The Wave Equation

The motion of a string of length L and constant density ρ (mass per unit length) is a simple example of a physical system described by the wave equation. See Figure 10.1.1. Assuming that the equilibrium position of the string and the interval [0, L] along the x-axis coincide, the equation of motion which describes the vertical displacement u(x, t) of the string follows by considering a short piece whose ends are at x and x+∆x and applying Newton’s second law.