Difference between revisions of "Energy Balance for Two Elastic Plates"

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</math></center>
 
</math></center>
 
where <math>n</math> denotes the outward plane normal to the boundary and <math>l</math> denotes the plane parallel to the boundary.
 
where <math>n</math> denotes the outward plane normal to the boundary and <math>l</math> denotes the plane parallel to the boundary.
As <math>\phi</math> and <math>\phi^*</math> satisfy \eqref{eqn:Laplace2}, the left hand side of \eqref{eqn:Green1} vanishes so that \eqref{eqn:Green1} reduces to  
+
As <math>\phi</math> and <math>\phi^*</math> satisfy the Laplace's equation, the left hand side of the Green theorem equation vanishes so that it reduces to  
 
<center><math>
 
<center><math>
 
  \Im\int_\mathcal{S}\phi\frac{\partial\phi^*}{\partial n} dl  =  0,
 
  \Im\int_\mathcal{S}\phi\frac{\partial\phi^*}{\partial n} dl  =  0,

Revision as of 00:36, 9 July 2008

Based on the method used in Evans and Davies 1968, a check can be made to ensure the solutions of the floating plate problem are in energy balance. This is simply a condition that the incident energy is equal to the sum of the radiated energy. When the first and final plates have different properties, the energy balance equation is derived by applying Green's theorem to [math]\displaystyle{ \phi }[/math] and its conjugate Evans and Davies 1968. We set up the problem as given in Figure 256 \begin{figure}[H] \begin{center} \includegraphics[width=.8\textwidth]{Figures/EnergyBalanceTricks} \end{center} \caption[A diagram depicting the area [math]\displaystyle{ \mathcal{U} }[/math] which is bounded by the rectangle [math]\displaystyle{ \mathcal{S} }[/math].] {A diagram depicting the area [math]\displaystyle{ \mathcal{U} }[/math] which is bounded by the rectangle [math]\displaystyle{ \mathcal{S} }[/math]. The rectangle [math]\displaystyle{ \mathcal{S} }[/math] is bounded by [math]\displaystyle{ -h\leq z \leq0 }[/math] and [math]\displaystyle{ -\infty\leq x \leq \infty }[/math].} (256) \end{figure}

Applying Green's theorem to [math]\displaystyle{ \phi }[/math] and its conjugate [math]\displaystyle{ \phi^* }[/math] gives

[math]\displaystyle{ (257) { \int\int_\mathcal{U}(\phi\nabla^2\phi^* - \phi^*\nabla^2\phi)dxdz = \int_\mathcal{S}(\phi\frac{\partial\phi^*}{\partial n} - \phi^*\frac{\partial\phi}{\partial n})dl }, }[/math]

where [math]\displaystyle{ n }[/math] denotes the outward plane normal to the boundary and [math]\displaystyle{ l }[/math] denotes the plane parallel to the boundary. As [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \phi^* }[/math] satisfy the Laplace's equation, the left hand side of the Green theorem equation vanishes so that it reduces to

[math]\displaystyle{ \Im\int_\mathcal{S}\phi\frac{\partial\phi^*}{\partial n} dl = 0, }[/math]

Expanding gives

[math]\displaystyle{ (258) \xi_1 +\xi_2 + \xi_3 = 0, }[/math]

where

[math]\displaystyle{ \xi_1 = { \Im\int_{-\infty}^{\infty}(\phi\frac{\partial\phi^*}{\partial z})\big|_{z=0}dx }, }[/math]
[math]\displaystyle{ \xi_2 = { \Im\int_{-h}^0(\phi\frac{\partial\phi^*}{\partial x})\big|_{x=\infty}dz }, }[/math]

and

[math]\displaystyle{ \xi_3 = { - \Im\int_{-h}^0(\phi\frac{\partial\phi^*}{\partial x})\big|_{x=-\infty}dz } = 0 , }[/math]

where [math]\displaystyle{ \Im }[/math] denotes the imaginary part.

Expanding [math]\displaystyle{ \mathbf{\xi_1} }[/math]

Near [math]\displaystyle{ x=-\infty }[/math], we approximate [math]\displaystyle{ \phi }[/math] by

[math]\displaystyle{ (259) \begin{matrix}{l} { \phi \approx e^{-\kappa_{1}(0)(x-r_1)}\frac{\cos{(k_1(0)(z+h))}}{\cos{(k_1(0)h)}} + R_1(0)e^{\kappa_{1}(0)(x-r_1)}\frac{\cos{(k_{1}(0)(z+h))}}{\cos{(k_{1}(0)h)}}}. \end{matrix} }[/math]

To simplify the derivation, we re-express \eqref{eqn:EnergyPhiLeft1} as

[math]\displaystyle{ \phi \approx \left(e^{-i\kappa^I_1(x-r_1)}+ R_1(0)e^{i\kappa^I_1(x-r_1)}\right)\frac{\cosh{(k^I_1(z+h))}}{\cosh{(k^I_1h)}}, }[/math]

where [math]\displaystyle{ k^I_1 = -\Im k_{1}(0) }[/math] and [math]\displaystyle{ \kappa^I_1=-\Im \kappa_1(0) }[/math], so that

[math]\displaystyle{ \frac{\partial\phi}{\partial x} \approx \left(-i\kappa^{I}_1e^{-i\kappa^{I}_1(x-r_1)} + i\kappa^{I}_1R_1(0)e^{i\kappa^{I}_1(x-r_1)}\right)\frac{\cosh{(k^I_1(z+h))}}{\cosh{(k^I_1h)}}. }[/math]

Therefore,

[math]\displaystyle{ \begin{matrix}{rl} \xi_1=& {\Im\int_{-h}^0 \biggl[\left(e^{-i\kappa^{I}_1(x-r_1)}+ R_1(0)e^{i\kappa^{I}_1(x-r_1)}\right)\biggr. }\\ & { \left(i\kappa^{I}_1e^{i\kappa^{I}_1(x-r_1)} - i\kappa^{I}_1R_1(0)^*e^{-i\kappa^{I}_1(x-r_1)}\right) }\\ & { \left.\left(\frac{\cosh^2{(k^I_1(z+h))}}{\cosh^2{(k^I_1h)}}\right)\right]dz, }\\ \\ =& { \Im\left[\frac{i\kappa^{I}_1\left(1 - |R_1(0)|^2\right)}{2\cosh^2{(k^I_1h)}} \int_{-h}^0 \left(\cosh{(2k^I_1(z+h))+1}\right)dz\right] ,}\\ \\ =& { \Im\left[\frac{i\kappa^{I}_1\left(1 - |R_1(0)|^2\right)}{2\cosh^2{(k^I_1h)}} \left[\frac{1}{2k^I_1}\sinh{(2k^I_1(z+h))} + z\right]_{-h}^0\right], }\\ \\ =& { \frac{\kappa^{I}_1\left(1 - |R_1(0)|^2\right)}{2\cosh^2{(k^I_1h)}} \left(\frac{1}{2k^I_1}\sinh{(2k^I_1h))} + h\right) ,}\\ \\ =& { \frac{\kappa^{I}_1\left(1 - |R_1(0)|^2\right)}{2k^I_1}\left(\tanh{(k^I_1h)}+\frac{hk^I_1}{\cosh^2{(k^I_1h)}}\right) , } \end{matrix} }[/math]

where [math]\displaystyle{ R_1(0)^* }[/math] is the conjugate of [math]\displaystyle{ R_1(0) }[/math]. \subsection*{Expanding [math]\displaystyle{ \mathbf{\xi_2} }[/math]} Near [math]\displaystyle{ x=\infty }[/math], we approximate [math]\displaystyle{ \phi }[/math] by

[math]\displaystyle{ (260) \begin{matrix}{l} { \phi \approx T_\Lambda(0)e^{-\kappa_{\Lambda}(0)(x-l_\Lambda)}\frac{\cos{(k_\Lambda(0)(z+h))}}{\cos{(k_\Lambda(0)h)}}, } \end{matrix} }[/math]

and re-express as

[math]\displaystyle{ \phi \approx T_\Lambda(0)e^{-i\kappa^I_\Lambda(x-l_\Lambda)})\frac{\cosh{(k^I_\Lambda(z+h))}}{\cosh{(k^I_\Lambda h)}}, }[/math]

where [math]\displaystyle{ k^I_\Lambda = -\Im k_{\Lambda}(0) }[/math] and [math]\displaystyle{ \kappa^I_\Lambda=-\Im \kappa_\Lambda(0) }[/math], so that

[math]\displaystyle{ \frac{\partial\phi}{\partial x} \approx -i\kappa^I_{\Lambda}T_\Lambda(0)e^{-i\kappa^I_{\Lambda}(x-l_\Lambda)})\frac{\cosh{(k^I_\Lambda(z+h))}}{\cosh{(k^I_\Lambda h)}}. }[/math]

Therefore, \begin{multline} \xi_2 = \Im\int_{-h}^0 \biggl[\biggr.(T_\Lambda(0)e^{-i\kappa^I_{\Lambda}(x-l_\Lambda)})(i\kappa^I_{\Lambda}T_\Lambda(0)^*e^{-i\kappa^I_{\Lambda}(x-l_\Lambda)}) \\ \frac{\cosh^2{(k^I_\Lambda(z+h))}}{\cosh^2{(k^I_\Lambda h)}} \biggl.\biggr]dz, \end{multline}

[math]\displaystyle{ \begin{matrix}{rl} =& { \frac{\kappa^I_{\Lambda}|T_\Lambda(0)|^2}{2\cosh^2{(k^I_\Lambda h)}} \left(\frac{1}{2k^I_\Lambda}\sinh{(2k^I_\Lambda h)} + h\right)},\\ \\ =& { \frac{\kappa^I_{\Lambda}|T_\Lambda(0)|^2}{2k^I_\Lambda}\left(\tanh{(k^I_\Lambda h)}+\frac{k^I_\Lambda h}{\cosh^2{(k^I_\Lambda h)}}\right)}. \end{matrix} }[/math]

\subsection*{Expanding [math]\displaystyle{ \mathbf{\xi_3} }[/math]} The ice-covered boundary condition, \eqref{Eq:IceCoveredCondition}, gives

[math]\displaystyle{ (261) \xi_3 = { \Im\int_{-\infty}^{\infty}\left(\frac{\beta}{\alpha} \left(\frac{\partial^2}{\partial x^2} - k_y^2\right)^2 - \gamma + \frac{1}{\alpha}\right) \frac{\partial\phi}{\partial z}\centerdot\frac{\partial\phi^*}{\partial z}\bigg|_{z=0}dx.} }[/math]

Since [math]\displaystyle{ {\frac{\partial\phi}{\partial z}\centerdot{\partial\phi^*}{\partial z}} }[/math] is real, [math]\displaystyle{ }[/math]\xi_3 = \Im\int_{-\infty}^{\infty}\left(\frac{\beta}{\alpha}\frac{\partial^2}{\partial x^2} \left(\frac{\partial^2}{\partial x^2} - 2k_y^2\right)^2 \right) \frac{\partial\phi}{\partial z}\centerdot\frac{\partial\phi^*}{\partial z}\bigg|_{z=0}dx.[math]\displaystyle{ }[/math] Integration by parts gives \begin{multline}(262) \xi_3 = \Im\biggl[\biggr.\left[\frac{\beta}{\alpha}\frac{\partial}{\partial x} \left(\frac{\partial^2}{\partial x^2}-2k_y^2\right) \frac{\partial\phi}{\partial z}\centerdot\frac{\partial\phi^*}{\partial z}\right]_{-\infty}^{\infty} \\ -\int_{-\infty}^{\infty}\frac{\beta}{\alpha}\frac{\partial}{\partial x} \left(\frac{\partial^2}{\partial x^2} - 2k_y^2\right) \frac{\partial\phi}{\partial z}\centerdot\frac {\partial}{\partial x}\frac{\partial\phi^*}{\partial z}dx \biggl.\biggr]. \end{multline} As [math]\displaystyle{ {2k_y^2\frac{\partial}{\partial x}\frac{\partial\phi}{\partial z}\centerdot \frac{\partial}{\partial x}\frac{\partial\phi^*}{\partial z}} }[/math] is real and by integration by parts, \eqref{Eqn:EnergyIceCover2} becomes, \begin{multline}(263) \xi_3 =\Im\left[ \left[\frac{\beta}{\alpha}\frac{\partial}{\partial x} \left(\frac{\partial^2}{\partial x^2}-2k_y^2\right) \frac{\partial\phi}{\partial z}\centerdot\frac{\partial\phi^*}{\partial z}\right]_{-\infty}^{\infty}\right.\\ - \left[\frac{\beta}{\alpha}\frac{\partial^2}{\partial x^2}\frac{\partial\phi}{\partial z}\centerdot \frac{\partial}{\partial x}\frac{\partial\phi^*}{\partial z}\right]_{-\infty}^{\infty} \\ + \left.\int_{-\infty}^\infty\frac{\beta}{\alpha}\frac{\partial^2}{\partial x^2}\frac{\partial\phi}{\partial z}\centerdot \frac{\partial^2}{\partial x^2}\frac{\partial\phi^*}{\partial z}dx\right]. \end{multline} As [math]\displaystyle{ {\frac{\partial^2}{\partial x^2}\frac{\partial\phi}{\partial z}\centerdot \frac{\partial^2}{\partial x^2}\frac{\partial\phi^*}{\partial z}} }[/math] is real, \eqref{Eqn:Energy4} becomes

[math]\displaystyle{ (264) \xi_3 = \Im\left[ \left[\frac{\beta}{\alpha}\frac{\partial}{\partial x} \left(\frac{\partial^2}{\partial x^2}-2k_y^2\right) \frac{\partial\phi}{\partial z}\centerdot\frac{\partial\phi^*}{\partial z}\right]_{-\infty}^{\infty}- \left[\frac{\beta}{\alpha}\frac{\partial^2}{\partial x^2}\frac{\partial\phi}{\partial z}\centerdot \frac{\partial}{\partial x}\frac{\partial\phi^*}{\partial z}\right]_{-\infty}^{\infty}\right]. }[/math]

Now breaking [math]\displaystyle{ \xi_3 }[/math] down, \begin{multline*} \frac{\partial}{\partial x}\left(\frac{\partial^2}{\partial x^2}-2k_y^2\right) \frac{\partial\phi(x_2,0)}{\partial z}\centerdot \frac{\partial\phi(x_2,0)^*}{\partial z} \\ \\ =\left((-i\kappa^I_{\Lambda})^3k^I_\Lambda T_\Lambda(0)e^{-i\kappa^I_{\Lambda}(x-l_\Lambda)}\tanh{(k^I_\Lambda h)}\right.\\ \left.- 2k_y^2(-i\kappa_{\Lambda}(0))k^I_\Lambda T_\Lambda(0)e^{-i\kappa^I_{\Lambda}(x-l_\Lambda)} \tanh{(k^I_\Lambda h)}\right)\\ \left(k^I_\Lambda T_\Lambda(0)^*e^{i\kappa^I_{\Lambda}(x-l_\Lambda)}\tanh{(k^I_\Lambda h)}\right), \end{multline*} \begin{flalign*} = i\kappa^I_{\Lambda} (k^I_\Lambda)^2\left((\kappa^I_{\Lambda})^2 + 2k_y^2 \right) \tanh^2{(k^I_\Lambda h)}|T_\Lambda(0)|^2. \end{flalign*} \begin{multline*} \frac{\partial}{\partial x}\left(\frac{\partial^2}{\partial x^2}-2k_y^2\right) \frac{\partial\phi(x_1,0)}{\partial z}\centerdot \frac{\partial\phi(x_1,0)^*}{\partial z}\\ \\ = \left[ i(\kappa^I_{1})^3k^I_1\left(e^{-i\kappa^I_{1}(x-r_1)}-R_1(0)e^{i\kappa^I_{1}(x-r_1)}\right) \tanh{(k^I_1 h)}\right.\\ \left.- 2ik_y^2\kappa^I_{1}k^I_1\left(-e^{-i\kappa^I_{1}(x-r_1)} + R_1(0)e^{i\kappa^I_{1}(x-r_1)}\right) \tanh{(k^I_1h)}\right]\\ \left[k^I_1\left(e^{i\kappa^I_{1}(x-r_1)} + R_1(0)^*e^{-i\kappa^I_{1}(x-r_1))}\right) \tanh{(k^I_1h)}\right],\\\\ = \left[i\kappa^I_{1}k_1(0)((\kappa^I_{1})^2 + 2k_y^2)\left(e^{-i\kappa^I_{1}(x-r_1)} - R_1(0)e^{i\kappa^I_{1}(x-r_1)}\right) \tanh{(k^I_1h)}\right]\\ \left[k^I_1\left(e^{i\kappa^I_{1}(x-r_1)} + R_1(0)^*e^{-i\kappa^I_{1}(x-r_1))}\right) \tanh{(k^I_1h)}\right], \end{multline*}

[math]\displaystyle{ = i\kappa^I_{1}(k^I_1)^2\left((\kappa^I_{1})^2 + 2k_y^2\right)\tanh^2{(k^I_1h)}\left(1 - |R_1(0)|^2\right). }[/math]

\begin{multline*} \frac{\partial^2}{\partial x^2}\frac{\partial\phi(x_2,0)}{\partial z}\centerdot \frac{\partial}{\partial x}\frac{\partial\phi^*}{\partial z}\\\\ = \left[-(\kappa^I_{\Lambda})^2k^I_\Lambda T_\Lambda(0)e^{-i\kappa^I_{\Lambda}(x-l_\Lambda)} \tanh{(k^I_\Lambda h)}\right]\\, \left[i\kappa^I_{\Lambda} k^I_\Lambda T_\Lambda(0)^*e^{i\kappa^I_{\Lambda}(x-l_\Lambda)} \tanh{(k^I_\Lambda h)}\right], \end{multline*} \begin{multline*}

= -i(\kappa^I_{\Lambda})^3(k^I_\Lambda)^2\tanh^2{(k^I_\Lambda h)}|T_\Lambda(0)|^2,

\end{multline*} and finally, \begin{multline*} \frac{\partial^2}{\partial x^1}\frac{\partial\phi(x_1,0)}{\partial z}\centerdot \frac{\partial}{\partial x}\frac{\partial\phi^*}{\partial z}\\\\ = \left[-(\kappa^I_{1})^2k^I_1\left(e^{-i\kappa^I_{1}(x-r_1)} + R_1(0)e^{i\kappa^I_{1}(x-r_1)}\right) \tanh{(k^I_1 h)}\right]\\ \left[i\kappa^I_{1} k^I_1\left(e^{i\kappa^I_{1}(x-r_1)} - R_1(0)e^{-i\kappa^I_{1}(x-r_1)}\right) \tanh{(k^I_1 h)}\right], \end{multline*}

[math]\displaystyle{ = -i(\kappa^I_{1})^3(k^I_1)^2\tanh^2{(k^I_1h)}\left(1 - |R_1(0)|^2\right). }[/math]

We can now express \eqref{Eqn:EnergyTerm1} as \begin{multline*} \Im\left[ \frac{\beta_\Lambda}{\alpha}\left[i\kappa^I_{\Lambda} (k^I_\Lambda)^2\left((\kappa^I_{\Lambda})^2 + 2k_y^2 \right) \tanh^2{(k^I_\Lambda h)}|T_\Lambda(0)|^2\right] \right.\\ -\left.\frac{\beta_1}{\alpha}\left[i\kappa^I_{1}(k^I_1)^2\left((\kappa^I_{1})^2 + 2k_y^2\right)\tanh^2{(k^I_1h)} \left(1 - |R_1(0)|^2\right)\right] \right.\\ -\left.\frac{\beta_\Lambda}{\alpha}\left[-i(\kappa^I_{\Lambda})^3(k^I_\Lambda)^2\tanh^2{(k^I_\Lambda h)}|T_\Lambda(0)|^2\right]\right.\\ +\left.\frac{\beta_1}{\alpha}\left[-i(\kappa^I_{1})^3(k^I_1)^2\tanh^2{(k^I_1h)} \left(1 - |R_1(0)|^2\right)\right]\right] \end{multline*} \begin{multline*} = \frac{\beta_\Lambda}{\alpha}\left[2\kappa^I_{\Lambda} (k^I_\Lambda)^2((\kappa^I_{\Lambda})^2 + k_y^2)\tanh^2{(k^I_\Lambda h)}|T_\Lambda(0)|^2\right]\\ -\frac{\beta_1}{\alpha}\left[2\kappa^I_{1}(k^I_1)^2((\kappa^I_{1})^2 + k_y^2)\tanh^2{(k^I_1h)}(1-|R_1(0)|^2)\right] . \end{multline*}

Solving the Energy Balance Equation

Pulling it all together, \eqref{Eqn:EnergyPhi} becomes

[math]\displaystyle{ \frac{\beta_\Lambda}{\alpha}\left[2\kappa^I_{\Lambda}(k^I_\Lambda)^2((\kappa^I_{\Lambda})^2 +k_y^2)\tanh^2{(k^I_\Lambda h)}|T_\Lambda(0)|^2\right]\\ -\frac{\beta_1}{\alpha}\left[2\kappa^I_{1}(k^I_1)^2((\kappa^I_{1})^2 +k_y^2)\tanh^2{(k^I_1h)}(1-|R_1(0)|^2)\right]\\ + \frac{\kappa^I_{\Lambda}|T_\Lambda(0)|^2}{2k^I_\Lambda}\left(\tanh{(k^I_\Lambda h)}+\frac{k^I_\Lambda h}{\cosh^2{(k^I_\Lambda h)}}\right)\\ - \frac{\kappa^I_{1}\left(1 - |R_1(0)|^2\right)}{2k^I_1}\left(\tanh{(k^I_1h)}+\frac{hk^I_1}{\cosh^2{(k^I_1h)}}\right)=0. \end{multline*} Re-arranging gives \begin{multline*} \kappa^I_{\Lambda}\tanh{(k^I_\Lambda h)} \left(\frac{\beta_\Lambda}{\alpha}2(k^I_\Lambda)^2((\kappa^I_{\Lambda})^2 +k_y^2)\tanh{(k^I_\Lambda h)}\right. \\ \left.+ \frac{1}{2k^I_\Lambda}+\frac{h}{2\sinh{(k^I_\Lambda h)}\cosh{(k^I_\Lambda h)}}\right)|T_\Lambda(0)|^2 \end{multline*} \begin{multline*} \quad - \kappa^I_{1}\tanh{(k^I_1 h)} \left(\frac{\beta_1}{\alpha}2(k^I_1)^2((\kappa^I_{1})^2 +k_y^2)\tanh{(k^I_1 h)}\right. \\ \left.+ \frac{1}{2it}+\frac{h}{2\sinh{(k^I_1 h)}\cosh{(k^I_1 h)}}\right)(1-|R_1(0)|^2)=0 }[/math]

which can be expressed as

[math]\displaystyle{ D |T_{\Lambda}(0)|^2 + |R_{1}(0)|^2 = 1, }[/math]

where [math]\displaystyle{ D }[/math] is given by

[math]\displaystyle{ D = \left(\frac{\kappa^I_{\Lambda} k^I_1\cosh^2{(k^I_1h)}}{\kappa^I_{1}k^I_\Lambda\cosh^2{(k^I_\Lambda h)}}\right) \left(\frac{\frac{\beta_\Lambda}{\alpha}4(k^I_\Lambda)^3((\kappa^I_{\Lambda})^2 +k_y^2)\sinh^2{(k^I_\Lambda h)} + \frac{1}{2}{\sinh{(2k^I_\Lambda h)}}+k^I_\Lambda h} {\frac{\beta_1}{\alpha}4(k^I_1)^3((\kappa^I_{1})^2 +k_y^2)\sinh^2{(k^I_1h)} + \frac{1}{2}{\sinh{(2k^I_1h)}}+k^I_1h}\right) }[/math]