Exact solutions for the reflectivity

Wave splitting in linear viscoelasticity 547

5. Exact solutions for the reflectivity

Here we assume that the plane x = 0 coincides with the interface S between a homogeneous elastic medium x 0 and a homogeneous viscoelastic medium modelled as in the previous sections x 0. We are interested in the reflectivity at this interface with respect to a transient wave impinging on S from the elastic medium. To this aim we impose the continuity of the field v, w T at x = 0, i.e. v w + , t = v w − , t, 36 for any t 0, where 0 + and 0 − refer respectively to the limiting values from the right and from the left of x = 0. In view of Eq. 3.19, Eq. 36 reads D v + v − + , t = D e v + v − − , t, 37 where D e = 1 1 − √ ρ e σ e √ ρ e σ e and where ρ e , σ e are quantities pertaining to the elastic medium. After the multiplication of Eq. 37 to the left by D −1 we arrive at v + v − + , t = 1 2 1 + σ −1 √ ρ e σ e S∗ 1 − σ −1 √ ρ e σ e S∗ 1 − σ −1 √ ρ e σ e S∗ 1 + σ −1 √ ρ e σ e S∗ v + v − − , t. 38 We define the reflectivity Rt at S as follows v − − , t = R· · v + − , · t. 39 The definition 39 differs from that given in He and Ström 1996 since it implicitly contains the reflection coefficient. As will be clear in the results of this section, this choice allows us to identify the reflection coefficients with the instantaneous reflectivity at the surface S. Substituting into 38, we obtain v + + , t = 1 2 v + − , t + 1 2 R· + √ ρ e σ e σ S· − √ ρ e σ e σ S · R· ∗ v + − , · t, 40 v − + , t = 1 2 v + − , t + 1 2 R· + √ ρ e σ e σ S· + √ ρ e σ e σ S · R· ∗ v + − , · t. 41 Invoking the causality principle we assume that no backward waves occur for x 0, whence v − + , t = 0 and Eq. 41 yields the following Volterra-type integral equation for Rt √ ρ e σ e σ St − Rt − √ ρ e σ e σ S ∗ Rt = δt. 42 Substitution into 40 yields v + + , t = v + − , t + R· ∗ v + − , · t 43 548 M. Romeo which, according to the definition 39 and Eq. 19, is a consequence of the continuity of vx, t at S. Once Eq. 42 has been solved, the result 43 gives the boundary condition for the propagation problem stated in the previous section. Incidentally, we observe that Eq. 42 can be conversely regarded as a Volterra-type integral equation for St if the reflectivity Rt is known for t ∈ R ++ . This inverse problem has been investigated numerically in Ammicht et al. 1987. An exact solution of Eq. 42 can be obtained for the viscoelastic models here considered. The use of the Laplace transform allows us to write Rt = L −1 √ ρ e σ e LSs − σ √ ρ e σ e LSs + σ t and, taking into account Eqs 25 and 27, we obtain Rt = L −1 1 − χ √ sLGs 1 + χ √ sLGs t, 44 where χ = √ρσ√ρ e σ e . According to 8 and 9, the evaluation of the inverse transform in 44 yields R M t = νδt − e R M t, R MK t = ν exp−tτ K δt − e R MK t with e R M t = Z ∞ N ξ L M ξ , t dξ, 45 e R MK t = Z ∞ N ξ L M ξ , t + L MK ξ , t + L M ξ , · ∗ L MK ξ , · t dξ 46 respectively for the Maxwell and the Maxwell–Kelvin models, and where N ξ = 2χ exp−ξ χ exp χ 2 ξ erfc χ p ξ − 1 √ π ξ , L M ξ , t = s ξ τ M t J 1 2 s ξ t τ M , L MK ξ , t = exp−tτ K s αξ τ K t J 1 2 s αξ t τ K , ν = Z ∞ N ξ dξ = 1 − χ 1 + χ J 1 being the Bessel function of order one. The quantity ν accounts for that part of the reflected amplitude which is present irrespective of the viscoelastic relaxation. Hence it represents the instantaneous reflectivity and corresponds to the reflection coefficient of the purely elastic case. The results 45 and 46 are illustrated in figure 2 for shear waves on a aluminiumelastic–copperviscoelastic interface. The reflectivity functions e R M t and e R MK t respectively show, a typical one- and two-step behaviour. The relaxation times τ M and τ K are placed within the time intervals at which the transitions occur. Hence if we are interested in the inverse problem, any function e Rt given by the reflection data yields only roughly approximated values of τ M and τ K . In addition, if we test reflection properties by means of sharp acoustic Wave splitting in linear viscoelasticity 549 Figure 2. Reflectivity e Rt as given by Eqs 45 and 46 for shear waves impinging on a aluminiumelastic–copperviscoelastic interface for the Maxwell model • — and the Maxwell–Kelvin model ◦ —. τ M , τ K and α are same as in figure 1. pulses, we expect to obtain accurate values of e Rt only for short time intervals 0, ¯t, ¯t ≪ τ M , τ K . In the sequel we show how Eqs 45 and 46 can be exploited to derive τ M and τ K from the reflection data at t = 0. In the Maxwell case we can obtain τ M directly from e R M 0. Since L M ξ , 0 = ξτ M , by 45 we get τ M = ν 1 e R M , 47-a where ν 1 = Z ∞ N ξ ξ dξ. 47-b In the Maxwell–Kelvin case we derive τ M and τ K from the quantities e R MK 0 and e R ′ MK 0 = d e R MK dt t =0 . Since L MK ξ , 0 = αξτ K , we easily obtain from 46 e R MK 0 = 1 τ M + α τ K ν 1 . 48-a 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