## 2 Incident Surface Wave and the Scattering Problem  
### Incident Surface Wave  
When a wave propagates in a viscoelastic material, viscoelasticity will cause the dissipation of wave energy, which is manifested as wave attenuation and wave dispersion. In the uniform isotropic viscoelastic two-dimensional half plane defined by the \(x\)-\(z\) axis, the displacement field of the incident surface wave can be expressed as:  
\[\left\{\begin{array}{c}u_{x}=\pm iA_{in}U^{R}(z)e^{\pm ik^{+}x}\\ u_{z}=A_{in}W^{R}(z)e^{ik^{+}x}\end{array}\right. \tag{1}\]  
where \(A_{in}\) and \(\omega\) are the amplitude and frequency of the incident surface wave, respectively. \(t\) is the time, and \(u_{x}\) and \(u_{z}\) represent the displacements in the \(x\)-axis and \(z\)-axis directions, respectively. Moreover, the positive and negative signs indicate that the propagation directions of the surface wave are the positive and negative directions of the \(x\)-axis, respectively. The time item \(e^{\pm i\omega t}\) is omitted to make the theoretical derivation convenient. For viscoelastic material, \(k^{*}\) is the complex wave number, which can be calculated by \(k=\frac{\omega}{c_{R}{}^{*}}\), where \(c_{R}{}^{*}\) is the complex phase velocity of the surface wave, which is the solution of the well-known Rayleigh wave phase velocity equation:  
\[(2-\frac{{(c_{R}{}^{*})}^{2}}{{(c_{T}{}^{*})}^{2}})^{2}-4(1-\frac{{(c_{R}{}^{* })}^{2}}{{(c_{L}{}^{*})}^{2}})^{1/2}=0 \tag{2}\]  
where \(c_{L}{}^{*}\) and \(c_{T}{}^{*}\) are the complex velocities of the longitudinal and shear waves, respectively, which can be written as:  
\[\left\{\begin{array}{c}{c_{L}{}^{*}}=\sqrt{(\lambda^{*}+2\mu^{*})/\rho}\\ {c_{T}{}^{*}}=\sqrt{\mu^{*}/\rho}\end{array}\right. \tag{3}\]  
where \(\lambda^{*}\) and \(\mu^{*}\) are the complex _Lam\(\acute{e}\)_ parameters of the viscoelastic material. When the viscoelastic model is considered as the Kelvin-Voigt model, the complex _Lam\(\acute{e}\)_ parameters can be represented as:  
\[\left\{\begin{array}{c}\lambda^{*}=\frac{E^{*}\sigma}{(1-2\sigma)(1+\sigma)} \\ \mu^{*}=\frac{E^{*}}{2(1+2\sigma)}\end{array}\right. \tag{4}\]  
where \(E^{*}\) is the complex [PERSON]'s modulus, which can be written as:  
\[E^{*}=E_{0}(1-j\eta_{E}) \tag{5}\]  
where \(E_{0}\) is the static [PERSON]'s modulus, and \(\eta_{E}\) is the loss factor of [PERSON]'s modulus, which can be expressed as the ratio of imaginary parts to real parts of the complex [PERSON]'s modulus.  
The functions \(U^{R}(z)\) and \(W^{R}(z)\) in Equation (1) are defined as follows:  
\[\left\{\begin{array}{c}U^{R}(z)=d_{1}e^{-pz}+d_{2}e^{-qz}\\ W^{R}(z)=d_{3}e^{-pz}-e^{-qz}\end{array}\right. \tag{6}\]  
where the parameters of \(d_{1}-d_{3}\) are determined by the following formula:  
\[\left\{\begin{array}{c}d_{1}=-\left(k^{2}+q^{2}\right)/\left(2 kp\right)\\ d_{2}=q/k\\ d_{3}=\left(k^{2}+q^{2}\right)/\left(2k^{2}\right)\end{array}\right. \tag{7}\]where the parameters \(p\) and \(q\) can be calculated by the following equations:  
\[\left\{\begin{array}{l}p^{2}=k^{2}\big{(}1-c_{\rm R}^{2}/c_{\rm T}^{2}\big{)}\\ q^{2}=k^{2}\big{(}1-c_{\rm R}^{2}/c_{\rm T}^{2}\big{)}\end{array}\right. \tag{8}\]  
The stress equation relative to Equation (1) is  
\[\left\{\begin{array}{l}\tau_{xx}=A_{in}T_{xx}(z)e^{\pm ikx}\\ \tau_{zx}=\tau_{zz}=\pm iA_{in}T_{xz}(z)e^{\pm ikx}\\ \tau_{zz}=A_{in}T_{zz}(z)e^{\pm ikx}\end{array}\right. \tag{9}\]  
where  
\[\left\{\begin{array}{l}T_{xx}(z)=\mu(d_{4}e^{-pz}+d_{5}e^{-qz})\\ T_{xz}(z)=\mu(d_{6}e^{-pz}+d_{7}e^{-qz})\\ T_{zz}(z)=\mu(d_{8}e^{-pz}+d_{9}e^{-qz})\end{array}\right. \tag{10}\]  
and  
\[\left\{\begin{array}{l}d_{4}=\big{(}k^{2}+q^{2}\big{)}\big{(}2p^{2}+k^{2}-q ^{2}\big{)}\big{/}\big{(}2 pk^{2}\big{)}\\ d_{5}=-2q\\ d_{6}=\big{(}k^{2}+q^{2}\big{)}/k\\ d_{7}=-\big{(}k^{2}+q^{2}\big{)}/k\\ d_{8}=-\big{(}k^{2}+q^{2}\big{)}/\big{(}2 pk^{2}\big{)}\\ d_{9}=2q\end{array}\right. \tag{11}\]