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
00
/T
=
c
2
[(
R
00
/R
) + (
R
0
/
(
rR
)) + (Θ
00
/
(
r
2
Θ)) + (
Z
00
/Z
)), and
Z
(0) =
Z
(1) =
R
0
(1) =
T
0
(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
(0) = Θ
0
(2
π
). Hence
Z
00
=

βZ
, Θ
00
=
