On generalized Holmgren’s principle to the Lamé operator with applications to inverse elastic problems

Abstract

Consider the Lamé operator ${\mathcal {L}}({\mathbf {u}}) :=\mu \Delta {\mathbf {u}}+(\lambda +\mu ) \nabla (\nabla \cdot {\mathbf {u}} )$ that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lamé eigenfunction ${\mathbf {u}}$, namely $-{\mathcal {L}}({\mathbf {u}})=\kappa {\mathbf {u}}$ with $\kappa \in {\mathbb {R}}_+$ and ${\mathbf {u}}\in L^2(\Omega )^2$, $\Omega \subset {\mathbb {R}}^2$. We introduce the so-called homogeneous line segments of ${\mathbf {u}}$ in $\Omega$, on which ${\mathbf {u}}$, its traction or their combination via an impedance parameter is vanishing. We give a comprehensive study on characterizing the presence of one or two such line segments and its implication to the uniqueness of ${\mathbf {u}}$. The results can be regarded as generalizing the classical Holmgren’s uniqueness principle for the Lamé operator in two aspects. We establish the results by analyzing the development of analytic microlocal singularities of ${\mathbf {u}}$ with the presence of the aforesaid line segments. Finally, we apply the results to the inverse elastic problems in establishing two novel unique identifiability results. It is shown that a generalized impedance obstacle as well as its boundary impedance can be determined by using at most four far-field patterns. Unique determination by a minimal number of far-field patterns is a longstanding problem in inverse elastic scattering theory.

Appendix

Proof of Lemma 2.4

We first prove (2.18). Recall that ${\nu }\big |_{\Gamma _h^+}$ is defined in (2.8):

$$\begin{aligned} \varvec{\tau }=(-\cos \varphi _0,-\sin \varphi _0). \end{aligned}$$

(6.1)

Substituting (6.1) into (1.5) yields

$$\begin{aligned} \begin{aligned} T_{\nu } {\mathbf {u}}\Big |_{\Gamma ^+_h}&=2 \mu \left[ \begin{array}{cc} \partial _1 u_1 &{} \partial _2 u_1\\ \partial _1 u_2 &{} \partial _2 u_2\\ \end{array}\right] \left[ \begin{array}{c} -\sin \varphi _0\\ \cos \varphi _0\\ \end{array}\right] +\lambda \left[ \begin{array}{c} -\sin \varphi _0\\ \cos \varphi _0\\ \end{array}\right] \left( \partial _1 u_1+\partial _2 u_2\right) \\&\quad + \mu \left[ \begin{array}{c} -\cos \varphi _0\\ -\sin \varphi _0\\ \end{array}\right] \left( \partial _2 u_1-\partial _1 u_2\right) :=\left[ \begin{array}{c}{T_1 ({\mathbf {u}}) }\big |_{\Gamma _h^+ }\\ {T_2 ({\mathbf {u}}) }\big |_{\Gamma _h^+ }\end{array}\right] , \end{aligned} \end{aligned}$$

(6.2)

where

$$\begin{aligned} \begin{aligned} T_1( {\mathbf {u}})\big |_{\Gamma _h^+ }=&2\mu (-\sin \varphi _0 \partial _1 u_1+\cos \varphi _0 \partial _2 u_1 )\\&-\lambda \sin \varphi _0 (\partial _1 u_1+\partial _2 u_2 )-\mu \cos \varphi _0 (\partial _2 u_1-\partial _1 u_2 ) , \\ T_2 ({\mathbf {u}})\big |_{\Gamma _h^+ }=&2\mu (-\sin \varphi _0 \partial _1 u_2+\cos \varphi _0 \partial _2 u_2 )\\&-\lambda \cos \varphi _0 (\partial _1 u_1+\partial _2 u_2 )-\mu \sin \varphi _0 (\partial _2 u_1-\partial _1 u_2 ). \end{aligned} \end{aligned}$$

Using (2.17), it is readily shown that

$$\begin{aligned} \begin{aligned} \frac{\partial u_1}{\partial r}&= \sum _{m=0} ^\infty \left\{ \frac{k_p^2}{4}a_m \left\{ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}\left( k_p r\right) - \left[ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi }+\mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi }\right] J_m \left( k_p r\right) \right\} \right. \\&\quad + \frac{{\mathrm {i}}k_s^2}{4}b_m \left\{ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}\left( k_s r\right) - \left[ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi }-\mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi }\right] J_m \left( k_s r\right) \right\} \\&\quad + \frac{k_p^2}{4}a_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2} \left( k_p r\right) -\frac{{\mathrm {i}}k_s^2}{4}b_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2} \left( k_s r\right) \bigg \},\\ \frac{\partial u_1}{\partial \varphi }&= \sum _{m=0} ^\infty \left\{ \frac{{\mathrm {i}}\left( m-1\right) }{2} k_p \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-1} \left( k_p r\right) a_m - \frac{{\mathrm {i}}\left( m+1\right) }{2} k_p \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+1} \left( k_p r\right) a_m \right. \\&\quad - \frac{\left( m-1\right) }{2} k_s \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-1} \left( k_s r\right) b_m - \frac{\left( m+1\right) }{2} k_s \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+1} \left( k_s r\right) b_m\bigg \}, \end{aligned} \end{aligned}$$

(6.3)

and

$$\begin{aligned} \frac{\partial u_2}{\partial r}&= \sum _{m=0} ^\infty \left\{ \frac{{\mathrm {i}}k_p^2}{4}a_m \left\{ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}\left( k_p r\right) - \left[ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi }-\mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi }\right] J_m \left( k_p r\right) \right\} \right. \nonumber \\&\quad + \frac{k_s^2}{4}b_m \left\{ -\mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}\left( k_s r\right) + \left[ \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi }+\mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi }\right] J_m \left( k_s r\right) \right\} \nonumber \\&\quad - \frac{{\mathrm {i}}k_p^2}{4}a_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2} \left( k_p r\right) -\frac{k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2} \left( k_s r\right) \bigg \},\end{aligned}$$

(6.4)

$$\begin{aligned} \frac{\partial u_2}{\partial \varphi }&= \sum _{m=0} ^\infty \left\{ -\frac{\left( m-1\right) }{2} k_p \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-1} \left( k_p r\right) a_m - \frac{\left( m+1\right) }{2} k_p \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+1} \left( k_p r\right) a_m \right. \nonumber \\&\quad - \frac{{\mathrm {i}}\left( m-1\right) }{2} k_s \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-1} \left( k_s r\right) b_m \nonumber \\&\quad + \frac{{\mathrm {i}}\left( m+1\right) }{2} k_s \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+1} \left( k_s r\right) b_m\bigg \}. \end{aligned}$$

(6.5)

Using the fact that

$$\begin{aligned} \begin{aligned} \frac{\partial u_i}{\partial x_1}&=\cos \varphi \cdot \frac{\partial u_i}{\partial r}- \frac{\sin \varphi }{r} \cdot \frac{\partial u_i}{\partial \varphi },\quad \frac{\partial u_i}{\partial x_2}=\sin \varphi \cdot \frac{\partial u_i}{\partial r}+ \frac{\cos \varphi }{r} \cdot \frac{\partial u_i}{\partial \varphi }, \end{aligned} i=1,2, \end{aligned}$$

(6.6)

as well as (6.3) and (6.4), by tedious but straightforward calculations, one can obtain that

$$\begin{aligned}&\partial _1 u_1 \cdot \left( -\sin \varphi _0\right) +\partial _2 u_1 \cdot \left( \cos \varphi _0\right) \nonumber \\&\quad = \sum _{m=0} ^\infty \Bigg \{ \sin (\varphi -\varphi _0) \Big [\frac{k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_p r) \nonumber \\&\qquad - \frac{k_p^2}{4} a_m J_m(k_p r)\left( \mathrm {e}^{{\mathrm {i}}(m-1)\varphi }+\mathrm {e}^{{\mathrm {i}}(m+1)\varphi }\right) \nonumber \\&\qquad + \frac{k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2}(k_p r) + \frac{{\mathrm {i}}k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_s r) \nonumber \\&\qquad - \frac{{\mathrm {i}}k_s^2}{4} b_m J_m(k_s r)\left( \mathrm {e}^{{\mathrm {i}}(m-1)\varphi }-\mathrm {e}^{{\mathrm {i}}(m+1)\varphi }\right) - \frac{{\mathrm {i}}k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2}(k_s r) \Big ] \nonumber \\&\qquad + \frac{ \cos (\varphi -\varphi ) }{r} \Big [\frac{{\mathrm {i}}(m-1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_p r) \nonumber \\&\qquad - \frac{{\mathrm {i}}(m+1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}(m+1)\varphi } J_{m+1}(k_p r) - \frac{(m-1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_s r)\nonumber \\&\qquad - \frac{(m+1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}(m+1)\varphi } J_{m+1}(k_s r) \Big ] \Bigg \},\nonumber \\&\partial _1 u_2 \cdot \left( -\sin \varphi _0\right) +\partial _2 u_2 \cdot \left( \cos \varphi _0\right) \nonumber \\&\quad = \sum _{m=0} ^\infty \Bigg \{ \sin (\varphi -\varphi _0 ) \Big [ \frac{{\mathrm {i}}k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_p r) \nonumber \\&\qquad - \frac{{\mathrm {i}}k_p^2}{4} a_m J_m(k_p r)\left( \mathrm {e}^{{\mathrm {i}}(m-1)\varphi }-\mathrm {e}^{{\mathrm {i}}(m+1)\varphi }\right) \nonumber \\&\qquad - \frac{{\mathrm {i}}k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2}(k_p r) - \frac{ k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_s r) \nonumber \\&\qquad + \frac{ k_s^2}{4} b_m J_m(k_s r)\left( \mathrm {e}^{{\mathrm {i}}(m-1)\varphi }+\mathrm {e}^{{\mathrm {i}}(m+1)\varphi }\right) \nonumber \\&\qquad -\frac{ k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m+1\right) \varphi } J_{m+2}(k_s r) \Big ] + \frac{\cos (\varphi -\varphi _0)}{r} \Big [ \frac{-(m-1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_p r) \nonumber \\&\qquad - \frac{(m+1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}(m+1)\varphi } J_{m+1}(k_p r) - \frac{{\mathrm {i}}(m-1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_s r)\nonumber \\&\qquad + \frac{{\mathrm {i}}(m+1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}(m+1)\varphi } J_{m+1}(k_s r) \Big ] \Bigg \}. \end{aligned}$$

(6.7)

Similarly, from (6.3) and (6.4), we have

$$\begin{aligned} \partial _1 u_1 +\partial _2 u_2= & {} \sum _{m=0} ^\infty \Bigg \{ \frac{k_p^2}{4} \mathrm {e}^{{\mathrm {i}}m \varphi } a_m \left( J_{m-2} \left( k_p r\right) -2 J_m \left( k_p r\right) + J_{m+2} \left( k_p r\right) \right) \nonumber \\&+\frac{{\mathrm {i}}k_s^2}{4} \mathrm {e}^{{\mathrm {i}}m \varphi } b_m \left( J_{m-2} \left( k_s r\right) -J_{m+2} \left( k_s r\right) \right) \nonumber \\&+ \frac{1}{r} \bigg [ -\frac{k_p}{2}\mathrm {e}^{{\mathrm {i}}m \varphi } a_m \big ( \left( m-1\right) J_{m-1} \left( k_p r\right) \nonumber \\&+\left( m+1\right) J_{m+1} \left( k_p r\right) \big ) \nonumber \\&- \frac{{\mathrm {i}}k_s}{2}\mathrm {e}^{{\mathrm {i}}m \varphi } b_m \left( \left( m-1\right) J_{m-1} \left( k_s r\right) -\left( m+1\right) J_{m+1} \left( k_s r\right) \right) \bigg ]\Bigg \}, \end{aligned}$$

(6.8)

and

$$\begin{aligned} \begin{aligned} \partial _2 u_1-\partial _1 u_2 =&\sum _{m=0} ^\infty \Bigg \{\frac{{\mathrm {i}}k_p^2}{4} \mathrm {e}^{{\mathrm {i}}m \varphi } a_m \left( -J_{m-2} \left( k_p r\right) +J_{m+2} \left( k_p r\right) \right) \\&+\frac{k_s^2}{4} \mathrm {e}^{{\mathrm {i}}m \varphi } b_m \left( J_{m-2} \left( k_s r\right) -2 J_m \left( k_s r\right) +J_{m+2} \left( k_s r\right) \right) \\&+ \frac{1}{r} \bigg [ \frac{{\mathrm {i}}k_p}{2}\mathrm {e}^{{\mathrm {i}}m \varphi } a_m \bigg ( \left( m-1\right) J_{m-1} \left( k_p r\right) \\&-\left( m+1\right) J_{m+1} \left( k_p r\right) \bigg ) - \frac{k_s}{2}\mathrm {e}^{{\mathrm {i}}m \varphi } b_m \left( \left( m-1\right) J_{m-1} \left( k_s r\right) \right. \\&\left. +\left( m+1\right) J_{m+1} \left( k_s r\right) \right) \bigg ] \Bigg \}. \end{aligned} \end{aligned}$$

(6.9)

Plugging (6.7)–(6.9) into (6.2), after tedious but straightforward calculations, we have

$$\begin{aligned} \begin{aligned}&T_1({\mathbf {u}})\big |_{\Gamma _h^+ }\\&\quad = \sum _{m=0} ^\infty \left\{ \frac{k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_p r) \left[ 2 \mu \sin (\varphi -\varphi _0)\right. \right. \\&\qquad \left. \left. -\lambda \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0 +{\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0\right] \right. \\&\qquad + \frac{k_p^2}{2} a_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_m(k_p r)\left[ \lambda \sin \varphi _0+2 \mu \cos \varphi \sin (\varphi _0-\varphi )\right] \\&\qquad + \frac{k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_{m+2}(k_p r) \left[ 2 \mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin (\varphi -\varphi _0) - \lambda \sin \varphi _0 - {\mathrm {i}}\mu \cos \varphi _0 \right] \\&\qquad + \frac{{\mathrm {i}}k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_s r) \left[ 2 \mu \sin (\varphi -\varphi _0)-\lambda \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0 +{\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0\right] \\&\qquad + \frac{ k_s^2}{2} b_m \mathrm {e}^{{\mathrm {i}}m\varphi } J_m(k_s r)\left[ \mu \cos \varphi _0+2 \mu \sin \varphi \sin (\varphi _0-\varphi )\right] \\&\qquad + \frac{{\mathrm {i}}k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_{m+2}(k_s r) \left[ 2 \mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin (\varphi _0-\varphi ) + \lambda \sin \varphi _0 + {\mathrm {i}}\mu \cos \varphi _0\right] \\&\qquad + \frac{1}{r} \left\{ \frac{(m-1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_p r)\left[ 2 {\mathrm {i}}\mu \cos (\varphi _0-\varphi ) + \lambda \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0 - {\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0\right] \right. \\&\qquad + \frac{(m+1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_{m+1}(k_p r)\left[ -2 {\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos (\varphi _0-\varphi ) + \lambda \sin \varphi _0 + {\mathrm {i}}\mu \cos \varphi _0\right] \\&\qquad + \frac{(m-1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_s r)\left[ -2 \mu \cos (\varphi _0-\varphi ) + {\mathrm {i}}\lambda \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0 + \mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0\right] \\&\qquad + \frac{(m+1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}m\varphi } J_{m+1}(k_s r) \left[ -2 \mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos (\varphi _0-\varphi ) - {\mathrm {i}}\lambda \sin \varphi _0 + \mu \cos \varphi _0\right] \bigg \}\bigg \}. \end{aligned} \end{aligned}$$

(6.10)

Substituting (2.13) into (6.10), we can obtain that

$$\begin{aligned} \begin{aligned} T_1( {\mathbf {u}})\big |_{\Gamma _h^+ } =&\sum _{m=0} ^\infty \bigg \{ \frac{{\mathrm {i}}k_p^2}{2} a_m \mathrm {e}^{{\mathrm {i}}(m-1) \varphi } \mathrm {e}^{-{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m-2} \left( k_p r\right) \\&+ k_p^2 a_m \mathrm {e}^{{\mathrm {i}}m \varphi } \left( \lambda +\mu \right) \sin \varphi _0 J_m \left( k_p r\right) \\&-\frac{{\mathrm {i}}k_p^2}{2} a_m \mathrm {e}^{{\mathrm {i}}(m+1) \varphi } \mathrm {e}^{{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m+2} -\frac{k_s^2}{2} b_m \mathrm {e}^{{\mathrm {i}}(m-1) \varphi } \mathrm {e}^{-{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m-2} \left( k_s r\right) \\&-\frac{k_s^2}{2} b_m \mathrm {e}^{{\mathrm {i}}(m+1) \varphi } \mathrm {e}^{{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m+2} \left( k_s r\right) \bigg \}. \end{aligned} \end{aligned}$$

(6.11)

Similarly, substituting (6.7) to (6.9) into (6.2), after tedious but straightforward calculations, we have

$$\begin{aligned} \begin{aligned}&T_2({\mathbf {u}} )\big |_{\Gamma _h^+ }\\&\quad = \sum _{m=0} ^\infty \left\{ \frac{k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_p r) \left[ 2 {\mathrm {i}}\mu \sin (\varphi -\varphi _0)\right. \right. \\&\left. \left. \qquad +\lambda \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0 +{\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0\right] \right. \\&\qquad + \frac{k_p^2}{2} a_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_m(k_p r)\left[ -\lambda \cos \varphi _0+2 \mu \sin \varphi \sin (\varphi _0-\varphi )\right] \\&\qquad + \frac{k_p^2}{4} a_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_{m+2}(k_p r) \left[ 2 {\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin (\varphi _0-\varphi ) + \lambda \cos \varphi _0 - {\mathrm {i}}\mu \sin \varphi _0 \right] \\&\qquad + \frac{ k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}\left( m-1\right) \varphi } J_{m-2}(k_s r) \left[ 2 \mu \sin (\varphi _0-\varphi )+ {\mathrm {i}}\lambda \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0 - \mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0\right] \\&\qquad + \frac{ k_s^2}{2} b_m \mathrm {e}^{{\mathrm {i}}m\varphi } J_m(k_s r)\left[ \mu \sin \varphi _0+2 \mu \cos \varphi \sin (\varphi -\varphi _0)\right] \\&\qquad + \frac{ k_s^2}{4} b_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_{m+2}(k_s r) \left[ 2 \mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin (\varphi _0-\varphi ) -{\mathrm {i}}\lambda \cos \varphi _0 - \mu \sin \varphi _0\right] \\&\qquad + \frac{1}{r} \left\{ -\frac{(m-1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_p r)\right. \\&\left. \qquad \left[ 2 \mu \cos (\varphi _0-\varphi ) + \lambda \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0 + {\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0\right] \right. \\&\qquad + \frac{(m+1)k_p}{2} a_m \mathrm {e}^{{\mathrm {i}}m \varphi } J_{m+1}(k_p r)\left[ -2 \mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos (\varphi _0-\varphi ) - \lambda \cos \varphi _0 + {\mathrm {i}}\mu \sin \varphi _0\right] \\&\qquad + \frac{(m-1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}(m-1)\varphi } J_{m-1}(k_s r)\left[ -2 {\mathrm {i}}\mu \cos (\varphi _0-\varphi ) - {\mathrm {i}}\lambda \mathrm {e}^{{\mathrm {i}}\varphi } \cos \varphi _0 + \mu \mathrm {e}^{{\mathrm {i}}\varphi } \sin \varphi _0\right] \\&\qquad + \frac{(m+1)k_s}{2} b_m \mathrm {e}^{{\mathrm {i}}m\varphi } J_{m+1}(k_s r) \left[ 2 {\mathrm {i}}\mu \mathrm {e}^{{\mathrm {i}}\varphi } \cos (\varphi _0-\varphi ) + {\mathrm {i}}\lambda \cos \varphi _0 + \mu \sin \varphi _0\right] \bigg \}\bigg \}, \end{aligned} \end{aligned}$$

(6.12)

Plugging (2.13) into (6.12), we can obtain that

$$\begin{aligned} \begin{aligned} T_2( {\mathbf {u}})\big |_{\Gamma _h^+ } =&\sum _{m=0} ^\infty \left\{ - \frac{ k_p^2}{2} a_m \mathrm {e}^{{\mathrm {i}}(m-1) \varphi } \mathrm {e}^{-{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m-2} \left( k_p r\right) \right. \\&\left. - k_p^2 a_m \mathrm {e}^{{\mathrm {i}}m \varphi } \left( \lambda +\mu \right) \cos \varphi _0 J_m \left( k_p r\right) \right. \\&-\frac{ k_p^2}{2} a_m \mathrm {e}^{{\mathrm {i}}(m+1) \varphi } \mathrm {e}^{{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m+2}(k_p r)\\&-\frac{{\mathrm {i}}k_s^2}{2} b_m \mathrm {e}^{{\mathrm {i}}(m-1) \varphi } \mathrm {e}^{-{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m-2} \left( k_s r\right) \\&+\frac{{\mathrm {i}}k_s^2}{2} b_m \mathrm {e}^{{\mathrm {i}}(m+1) \varphi } \mathrm {e}^{{\mathrm {i}}(\varphi -\varphi _0)} \mu J_{m+2} \left( k_s r\right) \bigg \}. \end{aligned} \end{aligned}$$

(6.13)

Using the fact

$$\begin{aligned} \left[ \begin{array}{c}{\sin \varphi }\\ {-\cos \varphi }\end{array}\right] =\frac{{\mathrm {i}}}{2}\left( \mathrm {e}^{-{\mathrm {i}}\varphi }{\mathbf {e}}_1-\mathrm {e}^{{\mathrm {i}}\varphi }{\mathbf {e}}_2\right) , \end{aligned}$$

substituting (6.11) and (6.13) into (6.2), we can prove (2.18). The proof of (2.19) is similar to (2.18), which is omitted here. $\square$