., “The Uses and Abuses of the Acoustic Analogy in Helicopter Rotor Noise Prediction”, Journal of the American Helicopter Society, 33(1), pp. 29-36 (1988)
[95] Lyell, M.J., “Hydrodynamic field due to forcing by modulated acoustic waves”, International Journal of Multiphase Flow, 22(1), pp. 45-54 (1996)
[96] Schram, C., “A boundary element extension of Curle's analogy for non-compact geometries at low-Mach numbers”, Journal of Sound and Vibration, 322(1-2), pp. 264-281 (2009)