# Modes Control of Lamb Wave in Plates Using Meander-Line Electromagnetic Acoustic Transducers
## 2 Lamb Waves  
It is known that the ultrasonic Lamb wave is a type of wave that remains guided between two parallel free boundaries. Lamb wave theory is fully documented in many textbooks [15; 16; 17]. Here, we only introduce the bare essentials. The acoustic field equation can be stated in terms of a particle displacement \(\mathbf{u}\) as follows:  
\[(\lambda+\mu)\
abla(\
abla\cdot\mathbf{u})+\mu\
abla^{2}\mathbf{u}=\rho\frac{ \partial^{2}\mathbf{u}}{\partial t^{2}} \tag{1}\]  
According to the Helmholtz decomposition principle, \(\mathbf{u}=\
abla\varphi+\
abla\times\psi\), the decomposition is then substituted into the field equation, and we obtain the two separated wave equations:  
\[ \begin{array}{l}\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}}=\frac{1}{c_{L}^{2}}\frac{\partial^{2}\phi}{\partial t^{ 2}}\\ \frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2} }=\frac{1}{c_{T}^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}\end{array}  \tag{2}\]  
where \(\varphi\) and \(\psi\) are two potential functions, \(C_{L}^{2}\)= \((\lambda+2\mu)/\rho\) and \(C_{T}^{2}\)= \(\mu/\rho\) are the longitudinal and transverse wave speeds, \(\lambda\) and \(\mu\) are the Lame constants, and \(\rho\) is the mass density of the material. Based on the plane strain assumption, we assume that the solutions of Equation (2) are  
\[ \begin{array}{l}\phi\ =\ \Phi(z)\exp[i(kx-\omega t)]\\ \psi=\Psi(z)\exp[i(kx-\omega t)]\end{array}  \tag{3}\]  
where \(k\) is the wavenumber. The time dependence is assumed harmonic in the form \(e^{-i\omega t}\). Substituting Equation (3) into Equation (2), the general solution of Equation (2) is  
\[ \begin{array}{l}\Phi(z)=A_{1}\sin(py)+A_{2}\cos(py)\\ \Psi(z)=B_{1}\sin(qy)\ +\ B_{2}\cos(qy)\end{array}  \tag{4}\]\[p^{2}=\frac{\omega_{x}^{2}}{c_{l}^{2}}-k^{2}\] \[q^{2}=\frac{\omega_{x}^{2}}{c_{T}^{2}}-k^{2} \tag{5}\]  
The four integration constants (\(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\)) are to be found from the boundary conditions. Since the exponential appears in all of the expressions, it does not play a further role in the determination of the frequency equation, and it is therefore omitted in the sequel. Using the relations between the potential functions and the displacements and stresses:  
\[ \begin{array}{l}u_{x}=\frac{\partial\phi}{\partial x}+\frac{\partial\psi}{ \partial y}\\ u_{y}=\frac{\partial\phi}{\partial y}-\frac{\partial\psi}{\partial x}\\ \tau_{yx}=\mu(2\frac{\partial^{2}\phi}{\partial x\partial y}-\frac{\partial^{ 2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}})\\ \tau_{yy}=\lambda(\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2} \phi}{\partial y^{2}})+2\mu(2\frac{\partial^{2}\phi}{\partial y^{2}}-\frac{ \partial^{2}\psi}{\partial x\partial y})\end{array}  \tag{6}\]  
and substituting Equations (3)-(5) into Equation (6), the resulting equations are obtained as  
\[u_{x}=[ikA_{2}\cos(py)+qB_{1}\cos(qy)]+[ikA_{1}\sin(py)-qB_{2}\sin(qy)] \tag{7}\]  
\[u_{y}=-[pA_{2}\sin(py)+ikB_{1}\sin(qy)]+[A_{1}p\cos(py)-ikB_{2}\cos(qy)] \tag{8}\]  
\[\tau_{yx}=\mu[-2 ikpA_{2}\sin(py)+(k^{2}-q^{2})B_{1}\sin(qy)]+\mu[2 ikpA_{1} \cos(py)+(k^{2}-q^{2})B_{2}\cos(qy)] \tag{9}\]  
\[\tau_{yy}=\mu[(k^{2}-q^{2})A_{2}\cos(py)-2 ikqB_{1}\cos(qy)]+\mu[(k^{2}-q^{2}) A_{1}\sin(py)+2 ikqB_{2}\sin(qy)] \tag{10}\]  
where \(u_{x}\) is the in-plane displacement component, \(u_{y}\) is the out-plane displacement component, \(\tau_{yy}\) is the normal stress component, and \(\tau_{yx}\) is the shear stress component.