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This time the infinite
plane is replace by the semi-infinite
plane, with a boundary condition imposed on
.
 |
(1.13) |
Consider the case
, so that the waves are travelling from
small
to large
. The characteristics are again given by
The characteristic through the origin, splits the
plane into
two regions, as shown in Figure 1.6.
Figure 1.6:
The characteristic,
, splits the
plane into
two regions
|
In region
,
,the solution is determined by the initial
condition at
. In this region the characteristics originate from
the
-axis.
In region
,
, the solution is given by the boundary condition
at
. Here the characteristics originate from the
-axis.
Remembering that the solution is constant along the characteristics,
this implies that the solution is determined by where the
characteristics come from. Thus, the solution is
 |
(1.14) |
Note that the characteristics are
It is clear that (1.14) satisfies the initial condition,
since
implies that
and
, and the boundary
condition since
implies that
and so
.
In addition, it is easily shown that
also satisfies the PDE.
Example 1. .9
with
The characteristics are given by
Therefore the plane is split into three regions and the solution is
A little bit of thought about this last example will soon illustrate a
major problem. If the wave speed is negative, i.e.
, then there
is in general no solution to the initial-boundary-value
problem. This is because the characteristics intersect both
boundaries and
cannot be defined by two different values
(resulting from values of
and
) on the same
characteristics. This is illustrated in Figure 1.7.
Figure 1.7:
The characteristic, with
, intersects both the
-axis and the
-axis.
|
Next: Method of Characteristics :
Up: Method of Characteristics
Previous: Initial-value Problems
Prof. Alan Hood
2000-10-30