Next: The wave equation --
Up: TIME DEPENDENT PROBLEMS
Previous: TIME DEPENDENT PROBLEMS
0
The wave equation,
is the prototype for PDE's of hyperbolic type. We study the pure
initial-value problem associated with (hyper.1), augmented with
2-periodic
boundary conditions and subject to prescribed initial conditions,
We can solve this equation using the method of characteristics,
which yields
We shall study the manner in which the solution depends on the
initial data. In this context the following features are of importance.
- : the principle of superposition holds.
- : influence propagates
with speed a. This is the essential feature of
hyperbolicity. In the wave equation it is reflected by the fact that the value
of w at (x,t) is
not influenced by initial values outside domain of dependence (x - at, x +
at).
- for large enough set of admissible initial data:
arbitrary initial data can be prescribed and the
corresponding solution is .
- : the solution is uniquely determined for by its initial data.
- . The wave equation (hyper.1)
describes the motion of a string with
kinetic energy, , and potential one,
. In order to show that
the total energy
is conserved in time we may proceed in one of two ways: either by the so called
energy method or by Fourier analysis.
Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998