### PDEs

A *partial* differential equation (PDE) extends ODEs by adding
one or more independent variables (usually spatial variables). For
example, the wave equation for the ideal vibrating string adds one
spatial dimension (along the axis of the string) and may be written as
follows:

(Restoring Force = Inertial Force) | (2.1) |

where denotes the

*transverse*displacement of the string at position along the string and time , and denotes the

*partial derivative*of with respect to .

^{2.7}The physical parameters in this case are string tension and string mass-density . This PDE is the starting point for both digital waveguide models (Chapter 6) and finite difference schemes (§C.2.1).

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ODEs