Dispersion and Energy Characteristics of Bending Waves in a Plate Lying on a Two-Parameter Elastic Foundation

Abstract

The propagation of bending waves in a plate resting on a two-parameter elastic foundation is considered. In contrast to the classical Kirchhoff model, the mathematical model used here takes into account not only the kinetic and potential energies of bending vibrations, but also the kinetic energy due to the inertia of rotation of the plate elements during bending. The dispersion equation, phase velocity, energy transfer velocity, and energy characteristics of waves propagating in the plate are analyzed depending on the ratio of the coefficients determining the shear and compression stiffness of the elastic foundation. Conditions are found under which waves with phase and group velocities having opposite directions (frequently called “backward” waves), can exist in the plate. It is demonstrated that such waves significantly change the behavior of the energy flux. In addition, relations are found that relate the kinematic and average energy characteristics of the waves.

INTRODUCTION

A plate on an elastic foundation is widely used as a calculation model that describes various elements of aviation, shipbuilding, mechanical engineering, construction, etc., objects.

The continuous increase in the speed, operating time, and power density of machines and mechanisms, concern for reducing the weight of a structure while improving its performance characteristics, and the widespread introduction of new composite materials into modern technology lead to increased requirements on the model and calculation accuracy. Modern trends in studying of vibrations of plates on an elastic foundation tend to complicate the model, which in the overwhelming majority of cases is performed numerically. For example, whereas earlier, the Kirchhoff model [1] was chosen to analyze bending waves in a plate, recently, the Timoshenko model has seen increasing use [1–3]. This also applies to the elastic foundation, which is described by more complex models [4–6] than with the Winkler linear model [7].

As is known [8–16], in plates and thin layers, there is a special type of waves characterized by phase and group velocities having opposite directions. Such waves are frequently called “backward” waves, in contrast to ordinary, “direct” waves, for which the directions of the phase and group velocities coincide. When studying backward acoustic waves in plates, it was found that they are very sensitive to the plate parameters [12, 17–19].

This study attempts to find the parameters of both the plate and elastic foundation for which backward bending waves can exist.

PLATE DYNAMICS EQUATIONS

Let us consider bending vibrations of a plate resting on an elastic foundation, the properties of which are characterized by two coefficients of the subgrade stiffness: for tension k₁ and shear k₂. Coefficient k₁ has the dimension H/m³, and the shear coefficient k₂, which takes into account the joint work of neighboring regions, has the dimension N/m. Such a model of an elastic foundation is called a generalized, two-parameter, non-Winkler, or Pasternak model [20–24]. The degenerate case (k₂ = 0) corresponds to the classical Winkler foundation model. The use of the Pasternak model makes it possible not only to preserve the simplicity of the mathematical apparatus inherent to the Winkler model, but also to obtain more reliable results.

For the Lagrangian of small plate oscillations, taking into account the inertia of rotation of its elements during bending, we have the expression [25]

$$\begin{gathered} \lambda = \frac{1}{2}\left\{ {\rho {{h}_{*}}u_{t}^{2} + \frac{{\rho h_{*}^{3}}}{{12}}\left( {u_{{xt}}^{2} + u_{{yt}}^{2}} \right)} \right. \\ - \,\,\frac{{h_{*}^{3}}}{6}\left( {\frac{{\Lambda + 2\mu }}{2}{{{\left( {{{u}_{{xx}}} + {{u}_{{yy}}}} \right)}}^{2}}} \right. \\ \left. {\left. { + 2\,\,\mu \left( {u_{{xy}}^{2} - {{u}_{{xx}}}{{u}_{{yy}}}} \right)} \right) - {{k}_{1}}{{u}^{2}} - {{k}_{2}}\left( {u_{x}^{2} + u_{y}^{2}} \right)} \right\}. \\ \end{gathered} $$

(1)

Here, $\rho {{h}_{*}}$ is the surface density, $\Lambda$, $\mu$ are the Lamé constants, ${{h}_{*}}$ is thickness, u(x, y, t) is the transverse displacement of the plate, and ${{k}_{1}}$ and ${{k}_{2}}$ are coefficients of subgrade stiffness for compression and shear of the foundation of the plate, respectively; subscripts t and x, y are the partial derivatives of the bias function with respect to time and the spatial coordinates.

Substituting (1) into the Euler–Ostrogradsky equation [1], we obtain the equation for bending vibrations of the plate:

$$\begin{gathered} \rho {{h}_{*}}{{\partial }_{{tt}}}u - \frac{{\rho h_{*}^{3}}}{{12}}{{\partial }_{{tt}}}\Delta u + \frac{{\left( {\Lambda + 2\mu } \right)h_{*}^{3}}}{{12}}\Delta \Delta u \\ - \,\,{{k}_{2}}\Delta u + {{k}_{1}}u = 0 \\ \left( {{{\partial }_{{tt}}} = \frac{{{{\partial }^{2}}}}{{\partial {{t}^{2}}}},\,\,\,\,\Delta = {{\partial }_{{xx}}} + {{\partial }_{{yy}}}} \right), \\ \end{gathered} $$

(2)

the solution to which can be represented as the superposition of plane waves

$$u\left( {x,y,t} \right) = A\exp \left[ {i\left( {{{\omega }}t - {{k}_{x}}x - {{k}_{y}}y} \right)} \right],$$

where $A$, $\omega$, ${{k}_{x}}$, ${{k}_{y}}$ are complex constants.

ENERGY TRANSFER AND WAVE MOMENTUM EQUATIONS

A feature of distributed elastic systems is the possibility of transfer with a finite velocity of energy and wave momentum. To obtain the corresponding transfer equations, we multiply the dynamics equation (1) by the partial derivatives of the transverse bias function u(x, y, t) of the plate u_t, u_x, u_y and bring the obtained relations to a divergent form. As a result, we have

$$\frac{{\partial h}}{{\partial t}} +\text{div}\mathbf{S} = 0,\,\,\,\,\frac{{\partial {\mathbf{p}}}}{{\partial t}} + \text{div}\mathbf{T} = 0.$$

(3)

Here

$$\begin{gathered} h = \frac{1}{2}\left\{ {\rho {{h}_{*}}u_{t}^{2} + \frac{{\rho h_{*}^{3}}}{{12}}\left( {u_{{xt}}^{2} + u_{{yt}}^{2}} \right)} \right. \\ + \,\,\frac{{h_{*}^{3}}}{6}\left( {\frac{{\Lambda + 2\mu }}{2}{{{\left( {{{u}_{{xx}}} + {{u}_{{yy}}}} \right)}}^{2}}} \right. \\ \left. {\left. { + 2\mu {{{\left( {u_{{xy}}^{2} - {{u}_{{xx}}}{{u}_{{yy}}}} \right)}}^{{^{{}}}}}} \right) + {{k}_{1}}{{u}^{2}} + {{k}_{2}}{{{\left( {u_{x}^{2} + u_{y}^{2}} \right)}}^{{{{{^{{}}}}^{{^{{}}}}}}}}} \right\} \\ \end{gathered} $$

is the energy density, ${\mathbf{S}} = \left\{ {{{S}_{x}},{{S}_{y}}} \right\}$ is the energy flux density vector, frequently called the Umov–Poynting vector [26, 27], with the components

$$\begin{gathered} S_{x}^{{}} = {{u}_{t}}\left( {\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{{12}}\left( {{{u}_{{xxx}}} + {{u}_{{xyy}}}} \right) - \frac{{\rho h_{*}^{3}}}{{12}}{{u}_{{xtt}}} - {{k}_{2}}{{u}_{x}}} \right) \\ - \,\,\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{{12}}{{u}_{{xt}}}{{u}_{{xx}}} \\ - \,\,\frac{{h_{*}^{3}\Lambda }}{{12}}{{u}_{{xt}}}{{u}_{{yy}}} - \frac{{h_{*}^{3}\mu }}{6}{{u}_{{yt}}}{{u}_{{xy}}},\,\,\,\,\left( {x \leftrightarrow y} \right) \\ \end{gathered} $$

${\mathbf{p}} = \left\{ {{{p}_{x}},{{p}_{y}}} \right\}$ is the wave momentum density vector with the coordinates

$$p_{x}^{{}} = - \rho {{h}_{*}}{{u}_{t}}{{u}_{x}} - \frac{{\rho h_{*}^{3}}}{{12}}{{u}_{{xt}}}{{u}_{{xx}}} - \frac{{\rho h_{*}^{3}}}{{12}}{{u}_{{yt}}}{{u}_{{xy}}},$$

${\mathbf{T}}$ is the wave momentum flux density tensor with the components

$$\begin{gathered} T_{{xx}}^{{}} = \left\{ {\rho {{h}_{*}}u_{t}^{2} + } \right.\frac{{\rho h_{*}^{3}}}{{12}}\left( {u_{{xt}}^{2} + u_{{yt}}^{2} + 2{{u}_{x}}{{u}_{{xtt}}}} \right) \\ + \,\,\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{{12}}\left( {u_{{xx}}^{2} - u_{{yy}}^{2} - 2{{u}_{x}}\left( {{{u}_{{xxx}}} + {{u}_{{xyy}}}} \right)} \right) \\ {{\left. { - \,\,{{k}_{1}}{{u}^{2}} + {{k}_{2}}\left( {u_{x}^{2} - u_{y}^{2}} \right)} \right\}} \mathord{\left/ {\vphantom {{\left. { - \,\,{{k}_{1}}{{u}^{2}} + {{k}_{2}}\left( {u_{x}^{2} - u_{y}^{2}} \right)} \right\}} 2}} \right. \kern-0em} 2}, \\ \end{gathered} $$

$$\begin{gathered} T_{{xy}}^{{}} = {{u}_{y}}\left( {\frac{{\rho h_{*}^{3}}}{{12}}{{u}_{{xtt}}} - \frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{{12}}\left( {{{u}_{{xxx}}} + {{u}_{{xyy}}}} \right) + {{k}_{2}}{{u}_{x}}} \right) \\ + \,\,\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{{12}}{{u}_{{xy}}}\left( {{{u}_{{xx}}} + {{u}_{{yy}}}} \right),\,\,\,\left( {x \leftrightarrow y} \right). \\ \end{gathered} $$

Calculating the average values of these quantities for a wave period 2πω^-1, we obtain

$$\begin{gathered} \left\langle h \right\rangle = \frac{1}{2}\rho {{h}_{*}}\left( {1 + \frac{{h_{*}^{2}}}{{12}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} \right){{\omega }^{2}}AA^{*}, \\ \left\langle {S_{x}^{{}}} \right\rangle \\ = \frac{1}{2}\left( {\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{6}\left( {k_{x}^{2} + k_{y}^{2}} \right) - \frac{{\rho h_{*}^{3}}}{{12}}{{\omega }^{2}} + {{k}_{2}}} \right)\omega {{k}_{x}}AA*, \\ \left\langle {{{p}_{x}}} \right\rangle = \frac{1}{2}\rho {{h}_{*}}\left( {1 + \frac{{h_{*}^{2}}}{{12}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} \right)\omega {{k}_{x}}AA^{*}, \\ \left\langle {T_{{xy}}^{{}}} \right\rangle \\ = \frac{1}{2}\left( {\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{6}\left( {k_{x}^{2} + k_{y}^{2}} \right) - \frac{{\rho h_{*}^{3}}}{{12}}{{\omega }^{2}} + {{k}_{2}}} \right){{k}_{x}}{{k}_{y}}AA^{*}, \\ \left( {x \leftrightarrow y} \right). \\ \end{gathered} $$

(4)

Note: * denotes complex conjugation.

On the kinematic characteristics of the wave—frequency ω, wave vector components κ ={k_x, k_y}, and parameters of the distributed system—the dynamics equations impose a relationship, which is usually called the dispersion equation [28].

ANALYSIS OF DISPERSION DEPENDENCES

Based on plate dynamics equation (2), we find that the frequency ω and wave vector components k_x, k_y are related by the dispersion relation

$$\begin{gathered} {{\omega }} = \pm {{\left( {\frac{{h_{*}^{3}\left( {\Lambda + 2\mu } \right)}}{{12}}{{{\left( {k_{x}^{2} + k_{y}^{2}} \right)}}^{2}} + {{k}_{2}}\left( {k_{x}^{2} + k_{y}^{2}} \right) + {{k}_{1}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \\ \times \,\,{{\left( {\rho {{h}_{*}} + \frac{{\rho h_{*}^{3}}}{{12}}\left( {k_{x}^{2} + k_{y}^{2}} \right)} \right)}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2}}}}. \\ \end{gathered} $$

(5)

Determining the lowest frequency of waves excited in the plate, we have

$${{{{\omega }}}_{{{\text{RP}}}}} = \frac{2}{{\sqrt \rho {{h}_{*}}}}{{\left( {6\left( {\Lambda + 2\mu } \right)\left( {\sqrt {1 - \frac{{{{k}_{2}}}}{{{{h}_{*}}\left( {\Lambda + 2\mu } \right)}} + \frac{{{{k}_{1}}{{h}_{*}}}}{{12\left( {\Lambda + 2\mu } \right)}}} + \frac{{{{k}_{2}}}}{{2{{h}_{*}}\left( {\Lambda + 2\mu } \right)}} - 1} \right)} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}$$

for $\frac{{{{k}_{2}}}}{{{{k}_{1}}h_{*}^{2}}} < \frac{1}{{12}}$. The low-frequency field with frequency ω < ω_RP does not propagate, but decays exponentially as it penetrates the system. By comparison with the frequency

$${\begin{gathered} {{{{\omega }}}_{{{\text{RV}}}}} \\ = 2{{\left( {6\left( {\Lambda \, + \,2\mu } \right)\left( {\sqrt {1\, + \,{{{{k}_{1}}h_{*}^{{}}} \mathord{\left/ {\vphantom {{{{k}_{1}}h_{*}^{{}}} {12\left( {\Lambda \, + \,2\mu } \right)}}} \right. \kern-0em} {12\left( {\Lambda \, + \,2\mu } \right)}}} - 1} \right)} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \\ \end{gathered} \mathord{\left/ {\vphantom {\begin{gathered} {{{{\omega }}}_{{{\text{RV}}}}} \\ = 2{{\left( {6\left( {\Lambda \, + \,2\mu } \right)\left( {\sqrt {1\, + \,{{{{k}_{1}}h_{*}^{{}}} \mathord{\left/ {\vphantom {{{{k}_{1}}h_{*}^{{}}} {12\left( {\Lambda \, + \,2\mu } \right)}}} \right. \kern-0em} {12\left( {\Lambda \, + \,2\mu } \right)}}} - 1} \right)} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \\ \end{gathered} {\sqrt \rho {{h}_{*}}}}} \right. \kern-0em} {\sqrt \rho {{h}_{*}}}}$$

for the Winkler foundation, it can be stated that ω_R.V. < ω_RP. Consequently, the area of impermeability for a plate, taking into account the inertia of rotation of its elements during bending, resting on a Winkler foundation is narrower than for the elastic foundation of the Pasternak model.

Figure 1 shows the dispersion curves (dependence of the frequency ω on the wavenumber κ is the modulus of the wave vector κ of the wave) for different shear coefficients of the foundation of the plate. Clearly, there is a portion of the dispersion curve where the slope of the tangent line is negative. Consequently, for this model, there is a frequency range $\left( {{{{{\omega }}}_{{{\text{RP}}}}},\sqrt {{{{{k}_{1}}} \mathord{\left/ {\vphantom {{{{k}_{1}}} {\rho {{h}_{*}}}}} \right. \kern-0em} {\rho {{h}_{*}}}}} } \right)$, where the backward wave effect is valid [29–31]. For such waves (as mentioned above), the phase and group velocities have opposite directions. This possibility was first pointed out by Lamb [32], who constructed a number of models of media with this property.

Fig. 1.

Dependence of dimensionless frequency on dimensionless wavenumber for different values of reduced shear coefficient (${{\tilde {k}}_{2}} = {{{{k}_{2}}} \mathord{\left/ {\vphantom {{{{k}_{2}}} {{{k}_{1}}h_{*}^{2}}}} \right. \kern-0em} {{{k}_{1}}h_{*}^{2}}}$) of elastic foundation: (1) ${{\tilde {k}}_{2}} = 0$; (2) ${{\tilde {k}}_{2}} = 0.01$; (3) ${{\tilde {k}}_{2}} = 0.04$; (4) ${{\tilde {k}}_{2}} = 0.2$.

The width of the frequency domain of the existence of backward waves depends on the ratio of the coefficients of the subgrade stiffness for shear and compression. For k₂ = 0, which corresponds to the Winkler foundation model, the frequency domain for the existence of backward waves is wider than for the Pasternak model $\left( {{{{{\omega }}}_{{{\text{RP}}}}},\sqrt {{{{{k}_{1}}} \mathord{\left/ {\vphantom {{{{k}_{1}}} {\rho {{h}_{*}}}}} \right. \kern-0em} {\rho {{h}_{*}}}}} } \right) \subset \left( {{{{{\omega }}}_{{{\text{RV}}}}},\sqrt {{{{{k}_{1}}} \mathord{\left/ {\vphantom {{{{k}_{1}}} {\rho {{h}_{*}}}}} \right. \kern-0em} {\rho {{h}_{*}}}}} } \right)$. An increase in the shear coefficient of the elastic foundation of the plate leads to a decrease in the frequency domain where backward waves take place and, for ${{{{k}_{2}}} \mathord{\left/ {\vphantom {{{{k}_{2}}} {{{k}_{1}}h_{*}^{2}}}} \right. \kern-0em} {{{k}_{1}}h_{*}^{2}}} > {1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}}$, makes their existence impossible. It should also be noted that in the plate of the Kirchhoff model transverse waves are direct (the directions of the phase and group velocities coincide) [33].

Defining the phase velocity of the waves as v_ph = ωκ/κ², in dimensionless variables

$$\begin{gathered} a = \sqrt {\frac{{\Lambda + 2\mu }}{{{{k}_{1}}{{h}_{*}}}}} ,\,\,\,\,{{{\tilde {k}}}_{2}} = \frac{{{{k}_{2}}}}{{{{k}_{1}}h_{*}^{2}}},\,\,\,\,{{\tilde {\omega }}} = {{\omega }}\sqrt {{{\rho {{h}_{*}}} \mathord{\left/ {\vphantom {{\rho {{h}_{*}}} {{{k}_{1}}}}} \right. \kern-0em} {{{k}_{1}}}}} , \\ {{{\tilde {k}}}_{x}} = {{k}_{x}}{{h}_{*}},\,\,\,\,{{{\tilde {k}}}_{y}} = {{k}_{y}}{{h}_{*}},\,\,\,\,{{{\tilde {v}}}_{{{\text{ph}}}}} = \left| {{{{\mathbf{v}}}_{{{\text{ph}}}}}} \right|\sqrt {\frac{\rho }{{{{k}_{1}}{{h}_{*}}}}} , \\ \end{gathered} $$

we obtain an expression for the phase velocity modulus:

$${{\tilde {v}}_{{{\text{ph}}}}} = \tilde {\omega }{{\left( {\frac{6}{{{{a}^{2}}}}\left( {\frac{{{{{\tilde {\omega }}}^{2}}}}{{12}} - {{{\tilde {k}}}_{2}} \pm \sqrt {{{{\left( {\frac{{{{{\tilde {\omega }}}^{2}}}}{{12}} - {{{\tilde {k}}}_{2}}} \right)}}^{2}} - \frac{{{{a}^{2}}}}{3}\left( {1 - {{{\tilde {\omega }}}^{2}}} \right)} } \right)} \right)}^{{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-0em} 2}}}}.$$

The “–” sign corresponds to the frequency interval ${{\tilde {\omega }}} \in \left[ {{{{{\omega }}}_{{{\text{RP}}}}}\sqrt {{{\rho {{h}_{*}}} \mathord{\left/ {\vphantom {{\rho {{h}_{*}}} {{{k}_{1}}}}} \right. \kern-0em} {{{k}_{1}}}}} ;\;1} \right]$, and the “+” sign, to the interval $\left[ {{{\omega }_{{{\text{RP}}}}}\sqrt {{{\rho {{h}_{*}}} \mathord{\left/ {\vphantom {{\rho {{h}_{*}}} {{{k}_{1}}}}} \right. \kern-0em} {{{k}_{1}}}}} ;\; + \infty } \right)$. Figure 2 shows the dependences of the phase velocity modulus on frequency for various shear coefficients. The calculated graphs are plotted for dimensionless variables at a = 0.2.

Fig. 2.

Frequency dependences of phase velocities for different values of shear coefficient of elastic foundation: (1) ${{\tilde {k}}_{2}} = 0$; (2) ${{\tilde {k}}_{2}} = 0.01$; (3) ${{\tilde {k}}_{2}} = 0.04$; (4) ${{\tilde {k}}_{2}} = 0.2$.

The phase velocity in dimensionless variables has a horizontal asymptote ${{\tilde {v}}_{{{\text{ph}}}}} = a$ (for ${{\tilde {\omega }}} \to \infty$) and the vertical asymptote $\tilde {\omega } = 1$ (Fig. 2). The frequency domain $\left( {{{{{\omega }}}_{{{\text{RP}}}}}\sqrt {{{\rho {{h}_{*}}} \mathord{\left/ {\vphantom {{\rho {{h}_{*}}} {{{k}_{1}}}}} \right. \kern-0em} {{{k}_{1}}}}} ;\;1} \right)$ is the domain of existence of backward waves, and $\left( {{{{{\omega }}}_{{{\text{RP}}}}}\sqrt {{{\rho {{h}_{*}}} \mathord{\left/ {\vphantom {{\rho {{h}_{*}}} {{{k}_{1}}}}} \right. \kern-0em} {{{k}_{1}}}}} ;\; + \infty } \right)$, for direct waves.

Determining the minimum phase velocity of waves, we obtain

$$\tilde {v}_{{{\text{ph}}}}^{{\min }} = {{\left( {{{\left( {\sqrt {1 + 12{{a}^{2}} - 12{{{\tilde {k}}}_{2}}} - 1 + 6{{{\tilde {k}}}_{2}}} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt {1 + 12{{a}^{2}} - 12{{{\tilde {k}}}_{2}}} - 1 + 6{{{\tilde {k}}}_{2}}} \right)} 6}} \right. \kern-0em} 6}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}.$$

This velocity is one of the most important among the critical velocities that are considered in the literature. For example, as shown in [34, 35], when an object moves uniformly at a velocity exceeding the minimum phase velocity of the emitted wave, its transverse oscillations, due to the anomalous Doppler effect [36, 37], can be unstable. Therefore, when studying oscillations of an elastic system bearing a high-speed moving load, determining the minimum phase velocity becomes a top priority [35, 38–40].

Note that velocity V, wave energy transfer is defined as S/h [28]. Clearly, for quasi-harmonic waves, the average V for the period is dω/dκ = grad_κω, which coincides with the known approximate expression for the group velocity [28].

Figure 3 shows the dependence of the component V_x of the wave energy transfer velocity (group velocity) on frequency at k_y = 0 and different values of the shear coefficient of the foundation of the plate in dimensionless variables (${{\tilde {V}}_{x}} = {{V}_{x}}\sqrt {\rho {{{\left( {{{k}_{1}}{{h}_{*}}} \right)}}^{{ - 1}}}}$).

Fig. 3.

Frequency dependences of group velocity for different values of shear coefficient of elastic foundation: (1) ${{\tilde {k}}_{2}} = 0$; (2) ${{\tilde {k}}_{2}} = 0.01$; (3) ${{\tilde {k}}_{2}} = 0.04$; (4) ${{\tilde {k}}_{2}} = 0.2$.

For frequency ω_RP, where the backward waves turn into direct waves, the group velocity vanishes and, consequently, the energy at this frequency is not transferred.

At high frequencies, the velocities V and v_ph are close to the velocity of a longitudinal wave in an boundless medium. It should be expected that, by introducing a correction factor to models (1)–(2), a more accurate description of the dispersion properties of a real plate can be achieved; in particular, in the limit for ω → ∞ the phase velocity will be close to the shear wave velocity.

Comparing the phase and group velocities (for k_y = 0, Fig. 4), two wavenumber domains should also be distinguished. For the first, the phase velocity is greater than the group velocity; for the second, the phase velocity is less than the group velocity. Consequently, in the first domain, we have normal wave dispersion, while in the second, anomalous. There is a wavenumber (and, accordingly, a frequency) at which vibrational energy transfer occurs at the maximum rate.

Fig. 4.

Dependence of phase (1) and group (2) velocities on wavenumber for reduced shear coefficient of elastic foundation ${{\tilde {k}}_{2}} = 0.01$.

ANALYSIS OF ENERGY RELATIONS

It is of interest to estimate the local energy flux carried by waves in a plate, the bending vibrations of which are described by Eq. (2). It should be noted that, apparently, the energy flux densities of backward waves in an elastic strip as a function of the transverse coordinate of the layer were studied in [41], where local energy fluxes in the opposite direction were found for normal waves. In [42], when studying the longitudinal vibrations of an infinite plate of constant width, it was found that at the frequency of occurrence of a backward wave, fundamental restructuring of the field of the energy flux vector occurs. In the case when the dispersion curve does not have a segment corresponding to the backward wave, the energy flux density retains a constant sign, although, in the composite model, as demonstrated in [43], local energy fluxes have alternating signs not only for backward, but also for direct waves, which are not related to backward waves by a single dispersion curve.

Figure 5 shows the dependences of the energy flux density ${{\tilde {S}}_{x}} = \left\langle {{{S}_{x}}} \right\rangle \sqrt \rho {{\left( {\sqrt {k_{1}^{3}{{h}_{*}}} AA{\text{*}}} \right)}^{{ - 1}}}$ on the wavenumber for different reduced shear coefficients of the elastic foundation (for the case k_y = 0).

Fig. 5.

Energy flux density as function of wavenumber for (1) ${{\tilde {k}}_{2}} = 0$; (2) ${{\tilde {k}}_{2}} = 0.01$; (3) ${{\tilde {k}}_{2}} = 0.04$; (4) ${{\tilde {k}}_{2}} = 0.2$.

It can be seen that the behavior of the energy flux densities has a sign-alternating character if ${{{{k}_{2}}} \mathord{\left/ {\vphantom {{{{k}_{2}}} {{{k}_{1}}h_{*}^{2}}}} \right. \kern-0em} {{{k}_{1}}h_{*}^{2}}} < {1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}}$; otherwise, when ${{{{k}_{2}}} \mathord{\left/ {\vphantom {{{{k}_{2}}} {{{k}_{1}}h_{*}^{2}}}} \right. \kern-0em} {{{k}_{1}}h_{*}^{2}}} > {1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}}$, no change in sign of the energy flux density is observed.

From (4) and (5) it follows that the relations below are true for the average values of the local energy characteristics:

—the scalar product of the phase velocity and the wave impulse density vector is equal to the wave energy density:

$$\left\langle h \right\rangle = {{{\mathbf{v}}}_{{{\text{ph}}}}}\left\langle {\mathbf{p}} \right\rangle ;$$

—the product of the column vector of the group velocity V =(V_x,V_y)^T multipied by the row vector of the wave momentum density is equal to the wave momentum flux density tensor:

$$\left( {\begin{array}{*{20}{c}} {{{V}_{x}}} \\ {{{V}_{y}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\left\langle {{{p}_{x}}} \right\rangle }&{\left\langle {{{p}_{y}}} \right\rangle } \end{array}} \right) = \left\| {\begin{array}{*{20}{c}} {\left\langle {{{T}_{{xx}}}} \right\rangle }&{\left\langle {{{T}_{{xy}}}} \right\rangle } \\ {\left\langle {{{T}_{{yx}}}} \right\rangle }&{\left\langle {{{T}_{{yy}}}} \right\rangle } \end{array}} \right\|.$$

The ratio of the moduli of the average values of the energy flux density to the wave momentum density is equal to the product of the moduli of the phase and group wave velocities:

$$\frac{{\left| {\left\langle {\mathbf{S}} \right\rangle } \right|}}{{\left| {\left\langle {\mathbf{p}} \right\rangle } \right|}} = \left| {{{{\mathbf{v}}}_{{{\text{ph}}}}}} \right|\left| {\mathbf{V}} \right|.$$

The ratio of the energy density to frequency is equal to the ratio of the modulus of the wave momentum density vector to the wavenumber.

$$\frac{{\left\langle h \right\rangle }}{{{\omega }}} = \frac{{\left| {\left\langle {\mathbf{p}} \right\rangle } \right|}}{\kappa }.$$

CONCLUSIONS

For a plate resting on an elastic foundation, taking into account the rotational inertia of its elements during bending, leads to the presence of a frequency region of the existence of backward waves, i.e., waves whose phase and group velocities are oppositely directed.

In the presence of two subgrade stiffness coefficients (for shear k₂ and for compression k₁) characteristic of the Pasternak model, the width of this domain depends on their ratio. With a decrease in the shear coefficient of the elastic foundation k₂ → 0, the frequency domain of the existence of backward waves increases. The threshold value at which their existence is impossible is determined by the equality ${{{{k}_{2}}} \mathord{\left/ {\vphantom {{{{k}_{2}}} {{{k}_{1}}h_{*}^{2}}}} \right. \kern-0em} {{{k}_{1}}h_{*}^{2}}} = {1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}}$.

At the frequency of occurrence of a backward wave, the wave energy transfer velocity vanishes and a fundamental restructuring of the field of the energy flux density vector occurs.

The wave energy density is related to the density of the momentum carried by a quasi-harmonic wave through the phase velocity.

The components of the wave momentum flux density tensor and the density of the momentum carried by the wave are related by the group velocity.