550 M. Romeo
We also have d
dt L
M
ξ , t|
t =0
= − 1
2 ξ
2
τ
2 M
, d
dt L
MK
ξ , t|
t =0
= − αξ
τ
2 M
1 + αξ
2 .
Substitution into the time derivative of 46 yields
e
R
′ MK
0 = − 1
2 1
τ
M
− α
τ
K 2
ν
2
− α
τ
2 K
ν
1
, 49
where ν
2
=
Z
∞
N ξ ξ
2
dξ. The system 48 and 49 admits the following solution for the couple 1τ
M
and 1τ
K
1 τ
M
=
e
R
MK
ν
1
− α
e
R
MK ν
2
ν
1
−
q
−
1 2
α
e
R
2 MK
ν
2
ν
1
− α
e
R
′ MK
02αν
2
+ ν
1
2αν
2
+ ν
1
, 1
τ
K
= α
e
R
MK ν
2
ν
1
−
q
−
1 2
α
e
R
2 MK
ν
2
ν
1
− α
e
R
′ MK
02αν
2
+ ν
1
α 2αν
2
+ ν
1
.
6. Transmission through a layer
As a final step we formulate the problem of transient transmission for a homogeneous viscoelastic layer
of thickness d. Here we assume that the domain V is given by {x ∈ R
3
| 0 6 x 6 d} and that the half-spaces x
0 and x d correspond to homogeneous elastic media with mass densities and elastic moduli denoted respectively by ρ
e
, σ
e
and ρ
¯e
, σ
¯e
. Similarly to Eq. 36 we impose the continuity of v and w across x = d and write
v w
d
−
, t = v
w d
+
, t which, in terms of v
+
, v
− T
, yields v
+
v
−
d
+
, t = D
−1 ¯e
D v
+
v
−
d
−
, t. 50
Here the matrix D
¯e
is defined in the same way as D
e
with respect to the medium at x d. Along with Eq. 29 we define the backward wave propagator P
−
x, t by the following relation
v
−
x, t = P
−
x, · ∗ v
−
d, · t.
51 It is worth remarking that our definition of P
−
differs from that given in Karlsson 1996 since we are essentially interested in homogeneous media where forward and backward modes propagate independently.
Accordingly, if a δt pulse is given at x = d it backward propagates with amplitude P
−
x, t 0 6 x 6 d,
which is equal to the amplitude of a forward propagating δt pulse generated at x = 0 when it reaches the
Wave splitting in linear viscoelasticity 551
position d − x. In other words, the forward and backward propagators must comply with the property P
−
x, t = P
+
d − x, t 52
for any x ∈ [0, d] and any t ∈ R
+
. According to 52, Eq. 51 gives, in particular v
− +
, t = P
+
d, · ∗ v
−
d, · t.
An inversion procedure, analogous to that used to work out Eq. 18, yields v
−
d
−
, t = L
−1
1 LP
+
d, s · ∗ v
−
0, · t.
53 Owing to Eqs 37 and 53, the continuity condition 50 can be cast in the form
v
+
v
−
d
+
, t = D
−1 ¯e
D P
+
d, ·∗ L
−1 1
LP
+
d,s
·∗ D
−1
D
e
v
+
v
− −
, · t
and accounting for the definitions 19 and 20 we obtain v
+
v
−
d
+
, t = 1
4
h
A · ∗
v
+
v
− −
, ·
i
t, 54
where the entries of the matrix At are given by
A
11
= 1 +
χ χ
+
+
∗ P
+ d
+ 1 +
χ χ
−
+
∗ L
−1
1 LP
+ d
, 55-a
A
12
= 1 −
χ χ
+
−
∗ P
+ d
+ 1 −
χ χ
−
−
∗ L
−1
1 LP
+ d
, 55-b
A
21
= 1 −
χ χ
−
−
∗ P
+ d
+ 1 −
χ χ
+
−
∗ L
−1
1 LP
+ d
, 55-c
A
22
= 1 +
χ χ
−
+
∗ P
+ d
+ 1 +
χ χ
+
+
∗ L
−1
1 LP
+ d
55-d with P
+ d
t = P
+
d, t , χ = √ρσ√ρ
¯e
σ
¯e
and
±
t = L
−1
χ LQs
t ± Qt
χ .
56 Consistently with the causality principle we have no backward field for x d, so that v
−
d
+
, t = 0 ∀t 0. Denoting by Rt the reflectivity function for the layer, from Eq. 54 we get
A
21
t + A
22
∗ Rt = 0. 57
We can also define a transmission function T t such that v
+
d
+
, t = T · ∗ v
+ −
, · t
552 M. Romeo
for t ∈ R
+
. Substitution into 6.5 allows us to obtain T t = A
11
t + A
12
∗ Rt. 58
Equation 57 represents the Volterra integral equation for Rt which is appropriate for the layer’s problem. Once it has been solved, the transmission function follows from Eq. 58.
In view of Eqs 27, 31, 55, 56 and 44, it is possible to write the solution of Eq. 57 in the form Rt = L
−1
LRs − LP
+
2d, sLRs 1 − LP
+
2d, sLRsLRs t,
where Rt has the form 44 with χ in the place of χ. Now we observe that, in view of 44 and 31, LRs
1, LRs
1, LP
+
2d, s 1 for any s ∈ R
+
. Hence the following expansion holds 1 − LP
+
2d, sLRsLRs
−1
= 1 +
∞
X
n=1
LP
+
2nd, s LRs
n
LRs
n
. Accordingly, we obtain
Rt = R t +
∞
X
n=1
R · ∗ P
+
2nd, · ∗ R·
∗n
· R·
∗n
t, 59
where R
t = Rt − P
+
2d, · ∗ R· t
and where the exponent ∗n denotes n-times convolution. Equation 59 expresses the reflectivity of a layer in terms of the reflectivities R and R at the interfaces
between an elastic medium and a viscoelastic half-space. It accounts for successive reflections within the layer by means of a series expansion. The proper reflectivities for a Maxwell layer or a Maxwell–Kelvin layer can
be obtained from 59 after substitution of Eqs 32, 45 and 35, 46 respectively. We finally note that, as is expected
R 0 = R0,
R
′
0 = R
′
hence the solution to the inverse problem described in the previous section remains unchanged in the layer’s case.
7. Conclusion
