## 2 Waves Propagation in Isotropic Homogeneous Elastic Media. Basic Theoretical Concepts
The second type, i.e., distortion waves, are associated with shifts perpendicular to the direction of propagation, consisting of distortions that involve only changes in shape: for this reason. The two types represent respectively longitudinal and transverse waves. Distortion waves are generally described by three functions, \(\psi_{i}\), which must be solutions of the equation just seen above. Only two types of elastic waves can be propagated in an infinitely extended isotropic solid. In presence of a boundary surface, however, a third type of wave also occurs: the so called elastic surface waves, due to their rapidly decreasing effect with depth. In order to describe these waves, it is necessary to consider the propagation of a plane wave through an elastic solid with a boundary plane, which is free of tensions. For simplicity, let us consider that the boundary is located in the plane \(xy\) with the positive direction of the axis \(z\) facing the inner part of the solid, and assuming that the plane waves are traveling in the direction \(x\). Since the displacements will therefore be independent of \(y\) it is possible to  
These two equations will be satisfied if  
\[ \begin{array}{c}\frac{\partial^{2}\phi}{\partial t^{2}}=\left[\frac{\lambda+2 \mu}{p}\right]\
abla^{2}\phi={c_{1}}^{2}\
abla^{2}\phi\\ \frac{\partial^{2}\phi}{\partial t^{2}}=\left[\frac{\mu}{p}\right]\
abla^{2} \psi={c_{2}}^{2}\
abla^{2}\psi.\end{array}  \tag{9}\]  
where \(c_{1}\) and \(c_{2}\) are the longitudinal and transverse waves velocities, respectively. Considering a sine wave of frequency \(\frac{p}{2\pi}\) that propagates in the direction \(x\) with velocity \(c\) and wavelength \(\frac{2\pi}{f}\) so that \(c=\frac{p}{f}\), it is possible to find a solution for the equations (9) in the form  
\[ \begin{array}{c}\phi=F(z)\exp[i(pt-fx)]\\ \psi=G(z)\exp[i(pt-fx)]\end{array}  \tag{10}\]  
where \(i=\sqrt{-1}\) and \(F\) and \(G\) are functions that determine the way in which the wave amplitude changes with \(z\). By substituting the expression for \(\phi\) in the differential equation of motion, the following expression is obtained  
\[F^{\prime\prime}(z)-\left(f^{2}-h^{2}\right)F(z)=0 \tag{11}\]  
The general solution of Equation (11) corresponds to  
\[F(z)=A\exp(-qz)+A^{\prime}\exp(qz) \tag{12}\]  
where \(q^{2}=f^{2}-h^{2}\). The second term in the equation is related to the disturbance that increases with increasing \(z\), and, for the type of wave considered, \(A^{\prime}\) must be zero. Similar results are obtained by substituting the function \(\psi\) in the differential equation of motion  
\[G^{\prime\prime}(z)-\left(f^{2}-k^{2}\right)G(z)=0 \tag{13}\]  
with the general solution expressed in the form  
\[G(z)=B\exp(-sz) \tag{14}\]  
where \(s^{2}=f^{2}-k^{2}\). Hence the expressions (10) for \(\phi\) and \(\psi\) become  
\[ \begin{array}{c}\phi=A\ \exp[-qz+i(pt-fx)]\\ \psi=B\exp[-sz+i(pt-fx)].\end{array}  \tag{15}\]  
Considering the boundary conditions, for which the stress components \(\sigma_{zz}\), \(\sigma_{zy}\) and \(\sigma_{zx}\) vanish at the surface where \(z=0\), the following relations are obtained  
\[ \begin{array}{c}A\left[(\lambda+2\mu)q^{2}-\lambda f^{2}\right]-2B\mu isf=0\\ 2 iqfA+\left(s^{2}+f^{2}\right)B=0\end{array}  \tag{16}\]  
from which, by eliminating the ratio \(\frac{A}{B}\) from Equation (16), squaring both sides and substituting for \(q\) and \(s\) the previous expressions, the following equation can be derived  
\[16\mu^{2}\left(f^{2}-h^{2}\right)\left(f^{2}-k^{2}\right)f^{4}=\left[-( \lambda+2\mu)h^{2}+2\mu f^{2}\right]^{2}\left(2f^{2}-k^{2}\right)^{2} \tag{17}\]  
then simplified in the form  
\[16\left(1-\frac{h^{2}}{f^{2}}\right)\left(1-\frac{k^{2}}{f^{2}}\right)=\left[ 2-\frac{(\lambda+2\mu)\mu^{-1}h^{2}}{f^{2}}\right]^{2}\left(2-\frac{k^{2}}{f^ {2}}\right)^{2}. \tag{18}\]