## 4 Higher-Order Equivalent Single Layer Approach
\begin{array}{cccc}\frac{1}{A_{1}}\frac{ \partial}{\partial\mathbf{x}_{1}}+\frac{1}{A_{1}}\frac{\partial A_{2}}{ \partial\mathbf{x}_{1}}&-\frac{1}{A_{1}}\frac{\partial A_{1}}{\partial \mathbf{x}_{2}}&-\frac{1}{R_{1}}\\ -\frac{1}{A_{1}}\frac{\partial A_{2}}{\partial\mathbf{x}_{1}}&\frac{1}{A_{2}} \frac{\partial}{\partial\mathbf{x}_{2}}+\frac{1}{A_{1}}\frac{\partial A_{1}}{ \partial\mathbf{x}_{2}}&-\frac{1}{R_{2}}\\ \frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\mathbf{x}_{2}}&\frac{1}{A_{ 1}}\frac{\partial}{\partial\mathbf{x}_{1}}+\frac{1}{A_{1}}\frac{\partial A_{2}} {\partial\mathbf{x}_{1}}&0\\ \frac{1}{A_{2}}\frac{\partial}{\partial\mathbf{x}_{2}}+\frac{1}{A_{1}A_{2}} \frac{\partial A_{1}}{\partial\mathbf{x}_{2}}&\frac{1}{A_{1}A_{2}}\frac{ \partial A_{2}}{\partial\mathbf{x}_{1}}&0\\ \frac{1}{R_{1}}&0&\frac{1}{A_{1}}\frac{\partial}{\partial\mathbf{x}_{1}}+ \frac{1}{A_{1}A_{2}}\frac{\partial A_{2}}{\partial\mathbf{x}_{1}}\\ 0&\frac{1}{R_{2}}&\frac{1}{A_{2}}\frac{\partial}{\partial\mathbf{x}_{2}}+ \frac{1}{A_{1}A_{2}}\frac{\partial A_{1}}{\partial\mathbf{x}_{2}}\\ -1&0&0\\ 0&0&-1\\ \end{array} \right]^{T} \tag{42}\]On the other hand, the mass matrix \(\mathbf{M}^{(\tau\eta)}\) is defined below:  
\[\mathbf{M}^{(\tau\eta)}=\left[ \begin{array}{ccc}I^{(\tau\eta)}&0&0\\ 0&I^{(\tau\eta)}&0\\ 0&0&I^{(\tau\eta)}\end{array} \right] \tag{43}\]  
for \(\tau,\eta=0,1,2,...,N,N+1\). If \(\rho^{(k)}\) stands for the mass density of the \(k\)-th layer, the inertia masses \(I^{(\tau\eta)}\) in Equation (43) are computed for each order of kinematic expansion \(\tau,\eta=0,1,2,...,N,N+1\) as follows  
\[I^{(\tau\eta)}=\sum\limits_{k=1}^{l}\int\limits_{\xi_{k}}^{\zeta_{k+1}}\rho^{ (k)}F_{\tau}F_{\eta}H_{1}H_{2}d\zeta \tag{44}\]  
With reference to the motion Equation (41), it should be specified that \(\ddot{\mathbf{u}}^{(\eta)}(\alpha_{1},\alpha_{2},t)\) is the algebraic vector that collects the second-order derivatives of the generalized displacements Equation (28) with respect to the time variable \(t\). The system of governing Equation (41) can be expressed as a function of the degrees of freedom by inserting Equation (36) into (41). The fundamental equations are obtained for each order of kinematic expansion \(\tau,\eta=0,1,2,...,N,N+1\)  
\[\sum\limits_{\eta=0}^{N+1}\mathbf{L}^{(\tau\eta)}\mathbf{u}^{(\eta)}=\sum \limits_{\eta=0}^{N+1}\mathbf{M}^{(\tau\eta)}\ddot{\mathbf{u}}^{(\eta)} \tag{45}\]  
where \(\mathbf{L}^{(\tau\eta)}\) is the fundamental operator defined as follows  
\[\mathbf{L}^{(\tau\eta)}=\left[ \begin{array}{ccc}L_{11}^{(\tau\eta)}&L_{12}^ {(\tau\eta)}&L_{13}^{(\tau\eta)}\\ L_{21}^{(\tau\eta)}&L_{22}^{(\tau\eta)}&L_{23}^{(\tau\eta)}\\ L_{31}^{(\tau\eta)}&L_{32}^{(\tau\eta)}&L_{33}^{(\tau\eta)}\end{array} \right] \tag{46}\]  
The terms collected into Equation (46) can be computed by applying the following definition  
\[\mathbf{L}^{(\tau\eta)}=\mathbf{D}_{\Omega}^{\tau}\mathbf{A}^{(\tau\eta)} \mathbf{D}_{\Omega} \tag{47}\]  
for \(\tau,\eta=0,1,2,...,N,N+1\). It should be pointed out that Equation (45) define a set of \(3\times(N+2)\) motion equations. Since it represents a set of partial differential equations, the proper boundary conditions must be enforced to find the solution. At this point, Figure 2 can be taken as a reference to identify the coordinates of each edge along which the boundary conditions are applied. Let us consider fully clamped edges (specified by \"C\") or free edges (denoted by \"F\"). The boundary conditions assume the following aspect  
\[ \begin{array}{lll}\text{C}&\rightarrow&u_{1}^{(\tau)}=u_{2}^{(\tau)}=u_{3} ^{(\tau)}=0\\ \text{F}&\rightarrow&N_{1}^{(\tau)}=N_{12}^{(\tau)}=T_{1}^{(\tau)}=0\end{array}  \tag{48}\]  
for \(\alpha_{2}^{0}\leq\alpha_{2}\leq\alpha_{2}^{1}\) and \(\alpha_{1}=\alpha_{1}^{0}\) or \(\alpha_{1}=\alpha_{1}^{1}\). This is the case of the South (S) and North (N) edges. The same boundary conditions can be written as follows:  
\[