Will Le
The one-dimensional wave equation for wave function u(t, x) is . Using subscript to denote partial differentiation, we have the equation with initial conditions at t = 0: the initial waveform and the intial velocity waveform . Jean le Rond d'Alembert has shown that the solution for this equation can be separated into two waveform-invariant waves sliding in opposite directions: , where , and . In the applet above, check "Split wave" to split u(t, x) into and .