Prove finite-speed uniqueness by an energy balance

Try

At a turning point, a vibrating string can have zero velocity while still storing energy in its slope. At another instant its displacement may be nearly flat while it moves quickly. A useful conserved quantity must include both contributions.

For uₜₜ=c²uₓₓ, define local energy density e=(uₜ²+c²uₓ²)/2. Assume enough smoothness to differentiate and integrate by parts. On the line, also assume finite energy and sufficient decay when using a whole-space integral.

Predict

Will the integral of u² itself be conserved? Test a single oscillating sine mode before answering. Predict instead what combination of its displacement and velocity amplitudes stays constant.

Experiment

Multiply the PDE by uₜ. The product rule gives eₜ=∂ₓ(c²uₜuₓ). Thus the change of energy in [a,b] is the endpoint difference of c²uₜuₓ.

Endpoint condition	Why the flux vanishes
Fixed displacement u=0 for all time	uₜ=0 there
Free slope uₓ=0	The spatial factor is zero
Periodic data	Endpoint contributions cancel

For u=Acos(ct)sin x on [0,π], integrate both energy terms. The factors sin²(ct) and cos²(ct) add to 1, giving constant total energy πc²A²/4. The integral of u² alone oscillates.

Understand

Uniqueness follows by applying conservation to the difference of two solutions with identical data. Its initial velocity and spatial derivative vanish, so its energy remains zero. The difference is spatially constant and time independent; zero initial displacement fixes that remaining constant. Energy zero alone would not distinguish different constant displacements on a domain without prescribed values.

Finite propagation uses a shrinking interval. Center it at x₀ and let its endpoints be a(t)=x₀−R+ct and b(t)=x₀+R−ct. Differentiate the energy integral using the moving-boundary rule. At the right endpoint, the contribution is c²uₜuₓ−ce=−c(uₜ−cuₓ)²/2. At the left, it is −c²uₜuₓ−ce=−c(uₜ+cuₓ)²/2. Both are nonpositive.

If the initial energy is zero in [x₀−R,x₀+R], it remains zero in the shrinking interval until its endpoints meet. Applied to a difference of solutions whose initial displacement and velocity agree there, this proves agreement inside the corresponding cone. The displacement constant is fixed by the common initial data, just as in uniqueness.

The proof tracks energy crossing the cone boundary. It extends to higher-dimensional balls with appropriate flux estimates, even when no simple explicit formula is available. Heat cannot satisfy the same finite-cone conclusion: its positive Gaussian kernel transmits nonzero influence everywhere at positive time.

Apply

For speed 2, determine the interval of initial data that can influence (x,t)=(3,1). Then compare two profiles agreeing on that interval but differing outside it. State which regularity and equation assumptions permit the energy argument. Finally explain why identical total energy is far weaker than identical solutions.

Continue

Boundaries reflect waves and change the available modes. The next lesson combines modal oscillations with both initial functions, including the special treatment required for a zero eigenvalue.