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M. J. Beeson and D. T. Murphy, ``RoomWeaver: A digital waveguide mesh based room acoustics research tool,'' in Proceedings of the Conference on Digital Audio Effects (DAFx-04), Naples, Italy, Oct. 2004.

J. Bensa, S. Bilbao, R. Kronland-Martinet, and J. O. Smith, ``The simulation of piano string vibration: from physical models to finite difference schemes and digital waveguides,'' Journal of the Acoustical Society of America, vol. 114(2), pp. 1095-1107, 2003.

S. Bilbao, Wave and Scattering Methods for the Numerical Integration of Partial Differential Equations,
PhD thesis, Stanford University, June 2001,
http: //

A. Chaigne, ``On the use of finite differences for musical synthesis. application to plucked stringed instruments,'' Journal d'Acoustique, vol. 5, no. 2, pp. 181-211, 1992.

A. Chaigne and A. Askenfelt, ``Numerical simulations of piano strings, parts I and II,'' Journal of the Acoustical Society of America, vol. 95, pp. 1112-1118, 1631-1640, Feb.-March 1994.

J. l. R. d'Alembert, ``Investigation of the curve formed by a vibrating string, 1747,'' in Acoustics: Historical and Philosophical Development (R. B. Lindsay, ed.), pp. 119-123, Stroudsburg: Dowden, Hutchinson & Ross, 1973.

C. Erkut and M. Karjalainen, ``Finite difference method vs. digital waveguide method in string instrument modeling and synthesis,'' in Proceedings of the International Symposium on Musical Acoustics (ISMA-02), Mexico City, 2002,
http: //

C. Erkut and M. Karjalainen, ``Virtual strings based on a 1-D FDTD waveguide model,'' Proceedings of the Audio Engineering Society 22nd International Conference, Espoo, Finland, pp. 317-323, June 15-17, 2002,
http: //

G. Essl, Jan 13, 2004,
http: //

B. Fornberg, A Practical Guide to Pseudo-Spectral Methods,
Cambridge University Press, 1998.

T. Kailath, Linear Systems,
Englewood Cliffs, NJ: Prentice-Hall, Inc., 1980.

M. Karjalainen, ``1-D digital waveguide modeling for improved sound synthesis,'' in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Orlando, Florida, USA, vol. 2, (New York), pp. 1869-1872, IEEE Press, May 2002.

M. Karjalainen, ``Mixed physical modeling: DWG + FDTD + WDF,'' in Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY, (New York), pp. 225-228, IEEE Press, Oct. 2003.

M. Karjalainen, ``Time-domain physical modeling and real-time synthesis using mixed modeling paradigms,'' in Proceedings of the Stockholm Musical Acoustics Conference (SMAC-03), http: //, (Stockholm), pp. 393-396, Royal Swedish Academy of Music, Aug. 2003.

M. Karjalainen and C. Erkut, ``Digital waveguides vs. finite difference schemes: Equivalance and mixed modeling,'' EURASIP Journal on Applied Signal Processing, vol. 2004, pp. 978-989, June 15, 2004.

A. Krishnaswamy and J. O. Smith, ``Methods for simulating string collisions with rigid spatial objects,'' in Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY, (New York), IEEE Press, Oct. 2003.

J. D. Markel and A. H. Gray, Linear Prediction of Speech,
New York: Springer Verlag, 1976.

P. Moin, Engineering Numerical Analysis,
Cambridge University Press, 2001.

P. M. Morse, Vibration and Sound,
http: // American Institute of Physics, for the Acoustical Society of America, 1948,
1st edition 1936, last author's edition 1948, ASA edition 1981.

A. W. Nayor and G. R. Sell, Linear Operator Theory in Engineering and Science,
New York: Springer Verlag, 1982.

R. Pitteroff and J. Woodhouse, ``Mechanics of the contact area between a violin bow and a string, part ii: Simulating the bowed string,'' Acta Acustica, vol. 84, pp. 744-757, 1998.

R. Pitteroff and J. Woodhouse, ``Mechanics of the contact area between a violin bow and a string, part iii: Parameter dependence,'' Acta Acustica, vol. 84, pp. 929-946, 1998.

P. M. Ruiz, A Technique for Simulating the Vibrations of Strings with a Digital Computer,
PhD thesis, Music Master Diss., Univ. Ill., Urbana, 1969.

J. O. Smith, ``Music applications of digital waveguides,'' Tech. Rep. STAN-M-39, CCRMA, Music Department, Stanford University, 1987,
CCRMA Technical Report STAN-M-39, http: //

J. O. Smith, ``Physical modeling using digital waveguides,'' Computer Music Journal, vol. 16, pp. 74-91, Winter 1992,
special issue: Physical Modeling of Musical Instruments, Part I. http: //

J. O. Smith, ``Principles of digital waveguide models of musical instruments,'' in Applications of Digital Signal Processing to Audio and Acoustics (M. Kahrs and K. Brandenburg, eds.), pp. 417-466, Boston/Dordrecht/London: Kluwer Academic Publishers, 1998.

J. O. Smith, Mathematics of the Discrete Fourier Transform (DFT),
http: // W3K Publishing, 2003,
http: //

J. O. Smith, Introduction to Digital Filters,
http: //, May 2004.

J. O. Smith, Physical Audio Signal Processing: Digital Waveguide Modeling of Musical Instruments and Audio Effects,
http: //, December 2005.

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations,
Pacific Grove, CA: Wadsworth and Brooks, 1989.

S. A. Van Duyne and J. O. Smith, ``Physical modeling with the 2-D digital waveguide mesh,'' in Proceedings of the 1993 International Computer Music Conference, Tokyo, pp. 40-47, Computer Music Association, 1993,
http: //

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``On the Equivalence of the Digital Waveguide and Finite Difference Time Domain Schemes'', by Julius O. Smith III, version published at (in PDF and PostScript formats only).
Copyright © 2005-12-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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