Physical modelling synthesis refers to sound synthesis methods in which the waveform of the sound to be generated is computed using a mathematical model, a set of equations and algorithms to simulate a physical source of sound, usually a musical instrument.
Modelling attempts to replicate laws of physics that govern 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 describing the player's interaction with the instrument, such as plucking a string, or covering toneholes.
For example, to model the sound of a drum, there would be a mathematical model of how striking the drumhead injects energy into a two-dimensional membrane. Incorporating this, a larger model would simulate 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), describing 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.
In addition, the same concept has been applied to simulate voice and speech sounds. In this case, the synthesizer includes mathematical models of the vocal fold oscillation and associated laryngeal airflow, and the consequent acoustic wave propagation along the vocal tract. Further, it may also contain an articulatory model to control the vocal tract shape in terms of the position of the lips, tongue and other organs.
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.
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:
While not purely a hardware synth, the Yamaha DS-XG sound cards included hardware-assisted software VL physical modelling along with the Yamaha XG, wave audio, and 3D gaming sound capabilities of the chipset. But as they were not fully compatible with the AC-97 and later AC-98 standards, these chipsets have not been manufactured in nearly a decade.
The 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. Launched in 1995 with an MSRP of $5,000 (USD), the WSA1 was not a commercial success; only about 600 keyboards and 300 rack models were ever made, and most were sold at highly discounted prices.
Various Roland synthesizer models (V-Synth, V-Combo, XV-5080, Fantom, etc.) use COSM ("Composite Object Sound Modeling") physical modeling techniques to replicate guitars, brass and other instruments. COSM has been superseded by "SuperNatural", also based on physical modeling techniques. Introduced first in 2008 as part of the ARX expansion boards for Fantom hardware synthesizers, "SuperNatural" modeling is used in Roland's V-Drums (TD-30, TD-15, TD-11), V-Accordions (FR-7, FR-8) and various synth models (Jupiter 80, Integra 7, FA-08, JD-Xi, etc.) Later this has been expanded to ACB ("Analogue Circuit Behaviour"), using similar physical modeling techniques as before, which were incorporated into Roland's latest line of AIRA hardware synthesizer products (TB-3, System-1, System-1m, System-8), as well as their 'Boutique' line of hardware modules (JP08, JX03, JU06). While the Roland ESC2 chip inside the TD-30 and Integra-7 sound modules were marketed as "SuperNatural" modelling, the same ESC2 chip inside latest Roland "AIRA" and Boutique Products (System-1, System-1m, System-8, SH-01A, D-05, etc) was marketed as "ACB" or DCB ("Digital Circuit Behaviour", in case of the D-05) modelling technology.