Partial Differential Equation (PDE) An ordinary differential equation is a differential equation that has only one independent variable. For example, the angular position of a swinging pendulum as a function of time:  =  (t). However, most physical systems cannot be modeled by an ordinary differential equation because they usually depends on more than one variables. A differential equation involving more than one independent variables is called a partial differential equation. For example, the equations governing tidal waves should deal with the description of wave propagation varying both in time and space. W front =W front (x,y,z,t).

The Wave Equation Mechanical vibrations of a guitar string, or in the membrane of a drum, or a cantilever beam are governed by a partial differential equation, called wave equation, since they deal with variations taking place both in time and space taking a form of wave propagation. To derive the wave equation we consider an elastic string vibrating in a plane, as the string on a guitar. Assume u(x,t) is the displacement of the string away from its equilibrium position u=0. We can derive a partial differential equation governing the behavior of u(x,t) by applying the Newton’s second law and several simple assumptions (see chapter 11.2 in textbook)

u(x,t) x u T(x+  x,t) T(x,t) x x+  x  