Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

9 Integral transform method
The integral transform method provides the most general framework for the analytical and
numerical treatment of elastodynamic problems in unbounded continua, provided that the
media exhibit specific geometric and material regularities. In particular, it can be used to
solve problems of sources in unbounded, homogeneous, and layered media formulated
in Cartesian, cylindrical, or spherical coordinates. We provide a brief introduction to this
method while picking up the fundamental tools needed for the powerful stiffness matrix
method described in the next chapter, and we illustrate these concepts by means of various
examples.
In a nutshell, the method consists in carrying out an appropriate integral transform on
the vector wave equation – including the source term – which changes the problem from
a set of partial differential equations in the space-time domain to a system of coupled
linear equations in the frequency–wavenumber domain. After solving the latter for the
displacements, an inverse integral transformation is applied, which returns the sought-after
displacements (i.e., the wave field) in space–time.
In principle, the method is exact, but only in the simplest of problems (e.g., Pekeris’s
or Chao’s problem) are the inverse transforms amenable to exact evaluation via contour
integration. In most other (more complicated) cases, the inversion must be carried out
numerically.
9.1 Cartesian coordinates
Consider a horizontally layered, laterally unbounded system subjected to a source (or body
load) b acting at some location. In addition, the system may be either finite or infinitely
deep in the vertical direction. As we have seen in Section 1.4.1, the wave equation in
Cartesian coordinates for such a system is given by
∂ 2 u
D xx
∂x + D ∂ 2 u
2 yy
∂y + D ∂ 2 u
2 zz
∂z + (D 2 xy + D yx) ∂2 u
∂x ∂y
+ (D yz + D zy) ∂2 u
∂y ∂z + (D xz + D zx) ∂2 u
+ b = ρü (9.1)
∂x ∂z
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