For second order
hyperbolic PDE problems, the vibrating string is most
frequently used as an example of a well posed problem.
Think of as representing the vertical displacement at
position **x** and time **t** of an ideal string which in static
equilibrium occupies the horizontal line joining **x = 0** and **x
= L**. Then the following IBVP models the movement of the
string subject to an initial displacement given by and
an initial velocity given by .

A dramatic example of an ill-posed,
second order hyperbolic PDE problem is given by the
following BVP for the one dimensional wave equation. It can
be shown that if **T** is irrational, then the only solution of
this BVP for the wave equation is u identically zero; whereas
if **T** is rational, the problem has infinitely many nontrivial
solution. Thus the solution fails to depend continuously on
the data - namely on the size of the region on which the
problem is stated.

Figure 2:Ill-posed, second order hyperbolic PDE problem described in Example 13.