Environmental Engineering Reference
In-Depth Information
Remark 2. Effects of initial conditions can sometimes be neglected. The initial-
condition driven free vibration of a string will diminish progressively in reality due
to unavoidable resistance. Therefore, when considering the vibration due to initial
conditions and external non-decaying forces such as periodic forces, after a suffi-
ciently long time t 0 , we may neglect the effect of the real initial conditions and take
u
0 as the initial conditions. For heat conduction in a solid rod,
the temperature due to an initial distribution
(
x
,
t 0 )=
u t (
x
,
t 0 )=
ϕ (
x
)
must tend to a constant u 0 (some
mean value of
. For the temperature distribution due
to both the initial conditions and the heat source which is non-decaying with time,
after a sufficiently long time t 0 , we can take u
ϕ (
x
)
over the rod) as t
+
0 (without loss
of generality) as the initial condition so that the temperature distribution depends
only on the internal heat source.
(
x
,
t 0
)=
u 0 or u
(
x
,
0
)=
1.4.2 Boundary Conditions
Boundary conditions describe situations of dependent variables on the system
boundary orconstraints on the boundary. The system boundary is the end points,
the boundary curve and the boundary surface in one-, two- and three-dimensional
space, respectively. We normally have three types of boundary conditions.
Boundary Conditions of the First Kind
Consider the vibration of a string of length l , fixed at the two end points. The bound-
ary conditions are u
(
0
,
t
)=
u
(
l
,
t
)=
0. If the end x
=
0 is fixed but the other end
x
=
l vibrates in the form of u
= ϕ (
t
)
, the boundary conditions become u
(
0
,
t
)=
0,
u
(
l
.
Consider heat conduction in a circle plate D : x 2
,
t
)= ϕ (
t
)
y 2
R 2 . If the temperature
+
distribution at the boundary
D is given as
ϕ (
M
,
t
)
, M
D , the boundary condition
is
D is the circle x 2
y 2
(
,
) | D = ϕ (
,
) ,
+
=
.
u
M
t
M
t
where
R
(1.78)
Consider heat conduction in a three-dimensional domain
Ω
of boundary surface
ϕ (
,
)
∂Ω
∂Ω
. If the temperature distribution on
∂Ω
is given as
M
t
, M
, the bound-
ary condition is
u
(
M
,
t
) | ∂Ω = ϕ (
M
,
t
) .
When heat conduction is steady, in particular, the temperature u satisfies the po-
tential equation
Δ
u
=
0. If the temperature on the boundary S is given as
ϕ (
M
)
,
M
S , the boundary condition is
u
(
M
) | S = ϕ (
M
) .
Search WWH ::




Custom Search