Math 212 , Fall 2007
Instructor : Friedemann Brock
Homework assignment, 14 December 2007 – 4 January 2008 ,
Solution
Exercise:
Analyse the vibrations of an elastic solid cylinder occupying the
region 0
≤
r
≤
1, 0
≤
z
≤
1, in cylindrical coordinates if its top and bottom
are held fixed, its circular surface is free, and the initial velocity
u
t
is zero.
that is, find the general solution of
v
tt
=
c
2
(
v
rr
+
r

1
v
r
+
r

2
v
θθ
+
v
zz
)
,
v
(
r, θ,
0
, t
) =
v
(
r, θ,
1
, t
) =
v
r
(1
, θ, z, t
) =
v
t
(
r, θ, z,
0) = 0
.
Solution:
First we look for separated solutions,
v
=
R
(
r
)Θ(
θ
)
Z
(
z
)
T
(
t
), of the
PDE, which satisfies the given boundary and initial conditions. This leads to
T /T
=
c
2
[(
R /R
) + (
R /
(
rR
)) + (Θ
/
(
r
2
Θ)) + (
Z /Z
)), and
Z
(0) =
Z
(1) =
R
(1) =
T
(0) = 0. Due to the periodicity w.r.t.
θ
, the function Θ satisfies
the periodic boundary conditions Θ(0) = Θ(2
π
), and Θ (0) = Θ (2
π
). Hence
Z
=

βZ
, Θ
=

α
Θ,
R
+
r

1
R

αr

2
R
+
γR
= 0, and
T
=

c
2
(
γ
+
β
)
T
, for some numbers
α, β, γ
.
The problem for Θ has solutions
α
=

n
2
,
(
n
= 1
,
1
,
2
, . . .
), Θ =
a
cos
nθ
+
b
sin
nθ
, (
a, b
∈
IR), and the problem for
Z
has solutions
β
=
m
2
π
2
, (
m
= 1
,
2
, . . .
),
Z
= sin
mπz
.
The problem for
R
is a singular SturmLiouville problem which has to be accomplished by the