Environmental Engineering Reference
In-Depth Information
If an equation that represents some physical laws is correct, the unit of each of
the terms in the equation must be the same. The same unit of each term can not,
however, ensure the correctness of the equation. Therefore the same unit of each
term is a necessary, but not a sufficient condition for the correctness of the equation.
Remark 2.
Many other physical problems also lead to wave equations. In general,
the wave equation describes the propagation of vibration. The dependent variable
u
is not necessarily the displacement and can be other physical quantities. For exam-
ple, the electrical current and voltage in a high-frequency network also satisfy the
wave equation.
(
,
)
be
the dependent variable. Two- or three-dimensional homogeneous and nonhomoge-
neous equations are given by
Remark 3
.Let
M
be a point in a plane or a three-dimensional space, and
u
M
t
a
2
a
2
u
tt
=
Δ
u
,
u
tt
=
Δ
u
+
f
(
M
,
t
)
,
where the Laplace operator
Δ
is
2
2
Δ
=
∂
x
2
+
∂
y
2
,
(for the two-dimensional case)
,
∂
∂
2
2
2
Δ
=
∂
x
2
+
∂
y
2
+
∂
z
2
,
(for the three-dimensional case)
.
∂
∂
∂
1.2.4 Heat-Conduction Equations
Heat Flux
The basic law defining the relationship between the heat flow and the temperature
gradient, based on experimental observations, is generally named after the French
mathematical physicist Joseph Fourier who used it in his analytic theory of heat.
For a homogeneous, isotropic solid (i.e. material in which thermal conductivity is
independent of direction), the
Fourier law
is given in the form
q
(
M
,
t
)=
−
k
∇
u
(
M
,
t
)
,
(1.17)
where the temperature gradient
∇
u
is a vector normal to the isothermal surface,
the
heat flux vector
q
represents heat flow per unit time, per unit area of the
isothermal surface in the direction of the decreasing temperature, and
k
is called the
thermal conductivity
of the material which is a positive, scalar quantity. In a Carte-
sian coordinate system, for example, Eq. (1.17) is written as
(
M
,
t
)
k
∂
z
k
u
+
∂
u
+
∂
u
q
(
x
,
y
,
z
,
t
)=
−
x
i
y
j
,
∂
∂
∂
where
i
,
j
and
k
are the unit direction vectors along the
x
,
y
and
z
directions, respec-
tively.
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