conservation of wave energy

A plucked string left alone keeps ringing — it does not slow down or fade on its own. Behind this is a conservation law: the total energy of a wave, in the idealized frictionless model, stays exactly constant for all time. Nothing is created, nothing is lost; energy just sloshes between two forms.

For the vibrating string the total energy at time t is E(t) = (1/2) times the integral over the string of [(u_t)^2 + c^2 (u_x)^2]. The first piece, (u_t)^2, is the kinetic energy — how fast each bit of string is moving. The second, c^2 (u_x)^2, is the potential energy stored in the stretch — how steeply the string is bent. The claim is that E(t) does not change with time: dE/dt = 0. You prove it by a beautiful short computation — multiply the wave equation by u_t, integrate over the string, and integrate by parts; the wave equation u_tt = c^2 u_xx makes every term cancel in pairs, leaving the time-derivative of E equal to zero (with fixed or free ends so no energy leaks out the boundary). As the wave oscillates, energy trades back and forth between kinetic (when the string is flat and fast) and potential (when it is bent and momentarily still), but the sum holds fixed.

Conservation of energy is far more than bookkeeping — it is one of the most powerful proof tools in PDE theory, the 'energy method'. It instantly gives UNIQUENESS: if two solutions had the same initial data, their difference starts with zero energy, must keep zero energy forever, and so must stay zero — there is exactly one solution. It also gives stability (the solution depends continuously on the data) and, with boundaries, controls how waves reflect. The same multiply-and-integrate trick scales up to give energy estimates for far more general hyperbolic equations, where explicit formulas like d'Alembert's are unavailable. Note the honest caveat: energy is conserved only in the idealized lossless equation; any real damping (air resistance, internal friction) drains it, which is why a real string eventually goes quiet.