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 $x$ (along the axis of the string) and may be written as follows:

$$K\, y''(x,t) = \epsilon \, {\ddot y}(t) , \protect$$ (2.1)

where $y(x,t)$ denotes the transverse displacement of the string at position $x$ along the string and time $t$ , and $y'(x,t)\isdeftext \partial y(x,t)/\partial x$ denotes the partial derivative of $y$ with respect to $x$ .^2.7 The physical parameters in this case are string tension $K$ and string mass-density $\epsilon$ . This PDE is the starting point for both digital waveguide models (Chapter 6) and finite difference schemes (§C.2.1).