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where Z is called the acoustic impedance of the wall. (A rigid wall has a very large
impedance and an open window has zero impedance.) Now p − p 0 is in turn proportional
to ρ − ρ 0 for small disturbances. Thus the system of four equations (5.4), (5.5) satisfies
the boundary condition
v · n = a(ρ − ρ 0 ),
where a is a constant proportional to 1/Z.
A different kind of boundary condition in the case of the Wave equation is

∂u ∂u
+ b = 0. (5.7)
∂n ∂t

This condition means that energy is radiated to (b > 0) or absorbed from (b < 0) the
exterior through the boundary. For instance, a vibrating string whose ends are immersed
in a viscous liquid would satisfy (5.7) with b > 0 since energy is radiated to the liquid.

5.6 Conditions at infinity and jump conditions

In case the domain D is unbounded, the physics usually provides conditions at infinity.
These can be tricky.

Boundary conditions

D 1
D 2

Boundary conditions Jump conditions

Figure 5.3: Boundary and jump conditions

An example is Schr¨odinger’s equation, where the domain D is all of space, and we
R
2
require that |u| dx = 1. The finiteness of this integral means, in effect, that u ”vanishes
at infinity.” A second example is afforded by the scattering of acoustic or electromagnetic
waves. If we want to study sound or light waves that are radiating outward (to infinity),
the appropriate condition at infinity is ”Sommeffeld’s outgoing radiation condition

∂u ∂u
lim r − = 0, (5.8)
r→∞ ∂r ∂t

where r = Ix| is the spherical coordinate. (In a given mathematical context this limit
would be made more precise.)
Jump conditions occur when the domain D has two parts, D = D 1 ∪ D 2 (see Figure

5.3), with different physical properties. An example is heat conduction, where D 1 and D 2
consist of two different materials.

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