Find the stationary temperature T(x,y)
in an infinite bar of rectangular cross-section
[0,
p]×[0,h] if three faces are held at
temperature 0, while, on the fourth face, T(x,b)=f(x)
where f when extended as an odd function is defined by
|
f(x) = signum ( x ) cos( x ) for -p < x £ p. |
|
(And, it is worth extending the definition of f to
the whole real line, as a 2
p-periodic function.)
In entering your answers, we will denote the coefficients in the
series by
|
T(x,y) = |
¥
å
m=1
|
( b2m-1 |
sinh((2m-1)y)
sinh((2m-1)h)
|
sin((2m-1) x) + b2m |
sinh(2my)
sinh(2mh)
|
sin(2m x) ) |
|
In this sum, m goes from 1 to infinity.
The various coefficients for which you are asked are
rational functions of m, and should be entered as such.
Use the facts that sin(m
p)=0, cos(2m
p)=1,
cos((2m
-1)
p)=
-1 to remove trig functions from your
expressions for the coefficients.
Enter
p as Pi.