Wave equation 1D
Wave equation (1D):
∂
2
u
∂x
2
=
1
c
2
(x)
∂
2
u
∂t
2
(1)
0.0.1 Weak Form
w(x)
∂
2
u
∂x
2
= w(x)
1
c
2
(x)
∂
2
u
∂t
2
(2)
∂
∂x
w(x)
∂u
∂x
−
∂w
∂x
∂u
∂x
=
w(x)
1
c
2
(x)
∂
2
u
∂t
2
(3)
Z
Ω
∂
∂x
w(x)
∂u
∂x
dx −
Z
Ω
∂w
∂x
∂u
∂x
dx =
Z
Ω
w(x)
1
c
2
(x)
∂
2
u
∂
2
t
dx (4)
Because w(Γ
Dirichlet
) = 0 we get
R
Ω
∂
∂x
w(x)
∂u
∂x
dx
= w(x)
∂u
∂x
Γ
Neumann
|
{z }
q(Γ)
. Rearranging yields:
Z
Ω
∂w
∂x
∂u
∂x
dx = w(x)
∂u
∂x
Γ
Neumann
|
{z }
q(Γ)
−
Z
Ω
w(x)
1
c
2
(x)
∂
2
u
∂t
2
dx (5)
0.0.2 Discretization
u(x) ≈ U(x, t) = Φ(x)
T
C (6)
w(x) ≈ W (x, t) = W
T
Φ(x) (7)
1