Computer Algebra Recipes

Chapter 6
Linear PDE Models. Part 2
In the previous chapter, the computer algebra recipes concentrated on mathematical
models formulated in terms of the di®usion and Laplace equations. In
this continuation of our study of linear PDE models, the wave equation and a
variety of models involving semi-in¯nite and in¯nite domains are presented.
6.1 Wave Equation Models
Thewaveequation,
r 2 Ã(~r; t) = 1
c2 @2Ã(~r; t)
@t2 ; (6.1)
with r2 the Laplace operator, Ã a scalar or vector function of position ~r and
time t, andcthe wave velocity, applies to many di®erent types of waves, e.g.,
² electromagnetic waves (with à a vector);
² sound waves;
² tidal waves in an incompressible °uid;
² elastic waves in a solid;
² vibrations of elastic strings and membranes.
In our ¯rst example, Vectoria plucks the humble string.
6.1.1 Vectoria Encounters Simon Legree
\We called him Tortoise because he taught us," said the Mock Turtle
angrily. \Really you are very dull!"
Lewis Carroll, Alice's Adventures in Wonderland, 1865
It is some time later in the undergraduate career of Vectoria, the physics student
who has been featured in several of our earlier stories. Recall that, inspired by
the movie Casablanca, Vectoria used Maple to check and animate a textbook
formula for the transverse motion of an initially horizontal piano string that has
been struck. Since then Vectoria has progressed in her studies and is currently
enrolled in a mathematical physics course. She has begun to learn how to solve
wave equation problems using the separation of variables method.
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