Vibration of a Rectangular Drum
Let us first consider the vibrations of a membrane in the shape of a rectangle
R
=
braceleftbig
0
< x < a,
0
< y < b
bracerightbig
,
with side lengths
a
and
b
, whose edges are fixed to the (
x, y
)–plane. Thus, we seek to solve
the wave equation
u
tt
=
c
2
Δ
u
=
c
2
(
u
xx
+
u
yy
)
,
0
< x < a,
0
< y < b,
(12
.
136)
subject to the initial and boundary conditions
u
(
t,
0
, y
) =
u
(
t, a, y
) = 0 =
u
(
t, x,
0) =
u
(
t, x, b
)
,
u
(0
, x, y
) =
f
(
x, y
)
,
u
t
(0
, x, y
) =
g
(
x, y
)
,
0
< x < a,
0
< y < b.
(12
.
137)
As we saw in Section 12.2 the eigenfunctions and eigenvalues for the associated Helmholtz
equation on a rectangle,
c
2
(
v
xx
+
v
yy
) +
λ v
= 0
,
(
x, y
)
∈
R,
(12
.
138)
when subject to the homogeneous Dirichlet boundary conditions
v
(0
, y
) =
v
(
a, y
) = 0 =
v
(
x,
0) =
v
(
x, b
)
,
0
< x < a,
0
< y < b,
(12
.
139)
are
v
m,n
(
x, y
) = sin
mπx
a
sin
nπy
b
,
where
λ
m,n
=
π
2
c
2
parenleftbigg
m
2
a
2
+
n
2
b
2
parenrightbigg
,
(12
.
140)
with
m, n
= 1
,
2
, . . .
. The fundamental frequencies of vibration are the square roots of the
eigenvalues, so
ω
m,n
=
radicalBig
λ
m,n
=
π c
radicalbigg
m
2
a
2
+
n
2
b
2
,
m, n
= 1
,
2
,
. . . .
(12
.
141)
The frequencies will depend upon the underlying geometry — meaning the side lengths —
of the rectangle, as well as the wave speed
c
, which is turn is a function of the membrane’s
density and stiffness. The higher the wave speed, or the smaller the rectangle, the faster
the vibrations. In layman’s terms, (12.141) quantifies the observation that smaller, stiffer
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Boundary value problem, homogeneous Dirichlet boundary, Peter J. Olver, Dirichlet boundary conditions

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