544 M. Romeo
P t = ± 1
c L
−1
p
sLGs t,
25-a St = ±c
L
−1
1 √
sLGs t
25-b and Eq. 23 becomes
∂
x
v
+
v
−
= −
1 c
Q·
1 c
Q· ∂
t
v
+
v
−
26 with
Qt = L
−1
1 √
sLGs t
27 and where the choice of the sign in 25 has been performed according to the meaning of v
+
and v
−
as forward and backward propagating disturbances. Explicit expressions for P t and Qt = Stc can be derived for
the Maxwell model and the Maxwell–Kelvin model described in Section 2. With the help of a table of Laplace transforms Roberts and Kaufman, 1966, we have
P t = δt exp−t2τ
M
+ E t,
1 2τ
M
, Qt = δt exp−t2τ
M
+ F t,
1 2τ
M
and P t = δt exp−t2τ
2
+ E t,
1 2τ
2
+ E t,
1 2
1 τ
1
+ 1
τ
K
+ E
·, 1
τ
2
· E ·,
1 2
1 τ
1
+ 1
τ
K
t, Qt = δt exp−t2τ
2
+ F t,
1 2τ
2
+ F t,
1 2
1 τ
1
+ 1
τ
K
+ F
·, 1
τ
2
· F ·,
1 2
1 τ
1
+ 1
τ
K
t respectively for the Maxwell model and the Maxwell–Kelvin model with t 0, where δt is the usual Dirac
delta distribution and the functions E and F are given by Et, β = β
I βt + I
1
βt exp−βt,
F t, β = β I
βt + I
1
βt exp−βt,
where I and I
1
are modified Bessel functions.
4. Evaluation of the wave propagator
Equation 26 allows us to solve the problem of the one-dimensional transient propagation in a viscoelastic homogeneous medium in terms of the superposition of two independent modes v
+
and v
−
propagating in opposite directions. According to the present model, if a wave pulse is given at x = 0, the function v
+
0, t can
Wave splitting in linear viscoelasticity 545
be obtained by 20 and the evolution of the transient for x 0 can be determined by the equation ∂
x
v
+
x, t = − 1
c Q· · ∂
·
v
+
x, · t,
28 where ∂
·
stands for derivative with respect to the argument involved in the convolution. A convenient form of the solution to Eq. 28 can be achieved in terms of a wave propagator P
+
x, t which is defined as
v
+
x, t = P
+
x, · · v
+
0, · t,
29 where v
+
x, t satisfies Eq. 28. A comprehensive treatment of wave propagators in the context of one-
dimensional problems is given in Karlsson 1996 and applied to the more general case of inhomogeneous media. According to Eq. 29 the propagator P
+
x, t can be viewed as the solution to Eq. 28 corresponding
to a δ-like pulse at x = 0, that is, under the boundary condition v
+
0, t = δt. Substitution of 29 into 28 yields
∂
x
P
+
x, · · v
+
0, · t = −
1 c
Q P
+
x, · · v
+
0, · t −
1 c
∂
·
Q· · P
+
x, · · v
+
0, · t,
where the condition 22 has been exploited. The use of Eq. 27 and the arbitrariness of v
+
0, t yield the following equation for P
+
x, t ∂
x
P
+
x, t = − 1
c L
−1
s
s LGs
· · P
+
x, · t.
30 Equation 30 can be solved by means of the Laplace transform. Sine P
+
0, t = δt, we have P
+
x, t = L
−1
exp −
x c
s
s LGs
t. 31
In the case of the Maxwell model, Gs is given by 8 and an exact solution 31 is obtained in the form P
+ M
x, t = exp−t2τ
M
δ t −
x c
+ H t −
x c
x 2τ
M
c I
1 1
2τ
M
p
t
2
− xc
2
p
t
2
− xc
2
, 32
where H t is the Heaviside unit step function. Equation 32 corresponds to a superposition of a directly transmitted wave and a transient wave propagating with speed c and whose amplitude decays exponentially
in time. In the case of a Maxwell–Kelvin model we are unable to work out the inverse transform in Eq. 31. However, a realistic approximation of the argument of the exponential in 31 allows us to bypass this difficulty.
For the Maxwell–Kelvin model it results in
s
s LGs
=
s
s s +
1 τ
M
s
1 + α
τ
K
s s +
1 τ
K
s +
1 τ
M
. 33
546 M. Romeo
Figure 1.
Wave propagators for shear waves in a copper half-space for the Maxwell model P
+ M
and the Maxwell–Kelvin model P
+ MK
as functions of x for t = 100 s, τ
M
= 6 · 10
5
s, τ
K
= 2s, α = 130.
As frequently occurs Caviglia and Morro, 1990, we have α ≪ 1 and τ
M
≫ τ
K
. Then an approximated expression of the second root on the right-hand side of 33 holds for any s 0, such that
s
s LGs
≃
s
s s +
1 τ
M
+ α
2τ
K
s s +
1 τ
K
. 34
In view of 34 we can work out the inverse transform in 31 to obtain P
+ MK
x, t ≃ exp −
x c
α 2τ
K
P
+ M
x, t + Ax, · ∗ P
+ M
x, · t
, 35
where P
+ M
is given by 32 and Ax, t = exp−tτ
K
1 τ
K
√ t
r
αx 2c
I
1
2 τ
K
s
αx τ
K
t .
In order to make a comparison between the two viscoelastic models, the solutions 32 and 35 are shown in figure 1 for a fixed time in correspondence to shear waves. A relevant difference between the two models
appears at large distances from the boundary x = 0.
Wave splitting in linear viscoelasticity 547
5. Exact solutions for the reflectivity
