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

4.9 Spatially harmonic line source (2 1 / 2 -D problem) 65
g zz = A [ ks 2 H 0β − k 2 z B ]
0
(4.86c)
g xy = g yx = γ x γ y AB 2
g xz = g zx = i k z γ x AB 1
g yz = g zy = i k z γ y AB 1
(4.86d)
(4.86e)
(4.86f)
Cylindrical coordinates
Let r be the source-receiver distance in the x, y plane, with θ being the angle between r
and x, so that cos θ = γ x , sin θ = γ y , γ z = 0. Also, define the parameters u, ν, w, U, W as
u = A
(k 2 S H 0β − 1 )
r B 1 + B 2 , (4.87a)
(
v = A k 2 S H 0β − 1 )
r B 1
(4.87b)
w = Ai k z B 1 ,
U = Ai k z B 1
W = A ( kS 2 H 0β − k 2 z B )
0
(4.87c)
(4.87d)
(4.87e)
The radial, tangential, and axial components of the Green’s functions are then
g rx = u (cos θ) , g ry = u (sin θ) , g rz = U (4.88a)
g θ x = v (−sin θ) , g θy = v (cos θ) , g θz = 0 (4.88b)
g zx = w (cos θ) , g zy = w (sin θ) , g zz = W (4.88c)
These can be used in turn to obtain the Green’s functions for 2.5-D dipoles, torsional
moments, and seismic moments, using for this purpose the expressions given in Section 2.6.
For example, the Green’s function for a 2.5-D blast load in a homogeneous full space is
{( ∂u
B 2.5-D =−
∂r + u − v + ∂U
r ∂z
) ( ∂w
ˆr +
∂r + w r + ∂W ) }
ˆk
∂z
(4.89)
except that we have omitted here the factor 3(λ+2µ)/4µ associated with the presence
of an infinitesimal cavity within which the blast load acts, because there is no such thing
as a 2.5-D cavity. Thus, when the above expression is specialized to the case of a line
load (k z = 0 ⇒ k β = k S ), we recover the Green function B 2-D for a line blast source of
Section 3.5, but multiplied by µ/(λ + 2µ). Similarly, a Fourier transform could be used to
recover the point blast source B 3-D of section 4.7, but multiplied by 4µ/3(λ + 2µ). Thus,
the above expression is for a full solid, including the infinitesimal particle. Observe also
that we have changed the axial derivatives with respect to the source into axial derivatives
with respect to the receiver, i.e., ∂/∂z ′ =−∂/∂z, since the space is homogeneous. All other
Green’s functions for dipoles are as in section 2.6, so they need not be repeated.