Physical modeling synthesis, which has developed more recently, involves a physical description of the musical instrument as the starting point for algorithm design. For most musical instruments, this will be a coupled set of partial differential equations, describing, e.g., the displacement of a string, or membrane, bar, plate, or the motion of the air in a tube, etc. The idea, then, is to solve the set of equations, invariably through a numerical approximation, to yield an output waveform, subject to some input excitation (such as glottal vibration, bow or blowing pressure, etc.). The issues mentioned above, namely those of the synthetic character and control of sounds are rather neatly dealt with in this case--there is a virtual copy of the musical instrument available to the algorithm designer or performer, embedded in the synthesis algorithm itself, which serves as a reference. For instance, simulating the plucking of a guitar string at a given location may be accomplished by sending an input signal to the appropriate location in computer memory, corresponding to an actual physical location on the string model; plucking it strongly involves sending a larger signal. The control parameters, for a physical modeling sound synthesis algorithm are typically few in number, and physically and intuitively meaningful, such as material properties, instrument geometry, and input forces and pressures.
The main drawback to using physical modelling algorithms is, and has been, their relatively large computational expense; in many cases, this amounts to hundreds, if not thousands of arithmetic operations to be carried out, usually at a high audio sample rate (such as 44.1 kHz). In comparison, a bank of six FM oscillators will require probably at most twenty floating point operations/table lookups per sample period. For this reason, research into such methods has been slower to take root, even though the first such work on musical instruments began, with Ruiz in the late 1960s and early 1970s , though digital speech synthesis based on physical models can be dated back even further, to the work of Kelly and Lochbaum . On the other hand, computer power has grown enormously in the past decades, and presumably will continue to do so, and, thus, efficiency (an obsession in the earlier days of digital sound synthesis) will become less and less of a concern.