Wave-equation energy and domain of dependence

Take a sufficiently smooth real solution of $u_{tt}-c^2\Delta u=0$ with c positive. Multiplying by the time derivative and using the product rule gives a local conservation equation. The density contains kinetic and gradient contributions, while the flux is minus c squared times the time derivative times the spatial gradient. This sign is important: the energy inside a fixed region changes by minus the outward flux. The formula extends to finite-energy weak solutions by an appropriate approximation argument.

On all of Euclidean space, sufficiently decaying smooth data give constant total energy; periodic boundary conditions also cancel the boundary flux. On a bounded physical domain, the boundary condition must instead be included. A reflecting homogeneous boundary can preserve energy, whereas an absorbing or driven boundary can remove or inject it. One should therefore derive the boundary term before declaring a numerical or physical experiment conservative. Energy conservation is a property of the equation together with its domain and boundary behavior.

Integrate the energy over a ball whose radius decreases at speed c. Differentiation of this moving integral produces both a PDE boundary flux and a negative term from the shrinking boundary. The elementary inequality bounding the product of the normal spatial derivative and the time derivative shows that the outgoing contribution cannot exceed c times the density. Hence the energy inside that shrinking ball is nonincreasing. This argument uses the same Euclidean normal estimate in any spatial dimension.

If the displacement and velocity vanish initially in the base ball, the cone energy starts at zero and remains zero. Zero energy controls the time derivative and spatial gradient; the zero initial displacement then fixes the remaining constant, so the solution itself is zero inside the cone. Apply this to the difference of two solutions to obtain local uniqueness and the domain of dependence. For energy-class solutions the corresponding statement is understood through their standard weak or continuous-in-time representatives, rather than arbitrary point values of an equivalence class.

For smooth compactly supported initial displacement and velocity, their joint support can expand by no more than distance ct after time t. The same statement follows for finite-energy compactly supported data in the appropriate almost-everywhere sense. This is an upper bound on where influence may arrive. It does not say that every point inside the expanding set becomes nonzero, because cancellation, special initial data, or dimension-specific propagation can leave parts of that region quiet.

Likewise two initial states that agree in a ball produce the same solution in its shrinking cone, regardless of how differently they behave outside that ball. This locality is useful for truncating a computational domain: before any boundary-reflected signal can return, interior data have a protected region of dependence. Variable coefficients, lower-order terms, curved backgrounds, and nonlinear equations need their own hypotheses and adapted energy estimates. The present constant-c result should not be applied unchanged to an equation with unbounded propagation speeds.