1
Physics 384 2016 Assignment 5 SOLUTION
Problem I
Compute the steady-state temperature distribution in a thermally-conducting thin wedge
of radius
a
and angle
β
, illustrated above. The rim is held at temperature
u
0
, while the edges
of the wedge are held at zero temperature.
(a)
Work through the separation of variables in 2D plane polar coordinates
1
r
∂
∂r
r
∂
∂r
+
1
r
2
∂
2
∂θ
2
u
(
r, θ
) = 0
where
u
(
r, θ
) =
R
(
r
)Θ(
θ
)
taking note that you
cannot use the usual argument that the azimuthal basis functions are
periodic in
θ
→
θ
+ 2
π
. Show that the boundary conditions rule out exponential functions
in
θ
, so that one still ends up with trigonometric functions
Θ(
θ
) =
A
cos
kθ
+
B
sin
kθ
where
k
is a separation constant. Be sure to identify the
two linearly independent solutions
for
k
= 0
. To find the radial functions, try the simplest possible ansatz ;)!
(b)
Taking the bottom edge of the wedge to be at polar angle
θ
= 0
, determine the
eigenvalues that are allowed by the boundary conditions
Θ(
θ
= 0) = Θ(
θ
=
β
) = 0
Which, if any, of the
k
= 0
solutions satisfy these boudary conditions?
(c)
Find the linear combination of basis functions that satisfies the remaining boundary
condition
u
(
r
=
a, θ
) =
u
0
,
θ
∈
[0
, β
]
This equation should take the form of a Fourier series, but in this case the trigonometric
functions are orthogonal over the interval
θ
∈
[0
, β
]
, which is just another case of Sturm-
Liouville orthogonality with Dirichlet boundary conditions. You will need to compute the
normalization integral that is required to invert the Fourier series, and then find a closed
expression for the coefficients of the expansion.

2
Solution I
First, a note regarding my original, incorrect specification of the boundary conditions
and angular range of the wedge. The configuration of the wedge in the original statment of
the problem was a disk with a “cut” of angular width
β
(giving the conducting region the
shape of a “pacman”). I suggested to take the lower edge of the cut to lie at
θ
= 0
, and the
upper edge at
θ
=
β
, and that the rim of the the wedge was to be taken between
θ
=
β
and
θ
= 2
π
; however, because of the cut, one cannot identify the angular coordinates
θ
= 0
and
θ
= 2
π
. The correct specification for that pacman configuration would have been for
the rim to run from

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