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

90 Two-dimensional problems in homogeneous plates and strata
0.50
0.25
µg yy
0
-0.25
-0.50
0 1 2 3 4 5
ωh
2πβ
Figure 7.4a: Homogeneous stratum, Green’s function at (x, z) =
(h, 0.75h) due to SH line load at elevation z 0 = h (i.e., upper surface).
Solid line, real part; dashed line, imaginary part.
7.2.2 Normal mode solution
ξ = x − x 0
,ζ = z h h , ζ 0 = z 0
h ,
S = ωh
β
(7.13)
l = 1 2 (2 j − 1) = j − 1 ,
2
j = 1, 2, 3 ... Modal indices (7.14)
√
κ j = 2 S − π 2 l 2 , Im(κ j ) ≤ 0 Modal wavenumber (7.15)
φ j = (−1) j sin lπξ Normal mode (7.16)
g yy(ω, x, z, x 0 , z 0) =− i µ
∞∑
j=1
−iκ
sin(π lζ ) sin(π lζ 0)
j |ξ|
e
(7.17)
κ j
Observe again that the high modes are evanescent and thus decay rapidly in x. Hence,
the summation can safely be truncated above l ≫ S /π. For small x, the index of the
first neglected mode may be established by setting a threshold of exponential decay, say,
exp(−iκ j ξ)