1
Tacoma Narrows Bridge
The equation of the wave equation is
y
tt
=
c
2
y
xx
. To discretise the motion, use the central difference stencils,
which means it is secondorder accurate in both time and space:
y
tt

t
=
n
x
=
i
=
y
n
+1
i

2
y
n
i
+
y
n

1
i
Δ
t
2
(1)
y
xx

t
=
n
x
=
i
=
y
n
i
+1

2
y
n
i
+
y
n
i

1
Δ
x
2
(2)
The overall equation of motion can be written as:
y
n
+1
i

2
y
n
i
+
y
n

1
i
Δ
t
2
=
c
2
y
n
i
+1

2
y
n
i
+
y
n
i

1
Δ
x
2
(3)
Simplifying gives:
y
n
+1
i
= 2
y
n
i

y
n

1
i
+
Δ
t
2
c
2
Δ
x
2
(
y
n
i
+1

2
y
n
i
+
y
n
i

1
)
(4)
The eigenvalues of the wave equation can be solved through the separation of variables:
y
=
F
(
x
)
G
(
t
)
F
(
x
)
G
tt
(
t
) =
c
2
F
xx
(
x
)
G
(
t
)
(5)
The equation can be split into two ODEs:
F
00
(
x
) +
p
2
F
(
x
) = 0
(6)
G
00
(
t
) +
c
2
p
2
G
(
t
) = 0
(7)
Due to the boundary conditions,
F
(
x
) has the form of
F
n
(
x
) =
B
n
sin(
px
), where
p
=
nπ/L
. The initial tem
poral BC of
y
(
x,
0) = 0
.
125 sin(3
πx/L
), with spatial boundary conditions on the ends
y
(0
, t
) =
y
(
L, t
) = 0.
Equating coefficients, gives the only nonzero
F
n
(
x
) at
n
equals to 3.