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

5.3 Half-plane, SV-P source and receiver at surface (Lamb’s problem) 71
g yy (˜r 1 , ˜r 2 ,ω) =−
i [
]
H (2)
0
( ˜ 1 ) + H (2)
0
( ˜ 2 )
4µ x
⎤
⎡
u yy (x, z, t) = 1 ⎣ H (t − ˜t 1)
√
2πµ x
t 2 − ˜t
1
2
+ H (t − ˜t 2)
√
t 2 − ˜t 2
2
(5.10)
⎦ (5.11)
5.3 Half-plane, SV-P source and receiver at surface (Lamb’s problem) 2
An impulsive in-plane line source P is applied on the surface of a lower half-space, and
displacements are observed there. For an upper half-space z > 0, reverse the sign of the
coupling terms. The source has dimensions [F][T]/[L] = [impulse]/[length in y direction].
τ = tβ
|x| , τ R = β , a = β C R α = 1 − 2ν , δ = Dirac delta (5.12)
2(1 − ν)
⎧
0 τ1
⎧
0 τ1
⎧
2τ (2τ 2 − 1) √ τ 2 − a 2√ 1 − τ 2
u xz =
Pβ ⎪⎨
a ≤ τ ≤ 1
(2τ 2 − 1) 4 + 16τ 4 (τ 2 − a 2 )(1 − τ 2 )
πµx π(2τR ⎪⎩
2 − 1)3
4 ( 1 − 4τR 2 + 8τ R 6(1 − a2 ) )δ(τ − τ R) else
u zx =−u xz
u yy =
Pβ 1
√ H (τ − 1)
πµ|x| τ 2
− 1 (5.13e)
(5.13a)
(5.13b)
(5.13c)
(5.13d)
Note: Observe that x in u xz has no absolute sign. Also,
(β/x)δ(τ − τ R ) = sgn x δ(t − t R ).
2 Eringen, A. C. and Suhubi, S. S., 1975, Elastodynamics, Academic Press, Vol. II, p. 617. Note: eq. 7.16.7 in that
book has an error that affects the coefficient of the Dirac delta term in u xz. This has been corrected here.