*2.1. Propagation of Ultrasonic Waves in Linear-Elastic Material*

Generally, an anisotropic body, e.g., crystal of defined symmetry, can be the solid medium. The propagation of waves in the anisotropic medium, particularly velocity, depends on the direction relative to the axis of coordinates usually related to the crystallographic arrangement that corresponds to the given symmetry. Hooke's law [15] describes elastic properties of the anisotropic arrangement in the linear relationship between the stress tensor σij and the deformation tensor εkl in the following way:

$$
\sigma\_{i\bar{j}} = \mathcal{c}\_{i\bar{j}k\bar{l}}\varepsilon\_{k\bar{l}} + \mathcal{c}\_{i\bar{j}k\bar{l}m\bar{u}}\varepsilon\_{k\bar{l}}\varepsilon\_{m\bar{u}} + \dots \tag{1}
$$

where: σi—components of stress state, εkl—components of deformation state.

Both quantities are symmetric tensors of second rank, which means they can have six independent components. Coefficients *c*ijkl and *c*ijklmn are constants of elasticity of second or third rank, respectively. They are symmetric tensors of fourth and sixth rank, respectively. The linear theory of elasticity assumes

materials are elastic, and the relationship between stress and deformation is linear. All constants of elasticity of third order or higher are neglected. Even for such a simplification, the number of tensor components *c*ijkl defining elastic properties is 36, but the number of independent components is 21. In the case of orthotrophic materials with three mutually perpendicular planes of symmetry, elastic properties are described by nine independent constants of elasticity in the following form:

$$c\_{ij} = \begin{bmatrix} c\_{11} & c\_{12} & c\_{13} & 0 & 0 & 0\\ c\_{12} & c\_{22} & c\_{23} & 0 & 0 & 0\\ c\_{13} & c\_{23} & c\_{33} & 0 & 0 & 0\\ 0 & 0 & 0 & c\_{44} & 0 & 0\\ 0 & 0 & 0 & 0 & c\_{55} & 0\\ 0 & 0 & 0 & 0 & 0 & c\_{66} \end{bmatrix} \tag{2}$$

Regarding isotropic materials with the infinite number of axes of symmetry planes, elastic properties can be comprehensively described by two independent constants of elasticity *c*<sup>12</sup> and *c*44. Other matrix coefficients (2) can be expressed as linear combinations using the Lamé coefficients:

$$\mathfrak{c}\_{11} = \mathfrak{c}\_{22} = \mathfrak{c}\_{33} = \lambda + 2\mu,\ \mathfrak{c}\_{12} = \mathfrak{c}\_{23} = \mathfrak{c}\_{23} = \lambda,\ \mathfrak{c}\_{44} = \mathfrak{c}\_{55} = \mathfrak{c}\_{66} = \mu. \tag{3}$$

The force acting on any volume element in the solid medium, in which the disturbance is observed, can be expressed as the gradient of stress caused by the disturbance [1]. The Equation of the particle motion representing the equilibrium state between the restoring force and the inertial force is expressed by the following Equation:

$$
\rho\_0 \frac{\partial^2 \underline{\xi}\_i}{\partial t^2} = \frac{\partial T\_{ij}}{\partial \mathbf{x}\_j} \to \rho\_0 \frac{\partial^2 \underline{\xi}\_i}{\partial t^2} = c\_{ijkl} \frac{\partial^2 \underline{\xi}\_k}{\partial \mathbf{x}\_j \partial \mathbf{x}\_l} \tag{4}
$$

where: ρ0—density of the body in the tensionless state. The expression (4) contains the equations of three components of the displacement, which describe components of the wave equation of vector quantity ξ described by three components. Assuming that coordinates of the plane harmonic wave are expressed by the relationship ξ = ξ0*ei*(ω*t*−*kr*), Equation (4) can be expressed as:

$$-\omega^2 \rho\_0 \underline{\varsigma}\_{0i} = -c\_{i\bar{j}k} k\_j k\_l \underline{\varsigma}\_{0k} \rightarrow \left( c\_{i\bar{j}k} k\_j k\_l - \delta\_{ik} \omega^2 \rho\_0 \right) \underline{\varsigma}\_{ok} = 0 \tag{5}$$

where: ω is the wave frequency, *kj*, *kl*,—wave vector (towards *j, l*), ξ0i, ξ0k—coordinates of the plane harmonic wave (towards *i*, *k*).

Expression (5) is the system of homogeneous algebraic equations, which due to unknown ξ0k is described in the following form:

$$\begin{cases} c\_{1\bar{\jmath}1}k\_ik\_l - \alpha^2 \rho\_0 \vert \not\xi\_{10} + c\_{1\bar{\jmath}2}k\_ik\_l \not\xi\_{20} + c\_{1\bar{\jmath}3}k\_ik\_l \not\xi\_{30} = 0\\ c\_{2\bar{\jmath}1}k\_ik\_l \not\xi\_{10} + \left(c\_{2\bar{\jmath}2}k\_ik\_l - \alpha^2 \rho\_0\right) \not\xi\_{20} + c\_{2\bar{\jmath}3}k\_ik\_l \not\xi\_{30} = 0\\ c\_{3\bar{\jmath}1}k\_ik\_l \not\xi\_{10} + c\_{3\bar{\jmath}2}k\_ik\_l \not\xi\_{20} + \left(c\_{3\bar{\jmath}21}k\_ik\_l - \alpha^2 \rho\_0\right) \not\xi\_{30} = 0 \end{cases} \tag{6}$$

The system of equations is fulfilled when the determinant of the coefficients is equal to 0. The equation of third degree relevant to ω<sup>2</sup> is the solution for the determinant. The equation contains three roots that correspond to three different waves with mutually perpendicular displacements. When the simplest case of the isotropic body and waves travelling along one axis (x3), the determinant of the Equation (6) takes the following form:

$$
\begin{vmatrix}
 c\_{44}k^2 - \alpha^2 \rho\_0 & 0 & 0 \\
 0 & c\_{44}k^2 - \alpha^2 \rho\_0 & 0 \\
 0 & 0 & c\_{11}k^2 - \alpha^2 \rho\_0
\end{vmatrix} = 0\tag{7}
$$

By solving the determinant, the following equation is obtained:

$$\left(c\_{44}k^2 - \alpha^2\rho\_0\right)^2 \left(c\_{11}k^2 - \alpha^2\rho\_0\right) = 0\tag{8}$$

It has two roots equal to ω<sup>2</sup> <sup>1</sup> <sup>=</sup> <sup>ω</sup><sup>2</sup> <sup>2</sup> <sup>=</sup> *<sup>c</sup>*44*k*<sup>2</sup> <sup>ρ</sup><sup>0</sup> , and the third one equal to <sup>ω</sup><sup>2</sup> <sup>3</sup> <sup>=</sup> *<sup>c</sup>*11*k*<sup>2</sup> <sup>ρ</sup><sup>0</sup> . Taking into account that *k* = ω/*C* (where *C* is wave velocity), the following roots are obtained:

$$\mathbf{C}\_1 = \mathbf{C}\_2 = \sqrt{\frac{c\_{44}}{\rho\_0}}, \mathbf{C}\_3 = \sqrt{\frac{c\_{11}}{\rho\_0}} \tag{9}$$

A solution to this issue indicates the propagation of three waves in the body. Two of them are characterized by mutually perpendicular oscillations and the same velocity *C*<sup>1</sup> = *C*<sup>2</sup> = *cT* is known as transverse waves as *c*<sup>44</sup> is the shear. The third wave with the velocity *c*<sup>p</sup> is the longitudinal wave because *c*<sup>11</sup> is constant related to the component of the normal deformation. Taking into account relationships between material constants, the following expression is obtained:

$$c\_T = \sqrt{\frac{\mu}{\rho\_0}}, c\_p = \sqrt{\frac{\lambda + 2\mu}{\rho\_0}}\tag{10}$$

Constants λ and μ can be introduced into the system of Equations (4) by replacing coefficients *c*ijkl. Then, the system of equations broken down into components is for the isotropic body as follows:

$$\begin{array}{l} \rho\_0 \frac{\partial^2 \xi\_1}{\partial t^2} = (\lambda + 2\mu) \frac{\partial^2 \xi\_i}{\partial x\_1 \partial x\_i} + \mu \frac{\partial^2 \xi\_1}{\partial x\_1 \partial x\_i} \\ \rho\_0 \frac{\partial^2 \xi\_2}{\partial t^2} = (\lambda + 2\mu) \frac{\partial^2 \xi\_i}{\partial x\_2 \partial x\_i} + \mu \frac{\partial^2 \xi\_2}{\partial x\_2 \partial x\_i} \\ \rho\_0 \frac{\partial^2 \xi\_3}{\partial t^2} = (\lambda + 2\mu) \frac{\partial^2 \xi\_i}{\partial x\_3 \partial x\_i} + \mu \frac{\partial^2 \xi\_2}{\partial x\_2 \partial x\_i} \end{array} \tag{11}$$

When the medium is incompressible (no changes in volume), the above equations give the wave equation for transverse waves in the following vector form:

$$\frac{\partial^2 \xi}{\partial t^2} = \frac{\mu}{\rho\_0} \nabla^2 \xi \tag{12}$$

where <sup>∇</sup><sup>2</sup> is the Laplace operator of the second order in *n-* dimensional Cartesian coordinate system expressed as: <sup>∇</sup><sup>2</sup> <sup>=</sup> <sup>Δ</sup> <sup>=</sup> <sup>∂</sup><sup>2</sup> ∂*x*<sup>2</sup> *i* + <sup>∂</sup><sup>2</sup> ∂*x*<sup>2</sup> *j* + <sup>∂</sup><sup>2</sup> ∂*x*<sup>2</sup> *k* + ... + <sup>∂</sup><sup>2</sup> ∂*x*<sup>2</sup> *n* .

Assuming the irrotational medium, the wave equation for longitudinal waves is as follows:

$$\frac{\partial^2 \xi}{\partial t^2} = \frac{\lambda + 2\mu}{\rho\_0} \nabla^2 \xi \tag{13}$$
