Physical modelling synthesis
In sound synthesis, physical modelling synthesis refers to methods in which the waveform of the sound to be generated is computed by using a mathematical model, being a set of equations and algorithms to simulate a physical source of sound, usually a musical instrument. Such a model consists of (possibly simplified) laws of physics that govern the sound production, and will typically have several parameters, some of which are constants that describe the physical materials and dimensions of the instrument, while others are time-dependent functions that describe the player's interaction with it, such as plucking a string, or covering toneholes.
For example, to model the sound of a drum, there would be a formula for how striking the drumhead injects energy into a two dimensional membrane. Thereafter the properties of the membrane (mass density, stiffness, etc.), its coupling with the resonance of the cylindrical body of the drum, and the conditions at its boundaries (a rigid termination to the drum's body) would describe its movement over time and thus its generation of sound.
Similar stages to be modelled can be found in instruments such as a violin, though the energy excitation in this case is provided by the slip-stick behavior of the bow against the string, the width of the bow, the resonance and damping behavior of the strings, the transfer of string vibrations through the bridge, and finally, the resonance of the soundboard in response to those vibrations.
Although physical modelling was not a new concept in acoustics and synthesis, having been implemented using finite difference approximations of the wave equation by Hiller and Ruiz in 1971, it was not until the development of the Karplus-Strong algorithm, the subsequent refinement and generalization of the algorithm into the extremely efficient digital waveguide synthesis by Julius O. Smith III and others, and the increase in DSP power in the late 1980s that commercial implementations became feasible.
The first commercially available physical modelling synthesizer made using waveguide synthesis was the Yamaha VL1 in 1994.
While the efficiency of digital waveguide synthesis made physical modelling feasible on common DSP hardware and native processors, the convincing emulation of physical instruments often requires the introduction of non-linear elements, scattering junctions, etc. In these cases, digital waveguides are often combined with FDTD, finite element or wave digital filter methods, increasing the computational demands of the model.
Examples of physical modelling synthesis:
List of Virtual instruments 
- Tension, Electric, Collision and Corpus included with Ableton Live Suite.
- Ultrabeat, EVP88, EVB3, EVD6 and Sculpture included with Logic Pro
- Native Instrument Reaktor
- Cycling`74 Max/MSP
- Modalys (IRCAM)
- Modartt Pianoteq - Pianos
- AAS String Studio - Guitars, basses, harps, clavinets, bowed instruments, percussion
- Arturia BRASS - Trumpet, trombone and saxophone
- Sculpture (part of Logic Studio) - Strings
- Keolab Spicy Guitar - acoustic guitars
- Kong Drum Designer (Part of Reason (software)) - Drums
- Yamaha S-YXG100 plus VL and S-YXG1000 plus PolyVL (the latter released in Japan only). These were basically software-only equivalents to the hardware (and hardware-assisted software) MIDI synth capabilities of the DS-XG cards / YMF chipsets mentioned in the next section. The PolyVL had eight voice polyphony for the physical modelling, whereas the VL and all of the hardware Yamaha VL synths only had one voice, or two for the original VL-1. Like the DS-XG .VxD drivers required for VL support of the DX-XG chipsets, these would work only on pre-NT kernel versions of Windows (9# and ME), and not on NT, 2000, XP, etc. Yamaha quietly discontinued these years ago.
- Sakura from Image-Line
List of Hardware synthesizers 
- Korg OASYS and Korg Kronos - STR-1 Plucked string
- Korg OASYS PCI
- Korg Prophecy
- Korg SOLO-TRI (an expansion board for the Trinity with the synth engine of the Prophecy)
- Korg Z1
- Korg MOSS-TRI (a expansion board for the Trinity with the synth engine of the Z1) and EXB-MOSS (a multi timbral expansion board for the Triton and the KARMA workstation with the synth engine of the Z1)
- Yamaha VL1, VP1 and VL7
- Yamaha VL70m, PLG-100VL and 150VL (VL70m in the form of a plug-in card that can be installed into any of several Yamaha keyboards, tone modules, and the SW1000XG high-end PC midi sound card)
- Yamaha EX5, EX5R
- Technics WSA1/WSA1R
- Clavia Nord Modular G2
- Alesis Fusion
- Physis Unico
- Physis Piano (which is made in Italy, it's only available with 88 Keys and a full touch controlled user interface, without buttons)
- Hartmann Neuron and Neuron VS
While not purely a hardware synth, the DS-XG sound cards based on the Yamaha YMF-7#4 family of audio chipsets (including 724, 744, 754, and 764), including the Yamaha WaveForce 192 (SW192XG) as well as many from other manufacturers and even some PC motherboards with such an audio chipset, included hardware-assisted software VL physical modelling (like a VL70m or PLG-VL, and compatible with same) along with the Yamaha XG, wave audio, and 3D gaming sound capabilities of the chipset. Unfortunately, only the VxD (Virtual Device Drivers) drivers for pre-NT kernel versions of Windows (3.x, 9#, and ME) support the physical modelling feature. Neither the WDM (Windows Device Model) drivers for Windows 98, 98SE, nor ME, nor any driver for any NT-kernel version of Windows (NT, 2000, XP, Vista, Windows 2003 Server, Windows 7, Windows 2008 Server, nor likely any future OSes) support this, nor can they due to OS limitations. Those OSes do support the other features of the card, though.
In their prime, the DS-XG sound cards were easily the most affordable way of obtaining genuine VL technology for anyone who already had a Windows 3.x, 9#, or ME PC. Such cards could be had brand new for as low as $12 USD (YMF-724 versions). But since they were not fully compatible with the AC-97 and later AC-98 standards, these chipsets faded from the market and have not been manufactured by Yamaha in nearly a decade.
Technics WSA1 and its rackmounted counterpart WSA1R was Technics' first and only try at high-end synthesizers. It featured 64 voices of polyphony with a combination of sample playback (for initial transients) and DSP acoustic modelling. Technics WSA1 was launched in 1995, but the musical community did not have enough confidence in Technics to buy a $5000 hardware synth. Only about 600 keyboards and 300 rack models were ever made, and most were sold at highly discounted prices.
- Hiller, L.; Ruiz, P. (1971). "Synthesizing Musical Sounds by Solving the Wave Equation for Vibrating Objects". Journal of the Audio Engineering Society.
- Karplus, K.; Strong, A. (1983). "Digital synthesis of plucked string and drum timbres". Computer Music Journal (Computer Music Journal, Vol. 7, No. 2) 7 (2): 43–55. doi:10.2307/3680062. JSTOR 3680062.
- Julius O. Smith III (December 2010). Physical Audio Signal Processing.
- Vicinanza , D: Astra Project. http://www.astraproject.org/project.html, 2007.
- Johnstone, B: Wave of the Future. http://www.harmony-central.com/Computer/synth-history.html, 1993.
- Wood, S G: Objective Test Methods for Waveguide Audio Synthesis. Masters Thesis - Brigham Young University, http://contentdm.lib.byu.edu/cdm4/item_viewer.php?CISOROOT=/ETD&CISOPTR=976&CISOBOX=1&REC=19, 2007.
- Julius. O Smith III's A Basic Introduction to Digital Waveguide Synthesis
- Music synthesis approaches sound quality of real instruments — Stanford University's 1994 news release
- Audities Foundation Yamaha VL-1
- SoundSynthesis project
- Virtual Violin using finite elements by Christian Geiger und Michael Schreiner (including sound samples)
- ACROE Physical modelling Sound Library
- Physical Modeling on the Nord Modular G2
- NUSofting - Innovative physical modelling VST/AU instruments for computer musicians
- Pianoid - Pianoid, a piano emulating synthesizer
- Acoustic Alchemy - 1995 Review of Technics WSA1 hardware synthesizer.
- www.gersic.com:Physical Modeling Synthesis
- Physical Modelling in Digital Organs