Theoretical analysis of forced Lamb waves using the method of multiple scales and Green’s function method

Abstract

We investigated forced Lamb waves using the method of multiple scales and Green’s function method. With the former method, we derived a solvability condition containing terms describing forced effects. In Lamb waves problems, the solvability condition is a first-order partial differential equation in contrast to the governing equations describing Lamb waves, which are second-order partial differential equations and boundary conditions. With the latter method, we obtained the Green’s function for the solvability condition. This way of deriving the Green’s function is easier than solving the governing equations for Lamb waves. Finally, we obtained the amplitudes of the forced Lamb waves using the Green’s function. The proposed method may be used to obtain the amplitudes from arbitrary forced distributions in both the time and space domains. Furthermore, the method may be used to obtain the frequency and wave number responses that take into consideration the detuning effects and may be used to consider dispersion effects. As a result, we found out that some kind of resonance occurs when the relation between input position and time delay corresponds to the group velocity of the input propagation mode. Solutions obtained by our method were verified using the frequency-domain-based integral transform method.

Appendices

Appendix A: Coefficients in the amplitude equations

The expressions for coefficients $C_{j1}$ and $C_{j2}$ are

$$\begin{aligned} C_{j1}&=-2\rho _{d}\omega _{j}\int _{-h}^{h}(\phi _{jx}^{2}+\phi _{jz}^{2})dz, \end{aligned}$$

(A1)

$$\begin{aligned} C_{j2}&=\mu \phi _{jx}(h)\phi _{jz}(h)-\mu \phi _{jx}(-h)\phi _{jz}(-h)-\lambda \phi _{jx}(h)\phi _{jz}(h)+\lambda \phi _{jx}(-h)\phi _{jz}(-h)\nonumber \\&\quad -2(\lambda +2\mu )k_j\int _{-h}^{h}\phi _{jx}^{2}dz-(\lambda +\mu )\int _{-h}^{h}\phi _{jx}\frac{d\phi _{jz}}{dz}dz\nonumber \\&\quad +(\lambda +\mu )\int _{-h}^{h}\frac{d\phi _{jx}}{dz}\phi _{jz}dz-2\mu k_j\int _{-h}^{h}\phi _{jz}^{2}dz. \end{aligned}$$

(A2)

Appendix B: Green’s function and integral representation

Green’s function for Eq. (11) $G(x_1,t_1;y, \tau )$ is defined as

$$\begin{aligned} \textrm{i} C_{j1}\frac{\partial G(x_1,t_1;y, \tau )}{\partial t_{1}} + \textrm{i} C_{j2}\frac{\partial G(x_1,t_1;y, \tau )}{\partial x_{1}} + (C_{j1}\hat{\sigma } -C_{j2}\hat{\rho })G(x_1,t_1;y, \tau ) = - \delta (x_{1}-y)\delta (t_{1}-\tau ), \end{aligned}$$

(B3)

where $\delta (\cdot )$ indicates the Dirac delta function. From a physical viewpoint, this Green’s function is interpreted as a propagating wave packet. Therefore, we impose a causality condition in which the wave packet propagates in the positive $t_1$ direction.

The Fourier transform of Eq. (B3) with respect to $x_1$ and $t_1$ is

$$\begin{aligned} \left\{ C_{j1}\omega _1 -C_{j2} k_1 +(C_{j1}\hat{\sigma }-C_{j2}\hat{\rho }) \right\} \hat{G}(k_1,\omega _1 ;y, \tau ) = - \exp {\left\{ -\textrm{i}( k_1 y - \omega _1 \tau ) \right\} }, \end{aligned}$$

(B4)

where $\omega _1$ and $k_1$ are the circular frequency and wave number, respectively. From Eq. (B4), $\hat{G}( k_1, \omega _1;y, \tau )$ becomes

$$\begin{aligned} \hat{G}( k_1 , \omega _1 ;y, \tau ) = - \frac{\exp {\left\{ -\textrm{i}( k_1 y - \omega _1 \tau ) \right\} }}{C_{j1} \omega _1 -C_{j2} k_1 + (C_{j1}\hat{\sigma }-C_{j2}\hat{\rho })}. \end{aligned}$$

(B5)

Using the inverse Fourier transform, $G(x_1,t_1;y, \tau )$ is

$$\begin{aligned} G(x_1,t_1;y, \tau )&=\frac{1}{(2\pi )^{2}}\int \int _{\mathbb {R}^2} \frac{- \exp { \left\{ \textrm{i} k_1 (x_1 - y) - \textrm{i} \omega _1 (t_1 - \tau ) \right\} }}{C_{j1} \omega _1 -C_{j2} k_1 + (C_{j1}\hat{\sigma }-C_{j2}\hat{\rho })} d \omega _1 d k_1 \nonumber \\&=\frac{\textrm{i}}{C_{j2}} \exp { \left\{ \textrm{i}\frac{C_{j1}\hat{\sigma }-C_{j2}\hat{\rho }}{C_{j2}}(x_{1} - y) \right\} } H(x_{1}-y) \delta \left( t_{1}-\tau -\frac{C_{j1}}{C_{j2}}(x_{1}-y) \right) , \end{aligned}$$

(B6)

where we have used the causality condition to evaluate the contour integral; here $H(\cdot )$ denotes the Heaviside function.

Multiplying Eq. (11) for $(y, \tau )$ by Green’s function $G(x_1,t_1;y, \tau )$ and then integrating over $\mathbb {R}^2$ with respect to y and $\tau$, we obtain

$$\begin{aligned}&\int \int _{\mathbb {R}^2} \left\{ \textrm{i} C_{j1}\frac{\partial Y_{j1}(y, \tau )}{\partial \tau } + \textrm{i} C_{j2}\frac{\partial Y_{j1}(y, \tau )}{\partial y} +(C_{j1}\hat{\sigma } - C_{j2}\hat{\rho })Y_{j1}(y, \tau ) \right\} \nonumber \\&\quad \times G(x_1,t_1;y, \tau )dyd\tau = - \int \int _{\mathbb {R}^2} \tilde{F_{j}}(y, \tau )G(x_1,t_1;y, \tau )dyd\tau . \end{aligned}$$

(B7)

An integration by parts yields

$$\begin{aligned}&Y_{1j}(x_{1},t_{1}) = Y_0 (x_{1},t_{1}) +\int \int _{\mathbb {R}^2}\tilde{F_{j}}(y, \tau )G(x_1,t_1;y, \tau )dyd\tau , \end{aligned}$$

(B8)

where $Y_0$ is the homogeneous solution of Eq. (11) and is given by

$$\begin{aligned} Y_{0}(x_{1},t_{1})=&-\textrm{i} C_{j1} \lim _{\tau \rightarrow - \infty } \left[ \int _{\mathbb {R}} Y_{j1}(y,\tau )G(x_1,t_1;y, \tau )dy \right] \nonumber \\&-\textrm{i} C_{j2} \lim _{y \rightarrow - \infty } \left[ \int _{\mathbb {R}} Y_{j1}(y, \tau )G(x_1,t_1;y, \tau )d\tau \right] . \end{aligned}$$

(B9)

The second term on the right-hand side of Eq. (B8) is the particular solution of Eq. (11). In this study, we set $Y_0=0$ because our focus is on the problem for forced Lamb waves.