Linear Methods of Applied Mathematics
Evans M. Harrell II and James V. Herod*
*(c) Copyright 1994,1995,1996 by Evans M. Harrell
II and James V. Herod. All rights reserved.
It has to be admitted that the method of d'Alembert is more physically convincing than
the method of separation of variables, and students could easily wonder at this stage why
one should bother with separation of variables at all. I tell them that the two methods
have complementary advantages. Separation of variables has advantages when the
interval is finite or when it is of interest to decompose a signal into pure frequencies,
say for signal processing or filtering. The method of d'Alembert is generally preferable
when the interest is in traveling waves, and when boundaries are not very important. It
more readily exhibits the physics of the wave equation.
Many texts derive one method from the other by using trigonometric identities. While this
is somewhat instructive, I prefer not to emphasize this exercise,
since the students may come
away with the impression that one of the methods is the "real" method and the other is
subsidiary. It is important to understand that the two methods are logically different,
and neither is inferior to the other.
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