## 2 Waves Propagation in Isotropic Homogeneous Elastic Media. Basic Theoretical Concepts  
In this Section the basic concepts on the theory of wave propagation through linear elastic, isotropic and homogeneous media are given. The main reference is made to the theoretical aspects derived from the seminal books of [PERSON], [PERSON] and [PERSON] [43; 44; 45]. Let us consider an elastic solid media in a Cartesian coordinate system \((x,y,z)\) and a generic point with displacement vector \((u,v,w)\), stress tensor components \(\sigma_{ij}\) and body forces \((X,Y,Z)\). Being \(\rho\) the mass density, the equations of motion are expressed as  
\[\rho\frac{d^{2}u}{dt^{2}}=\rho X+\frac{\partial\sigma_{xx}}{\partial x}+\frac{ \partial\sigma_{yx}}{\partial y}+\frac{\partial\sigma_{zx}}{\partial z}\]  
\[\rho\frac{d^{2}v}{dt^{2}}=\rho Y+\frac{\partial\sigma_{xy}}{\partial x}+\frac {\partial\sigma_{yy}}{\partial y}+\frac{\partial\sigma_{zy}}{\partial z} \tag{1}\]  
\[\rho\frac{d^{2}w}{dt^{2}}=\rho Z+\frac{\partial\sigma_{xz}}{\partial x}+\frac {\partial\sigma_{yz}}{\partial y}+\frac{\partial\sigma_{zz}}{\partial z}\]  
For an isotropic elastic solid Equation (1) can be expressed in terms of displacements using Lame constants, \(\lambda\) and \(\mu\), as following  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=(\lambda+\mu)\frac{\partial\theta}{ \partial x}+\mu\
abla^{2}u+\rho X\]  
\[\rho\frac{\partial^{2}v}{\partial t^{2}}=(\lambda+\mu)\frac{\partial\theta}{ \partial y}+\mu\
abla^{2}v+\rho Y \tag{2}\]  
\[\rho\frac{\partial^{2}w}{\partial t^{2}}=(\lambda+\mu)\frac{\partial\theta}{ \partial z}+\mu\
abla^{2}w+\rho Z\]in which \(\theta=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}\). The expression \(\frac{d^{2}}{dt^{2}}\) has been replaced with \(\frac{\partial^{2}}{\partial t^{2}}\) as their difference involves squares or products of components assumed to be very small; linearizing the differential equations is possible through this operation. By introducing a scalar potential \(\varphi\) and a vector potential \(\psi(\psi_{1},\psi_{2},\psi_{3})\), with  
\[ \begin{array}{l}u=\frac{\partial\varphi}{\partial x}+\frac{\partial\psi_{3} }{\partial y}-\frac{\partial\psi_{2}}{\partial z}\\ \\ v=\frac{\partial\varphi}{\partial y}+\frac{\partial\psi_{1}}{\partial z}- \frac{\partial\psi_{3}}{\partial x}\\ \\ w=\frac{\partial\varphi}{\partial z}+\frac{\partial\psi_{2}}{\partial x}- \frac{\partial\psi_{3}}{\partial y}\\ \end{array}  \tag{3}\]  
and neglecting volume forces, the following expression can be obtained  
\[\begin{array}{l}\frac{\partial}{\partial x}\left(\rho\frac{\partial^{2} \varphi}{\partial t^{2}}\right)+ \begin{array}{l}\frac{\partial}{\partial y} \left(\rho\frac{\partial^{2}\psi_{2}}{\partial t^{2}}\right)-\frac{\partial} {\partial z}\left(\rho\frac{\partial^{2}\psi_{3}}{\partial t^{2}}\right)\\ =(\lambda+\mu)\frac{\partial}{\partial x}\
abla^{2}\varphi+\mu\frac{\partial} {\partial x}\
abla^{2}\varphi+\mu\frac{\partial}{\partial y}\
abla^{2}\psi_{ 3}-\mu\frac{\partial}{\partial z}\
abla^{2}\psi_{2}.\end{array}  \tag{4}\]  
The previous equation will be satisfied if the functions \(\varphi\) and \(\psi_{i}\) are solutions of the equations  
\[\
abla^{2}\varphi=\frac{1}{\alpha^{2}}\frac{\partial^{2}\varphi}{\partial t^{ 2}}\qquad\
abla^{2}\psi_{i}=\frac{1}{\beta^{2}}\frac{\partial^{2}\psi_{i}}{ \partial t^{2}}\qquad i=1,2,3 \tag{5}\]  
in which  
\[\alpha=\sqrt{\frac{\lambda+2\mu}{\rho}}\qquad\beta=\sqrt{\frac{\mu}{\rho}}. \tag{6}\]  
The obtained wave equations indicate that two types of disturbances with different velocities \(\alpha\) and \(\beta\) can be propagated through an elastic solid. The first type of waves found, i.e., the expansion waves, are associated with displacements in a direction that is equal to the direction of propagation, consisting of compressions and rarefactions that only lead to volume changes.