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.
