Civil Engineering Reference
In-Depth Information
1
Plane waves in isotropic fluids
and solids
1.1
Introduction
The aim of this chapter is to introduce the stress - strain relations, the basic equations
governing sound propagation which will be useful for the understanding of the Biot the-
ory. The framework of the presentation is the linear theory of elasticity. Total derivatives
with respect to time d/d
t
are systematically replaced by partial derivatives
∂/∂t
.The
presentation is carried out with little explanation. Detailed derivation can be found in
the literature (Ewing
et al
. 1957, Cagniard 1962, Miklowitz 1966, Brekhovskikh 1960,
Morse and Ingard 1968, Achenbach 1973).
1.2
Notation - vector operators
A system of rectangular cartesian coordinates
(x
1
,x
2
,x
3
)
will be used in the following,
having unit vectors
i
1
,
i
2
and
i
3
. The vector operator del (or nabla) denoted by
∇
can be
defined by
∇
=
i
1
∂
∂x
1
+
i
2
∂
∂x
2
+
i
3
∂
(1.1)
∂x
3
When operating on a scalar field
ϕ(x
1
,x
2
,x
3
)
the vector operator
∇
yields the gradient
of
ϕ
i
1
∂ϕ
i
2
∂ϕ
i
3
∂ϕ
∂x
3
grad
ϕ
=∇
ϕ
=
∂x
1
+
∂x
2
+
(1.2)
