# Application of the phase gradient method to the study of the frequency and angular resonances of a water-loaded anisotropic plate

Lionel Guénégou, Olivier Lenoir, Jean-Louis Izbicki

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Lionel Guénégou, Olivier Lenoir, Jean-Louis Izbicki. Application of the phase gradient method to the study of the frequency and angular resonances of a water-loaded anisotropic plate. 17e Congrès Français de Mécanique, Aug 2005, Troyes, France. hal-04105584

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Application of the phase gradient method to the study of the frequency
and angular resonances of a water-loaded anisotropic plate

Lionel Guénégou(a), Olivier Lenoir, Jean-Louis Izbicki

Laboratoire d'Acoustique Ultrasonore et d'Electronique (LAUE)
Université du Havre - Place Robert Schuman
76610 Le Havre

(a) e-mail : lionel.guenegou@univ-lehavre.fr

Résumé :

La méthode du gradient de phase est appliquée à des plaques anisotropes de symétrie cubique. Elle consiste à étudier les dérivées fréquentielle et angulaires de la phase du coefficient de réflexion. Les tracés de ces dérivées permettent d'obtenir facilement la position et la largeur des résonances fréquentielles et angulaires. Les résultats obtenus par la méthode du gradient de phase et ceux obtenus par le calcul des pôles du coefficient de réflexion sont en bon accord. De plus, ces dérivées permettent d'obtenir les facteurs de qualité fréquentiel et angulaires associés à une onde donnée et d'estimer les composantes de sa vitesse d'énergie.

Abstract :

The phase gradient method is applied to anisotropic plates of cubic symmetry. It consists in studying the frequency and angular derivatives of the phase of the reflection coefficient. The plots of these derivatives permit to simply obtain the location and the width of the frequency and angular resonances. Results from the PGM and from the exact calculation of the plate resonances are in good agreement. Moreover, these derivatives allow us to obtain frequency and angular quality factors and to estimate the components of the energy velocity.

Keywords : Resonances ; reflection coefficient ; anisotropy.

1 Introduction

The phase gradient method (PGM) deals with the study of the partial derivatives of the phase of the reflection coefficient with respect to the frequency, to the incidence angle, to the parameters of the plate and of the surrounding fluid (densities, phase velocities or stiffness components). In the isotropic case, the study of the frequency and angular phase derivatives was proved to be efficient to simply and accurately obtain the location and the width of the frequency and angular resonances of the plate [1]. These resonances correspond to the poles of the reflection coefficient or to the modes of the immersed plate. It avoids us to perform tedious calculations of roots in the complex frequency or angular planes. Indeed the plots of these derivatives, versus frequency or incidence angle, exhibit maximums located at the real parts of the frequency and angular roots and whose amplitudes are inversely proportional to their imaginary parts. Moreover, resonances associated either with symmetric waves or with antisymmetric waves can easily be separated. In this study, the phase gradient method is applied to anisotropic plates of cubic symmetry. The phase gradient method can be applied to such water-loaded anisotropic plates because the expression of their reflection coefficient was shown to be formally identical to the one of an isotropic plate [2, 3] ; it is recalled in the first section. Results dealing with a cubic (GaAs) plate studied in a non symmetry plane are presented in the second section : the frequency and angular phase derivatives are studied. Exact calculations of modes of the immersed plate are compared with the results obtained from the plots of frequency or angular phase derivatives. The phase derivative with respect to the rotation angle between the sagittal plane and a reference symmetry plane is also studied. A frequency quality factor and two angular ones can be
derived from those derivatives. In addition, it is shown that the components of the energy velocity of a given mode can be simply estimated from these quality factors.

## 2 Factorized expression of the reflection coefficient of an anisotropic plate

The factorized expression of the reflection coefficient is needed in the phase gradient method to separate the resonances associated with symmetric modes from those associated with antisymmetric modes. Consider a water-loaded plate of thickness h. The parameters for water are : density ρ<sub>F</sub> = 1000 kg/m<sup>3</sup> and sound speed c<sub>F</sub> = 1470 m/s. The geometry of the problem is given in FIG. 1. The coordinate system (x', y', z') is the crystallographical one. The x'-y', x'-z' and y'-z' planes are symmetry planes. The z'-axis is normal to the plate-water boundaries. In the coordinate system (x', y', z') the stiffness tensor is denoted as c'<sub>αβ</sub> in abbreviated subscript notation. In a transformed coordinate system (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub> = z') obtained by a counterclockwise rotation through an angle ψ about the z'-axis the stiffness tensor is c<sub>αβ</sub>.

<figure><img src="image_2.png" /><figcaption>FIG. 1 – Geometry of the problem.</figcaption></figure>

The quantity of interest is the reflection coefficient of an incident monochromatic plane wave of time dependence e<sup>-jωt</sup>, propagating in the x₁-x₃ sagittal plane, which insonifies the upper plate surface at x₃ = h/2. To obtain the reflection coefficient R, the continuity of the normal displacement and stress, and the nullity of the tangential stresses are derived at the interfaces x₃ = ±h/2. Its expression is given by :

$$R = \frac{C_A C_S - \tau^2}{(C_A + j\tau)(C_S - j\tau)} = \frac{N_R}{A S} \quad (1)$$

It is formally identical to the one used by Fiorito *et al.* in their application of the Resonant Scattering Theory (RST) to isotropic elastic plates [2]. The expression of R is also similar to the one obtained by A. H. Nayfeh [3]. The roots of the C<sub>A,S</sub> functions correspond to the antisymmetric and symmetric modes of the plate in vacuum. Those of functions A and S correspond to the antisymmetric and symmetric modes of the plate immersed in water.

## 3 The phase gradient method (PGM)

In this section, results derived from the derivatives with respect to the frequency, to the sine of the incidence angle and to the rotation angle are presented. Numerator N<sub>R</sub> in Eq. (1) is real, so the global phase of the reflection coefficient R can be written as :
$$ \varphi = \arctan(\tau/C_s) - \arctan(\tau/C_\Lambda) = \varphi_s + \varphi_\Lambda. \quad (2) $$

The phase terms due to symmetric and antisymmetric modes can be separated easily. From Eq. (2), the analytical expressions of the differentials of the phases $\varphi_{A,S}$ are given by :

$$ \partial\varphi_{A,S} = \mp \frac{C_{A,S}\partial\tau - \tau\partial C_{A,S}}{C_{A,S}^2 + \tau^2}. \quad (3) $$

In the following, the plots of interest are those of the $\pm z\partial\varphi_{A,S}/\partial z$ functions versus $z$ ($z$ is either a frequency variable, generally the frequency-thickness product $fh$ in MHz mm, or an angular variable, classically sine $y$ of the incident angle). A frequency resonance related to a A or S mode is denoted as $fh_p = fh_{Res} - j\Gamma_{A,S}/2$ and corresponds to a frequency pole of R. An angular resonance is denoted as $y_p = y_{Res} + j\gamma_{A,S}/2$ (associated with the complex reemission angle $\theta_p = \arcsin(y_p)$) and corresponds to an angular pole of R. In the vicinity of an isolated resonance, the $fh\partial\varphi_{A,S}/\partial fh$ and $y\partial\varphi_{A,S}/\partial y$ functions have the following approximate Breit-Wigner resonant type expressions :

$$ (fh \partial\varphi_{A,S} / \partial fh)_{\text{app}} = fh_{\text{Res}} \Gamma_{A,S} / 2 / \left( (fh - fh_{\text{Res}})^2 + (\Gamma_{A,S} / 2)^2 \right) \quad (4) $$

$$ (-y \partial\varphi_{A,S} / \partial y)_{\text{app}} = y_{\text{Res}} \gamma_{A,S} / 2 / \left( (y - y_{\text{Res}})^2 + (\gamma_{A,S} / 2)^2 \right). \quad (5) $$

Their maximums are obtained when $fh = fh_{Res}$ or $y = y_{Res}$ and are given by :

$$ (fh \partial\varphi_{A,S} / \partial fh)_{\text{app}} = 2fh_{\text{Res}} / \Gamma_{A,S} = \text{Re}(fh_p) / \text{Im}(fh_p) = 2 Q_x \quad (6) $$

$$ (-y \partial\varphi_{A,S} / \partial y)_{\text{app}} = 2 y_{\text{Res}} / \gamma_{A,S} = \text{Re}(y_p) / \text{Im}(y_p) = 2 Q_y. \quad (7) $$

Eqs. (6) and (7) define the frequency and angular quality factors $Q_x$ and $Q_y$. For an isolated resonance, Eqs. (4) and (5) provide very good approximate expressions of the exact $\pm z\partial\varphi_{A,S}/\partial z$ functions. Hence, the plots of the exact functions give very good estimates of the quality factors, with no need of calculations of the poles.

Another interest of the phase gradient method is to estimate the energy velocity. It is well known in the anisotropic case, contrary to the isotropic one, that the energy direction is different from the phase propagation direction [7]. The component of the energy velocity in the sagittal plane $V_{E\parallel}$ of a given mode is deduced from the calculation of the radiation quality factors obtained from the study of the frequency and angular derivatives.

$$ \text{The approximate expression is given by: } V_{E\parallel}^{\text{PGM}} = \frac{Q_x}{Q_x + Q_y} v_{\varphi}. \quad (8) $$

$$ v_{\varphi} \text{ is the phase velocity. The exact energy expression is given by: } V_{E\parallel} = \frac{\langle P_{x1} \rangle}{\langle E_{\text{tot}} \rangle}. \quad (9) $$
$\langle P_{x_1} \rangle$ stands for the mean $x_1$-component of the Poynting vector, $\langle E_{tot} \rangle$ for the mean total energy (sum of the kinetic and strain energies). In addition, the phase can be derivated with respect to the angle $\psi$ indicating the phase front direction with regard to a symmetry plane. A radiation $Q_{\psi}$ factor can be introduced. It can be shown that the component of the energy velocity normal to the sagittal plane $V_{E\perp}$ is deduced from the three radiation Q factors [6].

$$ \text{The approximate expression is: } V_{E\perp}^{\text{PGM}} = \frac{Q_{\psi}}{Q_x + Q_y} v_{\phi}, \qquad (10) $$

$$ \text{where } Q_{\psi} \text{ is given at a fixed resonance frequency } fh_{\text{Res}} \text{ by: } Q_{\psi} = \left(\frac{\partial\varphi}{\partial\psi}\right)(fh_{\text{Res}}). \qquad (11) $$

$$ \text{The exact energy expression of } V_{E\perp} \text{ is as follows: } V_{E\perp} = \frac{\langle P_{x_2} \rangle}{\langle E_{tot} \rangle}, \qquad (12) $$

where $\langle P_{x_2} \rangle$ stands for the mean $x_2$-component of the Poynting vector.

# 4 Numerical results

## 4.1 Frequency and angular derivatives

The material considered is Gallium Arsenide (GaAs) whose density is $\rho = 5307 \text{ kg/m}^3$. The values of the components of the stiffness tensor are $c'_{11} = 118.8 \text{ GPa}$, $c'_{12} = 53.8 \text{ GPa}$ and $c'_{44} = 59.4 \text{ GPa}$. In the following, only plots for $\psi = 30^\circ$ are presented at the incidence angle $\theta = 5^\circ$.

<figure><img src="image_3.png" /><figcaption>FIG. 2 – (a) Plots of the exact (solid line) and approximate (cross line) $fh\partial\varphi_{\Lambda}/\partial\text{fh}$ functions in the vicinity of $A_2$ mode ; (b) Plots of the exact (solid line) and approximate (cross line) $fh\partial\varphi_{\Lambda}/\partial\text{fh}$ functions in the vicinity of $A'_2$ mode.</figcaption></figure>

First, the plots of the exact $fh\partial\varphi_{\Lambda,S}/\partial\text{fh}$ functions are compared to those of the approximate ones. For $\psi \neq 0^\circ$, quasi-SH waves are detected in addition to Lamb type waves. In FIG. 2, the plots of the exact and approximate $fh\partial\varphi_{\Lambda}/\partial\text{fh}$ functions are compared in the vicinity of isolated resonances due to antisymmetric modes, either mainly quasi-L ($f_{\Lambda}h = 4.47 - j0.04$) in FIG. 2a or quasi-SH ($f_{\Lambda}h = 5.09 - j1.25 \times 10^{-4}$) in FIG. 2b.
The angular phase derivatives $-y\partial\varphi_{A,S}/\partial y$ function are also studied. These functions can take either positive or negative values. Each negative peak is associated with a mode whose group velocity is negative [5]. Let us consider a negative peak at $fh = 4.47$ associated with the root $y_p = 0.087 - j0.015$ ($\theta_p = (5.01 - j0.86)^\circ$) and a positive one at $fh = 5.60$ associated with the root $y_p = 0.087 + j6.02 \times 10^{-3}$ ($\theta_p = (4.99 + j0.35)^\circ$). In FIG. 3, the angular evolution of the exact and approximate $-y\partial\varphi_A/\partial y$ functions are compared at those frequencies. The curves exhibit Breit-Wigner shapes whose maximum (or minimum) are located around 5°. The amplitudes are nearly equal to the ratio of the real part to the imaginary part of $y_p$: the values of the angular quality factors $Q_y$ are obtained from the exact plot with less than 2% of error.

<figure><img src="image_4.png" /><figcaption>FIG. 3 – Plots of the exact (solid lines) and approximate $-y\partial\varphi_A/\partial y$ functions versus incidence angle. Negative peak (point line), $fh = 4.47$. Positive peak (cross line), $fh = 5.6$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 4 – Plots of the exact $\partial\varphi_A/\partial\Psi$ function versus frequency in the vicinity of $A_2$ mode. $\theta = 5^\circ$.</figcaption></figure>

The plot of the exact $\partial\varphi_A/\partial\Psi$ function also exhibits a Breit-Wigner shape as shown in FIG. 4. Calculation in the corresponding complex plane have not been performed. No comparison between exact and approximate functions are presented for this derivative.

## 4.2 Estimation of the components of the energy velocity

Comparisons between the results obtained from the phase gradient method and from energy calculations are presented for the antisymmetric mode $A_8$ in FIG. 5. In the energy calculations, the symmetric and antisymmetric contributions are not separated, so, applying the PGM, the phase considered is the sum of the antisymmetric and the symmetric terms from Eq. (2). The plots of $V_{E\parallel}$ obtained from the approximate expression (Eq. (8)) and from the exact one (Eq. (9)) are compared. We can observe that the exact and the approximate curves in FIG. 5 are nearly an exact match at low frequencies. At higher frequencies, there is a small mismatch between the two plots, nevertheless the evolutions are similar. The plots of $V_{E\perp}$ are also compared : the approximate expression (Eq. (10)) to the exact one (Eq. (12)) and the curves are nearly superimposed. We can observe transitions (minimums or maximums) on all these plots. They can be attributed to the closeness or the crossing of the dispersion curve of $A_8$ mode with the ones of other modes (symmetric or antisymmetric ones). For the cubic plate considered, we observe that the component $V_{E\perp}$ of the energy velocity can be negative.