66 2. FOUR IMPORTANT LINEAR PDE x = (x1,...,xn). In (2) the function f : U ×[0, ∞) → R is given. A common abbreviation is to write u := utt − Δu. We shall discover that solutions of the wave equation behave quite differ- ently than solutions of Laplace’s equation or the heat equation. For example, these solutions are generally not C∞, exhibit finite speed of propagation, etc. Physical interpretation. The wave equation is a simplified model for a vibrating string (n = 1), membrane (n = 2), or elastic solid (n = 3). In these physical interpretations u(x, t) represents the displacement in some direction of the point x at time t ≥ 0. Let V represent any smooth subregion of U. The acceleration within V is then d2 dt2 V u dx = V utt dx and the net contact force is − ∂V F · ν dS, where F denotes the force acting on V through ∂V and the mass density is taken to be unity. Newton’s law asserts that the mass times the acceleration equals the net force: V utt dx = − ∂V F · ν dS. This identity obtains for each subregion V and so utt = − div F. For elastic bodies, F is a function of the displacement gradient Du, whence utt + div F(Du) = 0. For small Du, the linearization F(Du) ≈ −aDu is often appropriate and so utt − aΔu = 0. This is the wave equation if a = 1. This physical interpretation strongly suggests it will be mathematically appropriate to specify two initial conditions, on the displacement u and the velocity ut, at time t = 0.

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