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.