On Ventcel-type transmission conditions for a Helmholtz problem with a non-uniform thin layer

Abstract

We study the asymptotic behaviour of the electric field in the transverse magnetic (TM) mode, propagating in a material composed of a two-dimensional object surrounded by a thin layer of non-constant thickness and embedded in an ambient medium. Using an asymptotic expansion of the solution $u^{\varepsilon }$ to the Helmholtz problem, we derive Ventcel-type transmission conditions on the limit interface $\Gamma$ modelling the effect of the thin layer with accuracy up to $O(\varepsilon ^{2})$.

Appendix A Calculation of the first three terms of the asymptotic expansion

1.1 A.1 Term of order 0

Equation (28) and conditions (25) and (27) give

$$\begin{aligned} \partial _{s}U_{m, \beta }^{0}=0, \ \ \ \ \beta \in \left\{ 1,2 \right\} . \end{aligned}$$

Using (22), (23) and (24), we get $\forall (t,s)\in \Omega _{m,\beta }$

$$\begin{aligned} U_{m,\beta }^{0}(t,s)=u_{-}^{0}|_{\Gamma }=u_{+}^{0}|_{\Gamma }. \end{aligned}$$

(A1)

Equation (29) implies

$$\begin{aligned} \partial _{s}^{2}U_{m,\beta }^{1}=A_{\beta ,1}U_{m,\beta }^{0}=0. \end{aligned}$$

Using (25), (26) and (27), we obtain $\forall (t,s)\in \Omega _{m,\beta }$

$$\begin{aligned} \partial _{s}U_{m,2}^{1}(t,s) = f(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma },\ \partial _{s}U_{m,1}^{1}(t,s)=\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }, \ \ \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }=\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }. \nonumber \\ \end{aligned}$$

(A2)

1.2 A.2 Term of order 1

Relation (A2) with conditions (22) and (24) yield $\forall (t,s)\in \Omega _{m,\beta }$

$$\begin{aligned}{} & {} U_{m,1}^{1}(t,s)= u_{-}^{1}|_{\Gamma } + \left( \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) s,\\{} & {} U_{m,2}^{1}(t,s)= u_{+}^{1}|_{\Gamma }+ \left( d+f(t)-f(t) \alpha _+ + sf(t) \alpha _+ \right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }. \end{aligned}$$

So (23) gives

$$\begin{aligned} u_{+}^{1}|_{\Gamma }-u_{-}^{1}|_{\Gamma }=\dfrac{f(t) \alpha _{+}\alpha _{-}-(d+f(t))\alpha _{-}+d\alpha _{+}}{2\alpha _{+}\alpha _{-}}(\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+ \alpha _{-}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }). \end{aligned}$$

(A3)

From (30) and (31), we have

$$\begin{aligned} \partial _{s}^{2}U_{m,1}^{2}= & {} A_{1,1}U_{m,1}^{1}+A_{1,2}U_{m,1}^{0}-k_{-}^{2}U_{m,1}^{0} =-c(t)\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }-\partial _{t}^{2}u_{-}^{0}|_{\Gamma }-k_{-}^{2}u_{-}^{0}|_{\Gamma }.\\ \partial _{s}^{2}U_{m,2}^{2}= & {} A_{1}U_{m,2}^{1}+A_{2}U_{m,2}^{0}-k_m^{2}U_{m,2}^{0}=-c(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\partial _{t} ^{2}u_{+}^{0}|_{\Gamma }-k_{m}^{2}u_{+}^{0}|_{\Gamma }. \end{aligned}$$

Using (25) and (27) , we obtain $\forall (t,s) \in \Omega _{m,\beta }$

$$\begin{aligned} \partial _{s}U_{m,1}^{2}(t,s)&=\left[ -c(t)\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma }-\partial _{t}^{2}u_{-}^{0}|_{\Gamma }-k_{-}^{2}u_{-}^{0} |_{\Gamma }\right] s+\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }. \end{aligned}$$

(A4)

$$\begin{aligned} \partial _{s}U_{m,2}^{2}(t,s)&=\left[ -c(t)\alpha _{+}\partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma }-\partial _{t}^{2}u_{+}^{0}|_{\Gamma }-k_{m}^{2}u_{-} ^{0}|_{\Gamma }\right] \left( s-1\right) +f(t)\alpha _{+}\partial _{n} u_{+}^{1}|_{\Gamma }\nonumber \\&\quad -c(t)f(t)f^{\prime }(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+f(t)f^{\prime }(t)\partial _{t}u_{+}^{0}|_{\Gamma }\nonumber \\&\quad +(d+f(t))f(t)\alpha _{+}\partial _{{\textbf{n}}}^{2}u_{+}^{0}|_{\Gamma }. \end{aligned}$$

(A5)

Now, from Condition (26) at order 1, we get

$$\begin{aligned} \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma }-\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }= & {} \frac{c(t)(f'(t)-d f(t)-1)}{f(t)}\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\frac{df(t) \alpha _- +1}{f(t)} \partial _{t}^{2}u_{+}^{0}|_{\Gamma } \nonumber \\{} & {} -\frac{dk_-^2\alpha _- + k_m^2}{f(t)} u_{+}^{0}|_{\Gamma }- f'(t) \partial _t u_{+}^{0}|_{\Gamma } \nonumber \\{} & {} -(d+f(t))\alpha _{+}\partial _n^2u_{+}^{0}|_{\Gamma }. \end{aligned}$$

(A6)

As

$$\begin{aligned} \bigtriangleup u_{+}&=\left[ \dfrac{(1+\eta c(t)(d+f(t))^{2}+\eta ^{2}f^{\prime 2}(t)}{(d+f(t))^{2}(1+\eta c(t)(d+f(t))^{2}}\partial _{\eta } ^{2}-\dfrac{2\eta f^{\prime }(t)}{(d+f(t))(1+\eta c(t)(d+f(t))^{2}}\partial _{t}\partial _{\eta }\right. \\&\quad +\dfrac{1}{(1+\eta c(t)(d+f(t))^{2}}\partial _{t}^{2}-\dfrac{\eta dc^{\prime }(t)+\eta c^{\prime }(t)f(t)}{(1+\eta c(t)(d+f(t))^{3}}\partial _{t}\\&\quad +\left( \dfrac{c(t)}{(d+f(t))(1+\eta c(t)(d+f(t))}+\dfrac{\eta (-f(t)f^{\prime \prime }(t)+2f^{\prime 2}(t))}{(d+f(t))^{2}(1+\eta c(t)(d+f(t))^{2}}\right. \\&\quad +\left. \left. \dfrac{\eta dc^{\prime }(t)f(t)f^{\prime 2}(t)c^{\prime 2}(t)f^{\prime }(t)}{(d+f(t))f^{2}(t)(1+\eta c(t)(d+f(t))^{3}}\right) \partial _{\eta }\right] \widetilde{u}_{+}. \end{aligned}$$

and

$$\begin{aligned} \partial _{\eta } u_+ (t,\eta )_{|\eta =0}=(d+f(t)) \partial _n u_+ |_{\Gamma }, \end{aligned}$$

it follows by taking the limit $\eta \rightarrow 0,$ we obtain

$$\begin{aligned} \partial _{t}^{2}u_{+}|_{\eta =0}+\frac{c(t)}{d+f(t)}\partial _{\eta }u_{+}^{n}|_{\eta =0}+\frac{1}{(d+f(t))^2}\partial _{\eta }^{2}u_{+}^{n}|_{\eta =0}+k_{+}^{2}u_{+}|_{\eta =0}=0, \end{aligned}$$

i.e.

$$\begin{aligned} \partial _{{\textbf{n}}}^{2}u_{+}|_{\Gamma }=-\partial _{t}^{2}u_{+}|_{\Gamma }-c(t)\partial _{{\textbf{n}}}u_{+}|_{\Gamma }-k_{+}^{2}u_{+}|_{\Gamma }. \end{aligned}$$

(A7)

So (A6) becomes

$$\begin{aligned} \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{1}-\alpha _{-}\partial _{{\textbf{n}}} u_{-}^{1}= & {} \dfrac{ (d+f(t))f(t) \alpha _+ - df(t) \alpha _- - 1}{2f(t)} \left( \partial _t^2 u^0_+ + \partial _t^2 u^0_- \right) \nonumber \\ {}{} & {} + \dfrac{ (d+f(t))f(t) k_+^2 \alpha _+ - d f(t) k_-^2 \alpha _- - k_m^2 }{2f(t)}\left( u^0_+ +u^0_- \right) \nonumber \\{} & {} + \dfrac{c(t)(f^2(t)-1)}{2f(t)} \left( \alpha _+ \partial _n u_+^0+ \alpha _- \partial _n u_-^0 \right) \nonumber \\ {}{} & {} + \dfrac{f'(t)(\alpha _+-1)}{2} \left( \partial _t u_+^0+ \partial _t u_-^0 \right) . \end{aligned}$$

(A8)

1.3 A.3 Term of order 2

Relations (A4) and (A5) with conditions (22) and (24) yield $\forall (t,s)\in \Omega _{m,\beta }$

$$\begin{aligned} U_{m,1}^{2}(t,s)= & {} \left[ -c(t)\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }-\partial _{t}^{2}u_{-}^{0}|_{\Gamma }-k_{-}^{2}u_{-}^{0}|_{\Gamma }\right] \frac{s^2}{2} + \partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }s + u_{-}^{2}|_{\Gamma }. \end{aligned}$$

(A9)

$$\begin{aligned} U_{m,2}^{2}(t,s)= & {} \left[ -c(t)\alpha _+ \partial _{{\textbf{n}}} u_{+}^{0}|_{\Gamma }-\partial _{t}^{2}u_{+}^{0}|_{\Gamma }-k_{m}^{2}u_{-} ^{0}|_{\Gamma }\right] \frac{s^2-2s+1}{2} + \left[ f(t) \alpha _+ \partial _n u_{+}^{1}|_{\Gamma } \right. \nonumber \\{} & {} \left. -c(t)f(t)f'(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+f(t)f'(t) \partial _t u_{+}^{0}|_{\Gamma }+(d+f(t))f(t)\alpha _{+}\partial ^2_{{\textbf{n}}} u_{+}^{0}|_{\Gamma }\right] (s-1) \nonumber \\ {}{} & {} +\left( d+f(t) \right) \partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma }+\frac{\left( d+f(t) \right) ^2}{2} \partial _{{\textbf{n}}}^2 u_{+}^{0}|_{\Gamma }+ u_{+}^{2}|_{\Gamma }. \end{aligned}$$

(A10)

So (23) with (A7) and (A8) we get

$$\begin{aligned} u_{+}^{2}|_{\Gamma }-u_{-}^{2}|_{\Gamma }= & {} -\frac{1}{2}\left[ -c(t)\alpha _+ \partial _{{\textbf{n}}} u_{+}^{0}|_{\Gamma }-\partial _{t}^{2}u_{+}^{0}|_{\Gamma }-k_{m}^{2}u_{-}^{0}|_{\Gamma }\right] + \left[ f(t) \alpha _+ \partial _n u_{+}^{1}|_{\Gamma } \right. \nonumber \\{} & {} \left. -c(t)f(t)f'(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+f(t)f'(t) \partial _t u_{+}^{0}|_{\Gamma }+(d+f(t))f(t)\alpha _{+}\partial ^2_{{\textbf{n}}} u_{+}^{0}|_{\Gamma }\right] \nonumber \\{} & {} - \left( d+f(t) \right) \partial _{{\textbf{n}}} u_{+}^{1}|_{\Gamma }-\frac{\left( d+f(t) \right) ^2}{2} \partial ^2_{{\textbf{n}}} u_{+}^{0}|_{\Gamma } + \left[ -c(t)\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }-\partial _{t}^{2}u_{-}^{0}|_{\Gamma } \right. \nonumber \\ {}{} & {} \left. -k_{-}^{2}u_{-}^{0}|_{\Gamma }\right] \frac{d^2}{2}+d \partial _n u_{-}^{1}|_{\Gamma }. \end{aligned}$$

(A11)

$$\begin{aligned} u_{+}^{2}|_{\Gamma }-u_{-}^{2}|_{\Gamma }= & {} \frac{ f(t) \alpha _+ \alpha _-+d \alpha _+-(d+f(t))\alpha _- }{\alpha _+ \alpha _-} \alpha _+ \partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma } \nonumber \\ {}{} & {} +\left( \frac{\left( f(t)-2f^2(t)f'(t)-2df^2(t)-2 f^3(t) \right) \alpha _+ \alpha _- -\left( d^2 f(t) +2df^2(t)-2d \right) \alpha _+ }{2f(t) \alpha _+ \alpha _-} \right. \nonumber \\{} & {} \left. + \frac{ (d+f(t))^2 f(t) \alpha _- }{2f(t) \alpha _+ \alpha _-} \right) c(t) \alpha _+ \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma } + \left( \frac{-2(d+f(t))f^2(t) \alpha _+^2 \alpha _- }{2f(t) \alpha _+ \alpha _-} \right. \nonumber \\{} & {} + \left. \frac{ -2d(d+f(t))f(t)\alpha _+^2 + (f(t)+2d^2f(t)+f^3(t)+2df^2(t))\alpha _+ \alpha _-+2d \alpha _+ }{2f(t) \alpha _+ \alpha _-} \right) \nonumber \\ {}{} & {} \times \partial _{t}^2u_{+}^{0}|_{\Gamma }+\left( \frac{-2(d+f(t))f^2(t)k_+^2\alpha _+^2 \alpha _- -2d(d+f(t))f(t) k_+^2\alpha _+^2 }{2f(t) \alpha _+ \alpha _-} \right. \nonumber \\ {}{} & {} +\left. \frac{ +[k_m^2+d^2k_-^2+(d+f(t))^2k_+^2]f(t) \alpha _+\alpha _- +2d k_m^2 \alpha _+}{2f(t) \alpha _+ \alpha _-} \right) u_{+}^{0}|_{\Gamma } \nonumber \\{} & {} +\frac{f(t)\alpha _+ \alpha _- - d \alpha _+^2 +d\alpha _+}{\alpha _+ \alpha _-} f'(t) \partial _t u_{+}^{0}|_{\Gamma }. \end{aligned}$$

(A12)

Using (A8) we get

$$\begin{aligned} u_{+}^{2}|_{\Gamma }-u_{-}^{2}|_{\Gamma }= & {} \frac{ f(t) \alpha _+ \alpha _-+d \alpha _+-(d+f(t))\alpha _- }{\alpha _+ \alpha _-} \alpha _- \partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma } \nonumber \\ {}{} & {} +\left( \frac{\left( -f(t)-2f^2(t)f'(t)-2df^2(t) \right) \alpha _+ \alpha _- +\left( -f^3(t)+d^2f(t)+2f(t)+2d \right) \alpha _- }{2f(t) \alpha _+ \alpha _-} \right. \nonumber \\{} & {} \left. - \frac{d^2 f(t) \alpha _+ }{2f(t) \alpha _+ \alpha _-} \right) c(t) \alpha _+ \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma } + + \left( \frac{-2df^2(t) \alpha _+ \alpha _-^2+2df(t)(d+f(t))\alpha _-^2 }{2f(t) \alpha _+ \alpha _-} \right. \nonumber \\{} & {} + \left. \frac{ - (f(t)+2d^2f(t)+f^3(t)+2df^2(t))\alpha _+ \alpha _-+2(d+f(t)) \alpha _- }{2f(t) \alpha _+ \alpha _-} \right) \partial _{t}^2 u_{+}^{0}|_{\Gamma }\nonumber \\{} & {} +\left( \frac{-2d f^2(t)k_-^2 \alpha _+ \alpha _-^2 +2d(d+f(t))f(t) k_-^2\alpha _-^2 }{2f(t) \alpha _+ \alpha _-} \right. \nonumber \\ {}{} & {} +\left. \frac{ -[k_m^2+d^2k_-^2+(d+f(t))^2k_+^2]f(t) \alpha _+ \alpha _- +2(d+f(t)) k_m^2 \alpha _- }{2f(t) \alpha _+ \alpha _-} \right) u_{+}^{0}|_{\Gamma } \nonumber \\{} & {} +\frac{f(t) \alpha _+^2 \alpha _- - (d+f(t))\alpha _+ \alpha _- +(d+f(t)) \alpha _-}{\alpha _+ \alpha _-} f'(t) \partial _t u_{+}^{0}|_{\Gamma }. \end{aligned}$$

(A13)

Then

$$\begin{aligned} u_{+}^{2}|_{\Gamma }&-u_{-}^{2}|_{\Gamma } =\frac{f(t)\alpha _{+}\alpha _{-}+d\alpha _{+}-(d+f(t))\alpha _{-}}{2\alpha _{+}\alpha _{-}}\left( \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}} }u_{-}^{1}|_{\Gamma }\right) \nonumber \\&+\left( \frac{\left( -2f^{2}(t)f^{\prime 2}(t)-f^{3}(t)\right) \alpha _{+}\alpha _{-}+\left( d-d^{2}f(t)-df^{2}(t)\right) \alpha _{+}}{4f(t)\alpha _{+}\alpha _{-}}\right. \nonumber \\&\left. -\frac{\left( d+f(t)+d^{2}f(t)+df^{2}(t)\right) \alpha _{-} }{4f(t)\alpha _{+}\alpha _{-}}\right) c(t)\left( \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma }\right) \nonumber \\&+\left( \frac{-(d+f(t))f^{2}(t)\alpha _{+}^{2}\alpha _{-}-df^{2}(t)\alpha _{+}\alpha _{-}^{2}-df(t)(d+f(t))(\alpha _{+}^{2}-\alpha _{-}^{2})}{4f(t)\alpha _{+}\alpha _{-}}\right. \nonumber \\&+\left. \frac{d\alpha _{+}+(d+f(t))\alpha _{-}}{4f(t)\alpha _{+}\alpha _{-} }\right) \left( \partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\partial _{t}^{2} u_{-}^{0}|_{\Gamma }\right) \nonumber \\&+\left( \frac{-(d+f(t))f^{2}(t)k_{+}^{2}\alpha _{+}^{2}\alpha _{-} -df^{2}(t)k_{-}^{2}\alpha _{+}\alpha _{-}^{2}+(d+f(t))k_{m}^{2}\alpha _{-} }{4f(t)\alpha _{+}\alpha _{-}}\right. \nonumber \\&+\left. \frac{dk_{m}^{2}\alpha _{+}-d(d+f(t))f(t)(k_{+}^{2}\alpha _{+} ^{2}-k_{-}^{2}\alpha _{-}^{2})}{4f(t)\alpha _{+}\alpha _{-}}\right) \left( u_{+}^{0}|_{\Gamma }+u_{-}^{0}|_{\Gamma }\right) \nonumber \\&+\frac{f(t)\alpha _{+}^{2}\alpha _{-}-d\alpha _{+}^{2}-d\alpha _{+}\alpha _{-}+d\alpha _{+}+(d+f(t))\alpha _{-}}{4\alpha _{+}\alpha _{-}}f^{\prime }(t)\left( \partial _{t}u_{+}^{0}|_{\Gamma }+\partial _{t}u_{-}^{0}|_{\Gamma }\right) . \end{aligned}$$

(A14)

Equations (32) et (33) implie that, $\forall (t,s)\in \Omega _{m,\beta },$

$$\begin{aligned} \partial _{s}U_{m,1}^{3}(t,s)= & {} \left[ c^{2}(t)\partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }+c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+c(t)k_{-}^{2}u_{-}^{0}|_{\Gamma }-k_{-}^{2}\left( \partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }\right) -\partial _{t}^{2}\left( \partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }\right) \right] \frac{s^{2}}{2} \\{} & {} +\left[ c^{2}(t)\left( \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) +2c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+c^{\prime }(t)\partial _{t}u_{-}^{0}|_{\Gamma }\right] \frac{s^{2}}{2} \\{} & {} +\left( -\partial _{t}^{2}u_{-}^{1}|_{\Gamma }-k_{-}^{2}u_{-}^{1}|_{\Gamma }-c\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }\right) s+\partial _{s}U_{m,1}^{3}\left( t,0\right) ,\\ \partial _{s}U_{m,2}^{3}(t,s)= & {} -\dfrac{c(t)}{f(t)}\left[ -c(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\partial _{t}^{2}u_{+} ^{0}|_{\Gamma }-k_{m}^{2}u_{-}^{0}|_{\Gamma }\right] \left( \frac{1}{2}s^{2}-s+\frac{1}{2}\right) \\{} & {} \quad +\left[ f^{\prime }(t)f^{-1}(t)\partial _{t}\left( \left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right) 2-\partial _{t}^{2}\left[ \left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+} ^{0}|_{\Gamma }\right] \right. \\{} & {} \quad \left. -f^{\prime 2}(t)f^{-2}(t)\left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }2+f^{\prime \prime }(t)f^{-1}(t)\left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right. \\{} & {} \quad \left. +c^{2}(t)\left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}} u_{+}^{0}|_{\Gamma }+c(t)f(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }2\right. \\{} & {} \quad \left. -f(t)c^{\prime }(t)\partial _{t}u_{+}^{0}|_{\Gamma }-k_{m}^{2}\left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right] \left( \frac{1}{2}s^{2}-\frac{1}{2}\right) \\{} & {} \quad +\left[ dc^{2}f^{-1}(t)\left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma }-\partial _{t}^{2}u_{+}^{1}|_{\Gamma }-\partial _{t} ^{2}\left[ \left( d+f(t)-f(t)\alpha _{+}\right) \partial _{{\textbf{n}}} u_{+}^{0}|_{\Gamma }\right] \right. \\{} & {} \quad \left. -c(t)\alpha _{+}\partial _{n}u_{+}^{1}|_{\Gamma }-c(t)(d+f(t))\alpha _{+}\partial _{n}^{2}u_{+}^{0}|_{\Gamma }\right. \\{} & {} \quad \left. +c(t)f^{\prime }(t)\alpha _{+}\partial _{t}u_{+}^{0}|_{\Gamma }-c(t)f^{\prime }(t)\partial _{t}u_{+}^{0}|_{\Gamma }+2dc(t)\partial _{t}^{2} u_{+}^{0}|_{\Gamma }\right. \\{} & {} \quad \left. -k_{m}^{2}u_{+}^{1}|_{\Gamma }-k_{m}^{2}\left( d+f(t)-f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-dc^{\prime } (t)\partial _{t}u_{+}^{0}|_{\Gamma }\right] \left( s-1\right) \\{} & {} \quad +\partial _{s}U_{m,2}^{3}(t,1) \end{aligned}$$

From Condition (27) at order 2, we get

$$\begin{aligned} \partial _{s}U_{m,1}^{3}(t,s)= & {} \left[ c^{2}(t)\partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }+c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+c(t)k_{-}^{2}u_{-}^{0}|_{\Gamma }-k_{-}^{2}\left( \partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }\right) -\partial _{t}^{2}\left( \partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }\right) \right] \frac{s^{2}}{2} \\{} & {} +\left[ c^{2}(t)\left( \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) +2c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+c^{\prime }(t)\partial _{t}u_{-}^{0}|_{\Gamma }\right] \frac{s^{2}}{2} \\{} & {} +\left( -\partial _{t}^{2}u_{-}^{1}|_{\Gamma }-k_{-}^{2}u_{-}^{1}|_{\Gamma }-c(t)\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }\right) s+\partial _{{\textbf{n}} }u_{-}^{2}|_{\Gamma }, \end{aligned}$$

It follows from Transmission conditions (26) at order 2

$$\begin{aligned}&\frac{d^{2}}{2}\alpha _{-}c^{2}(t)\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma } +\frac{d^{2}}{2}\alpha _{-}c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+\frac{d^{2}}{2}\alpha _{-}c(t)k_{-}^{2}u_{-}^{0}|_{\Gamma }-\frac{d^{2}}{2}\alpha _{-}k_{-} ^{2}\left( \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) \nonumber \\&\qquad -\frac{d^{2}}{2}\alpha _{-}\partial _{t}^{2}\left( \partial _{{\textbf{n}}} u_{-}^{0}|_{\Gamma }\right) +\frac{d^{2}}{2}\alpha _{-}c^{2}(t)\left( \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) +d^{2}\alpha _{-} c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+\frac{d^{2}}{2}\alpha _{-}c^{\prime }(t)\partial _{t}u_{-}^{0}|_{\Gamma }\nonumber \\&\qquad -d\alpha _{-}\partial _{t}^{2}u_{-}^{1}|_{\Gamma }-d\alpha _{-}k_{-}^{2} u_{-}^{1}|_{\Gamma }-d\alpha _{-}c(t)\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{2}|_{\Gamma }\nonumber \\&\quad =\frac{c^{2}(t)}{2f^{2}(t)}\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\frac{c(t)}{2f^{2}(t)}\partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\frac{c(t)}{2f^{2}(t)} k_{m}^{2}u_{-}^{0}|_{\Gamma }\nonumber \\&\qquad -f^{\prime }(t)f^{-2}(t)\partial _{t}\left( \left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right) +\frac{1}{2}\frac{1}{f(t)}\partial _{t}^{2}\left[ \left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right] \nonumber \\&\qquad +f^{\prime 2}(t)f^{-3}(t)\left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\frac{1}{2}f^{\prime \prime } (t)f^{-1}(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\frac{1}{2}c^{2}(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\nonumber \\&-c(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\frac{1}{2}c^{\prime } (t)\partial _{t}u_{+}^{0}|_{\Gamma }+\frac{1}{2}k_{m}^{2}\alpha _{+} \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -dc^{2}(t)f^{-1}(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\frac{1}{f(t)}\partial _{t}^{2}u_{+}^{1}|_{\Gamma }+\frac{1}{f(t)}\partial _{t}^{2}\left[ \left( d+f(t)-f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right] \nonumber \\&\qquad +\frac{1}{f(t)}c(t)\alpha _{+}\partial _{n}u_{+}^{1}|_{\Gamma }+\frac{1}{f(t)}c(t)\left( d+f(t)\right) \alpha _{+}\partial _{n}^{2}u_{+}^{0}|_{\Gamma }-\frac{1}{f(t)}c(t)f^{\prime }(t)\alpha _{+}\partial _{t}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad +\frac{1}{f(t)}c(t)f^{\prime }(t)\partial _{t}u_{+}^{0}|_{\Gamma }-\frac{1}{f(t)}2dc(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }+k_{m}^{2}\left( d+f(t)-f(t)\alpha _{+}\right) \frac{1}{f(t)}\partial _{{\textbf{n}}}u_{+} ^{0}|_{\Gamma }\nonumber \\&\qquad +\frac{1}{f(t)}k_{m}^{2}u_{+}^{1}|_{\Gamma }+\frac{1}{f(t)}dc^{\prime } (t)\partial _{t}u_{+}^{0}|_{\Gamma }+\frac{1}{f(t)}\partial _{s}U_{m,2} ^{3}(t,1). \end{aligned}$$

(A15)

Now calculate the term $\partial _{s}U_{m,2}^{3}(t,1)$ in (A15). From (25) at order 2, we have

$$\begin{aligned}&\frac{1}{f(t)}\partial _{s}U_{m,2}^{3}(t,1) \nonumber \\&\quad =\left( f^{\prime }(t)\right) ^{2}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+2c(t)f^{\prime }(t)(d+f(t))\left( \alpha _{+}-1\right) \partial _{t}u_{+}^{0}|_{\Gamma } \nonumber \\&\qquad + \alpha _{+}\partial _{n}u_{+}^{2}|_{\Gamma }+\left( 1-\alpha _{+}\right) f^{\prime }(t)\partial _{t}u_{+}^{1}|_{\Gamma }+f^{\prime }(t)\left( d+f(t)\right) \left( 1-\alpha _{+}\right) \partial _{t}\partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma } \nonumber \\&\qquad +\frac{(d+f(t))^{2}}{2}\alpha _{+}\partial _{n}^{3}u_{+}^{0}|_{\Gamma }+(d+f(t))\alpha _{+}\partial _{n}^{2}u_{+}^{1}|_{\Gamma }. \end{aligned}$$

(A16)

Recall that

$$\begin{aligned} \Delta u_{+}^{n}+k_{+}^{2}u_{+}^{n}=\dfrac{-\eta c^{\prime }(t)}{\left[ 1+\eta c(t)\right] ^{3}}\partial _{t}u_{+}^{n}+\frac{1}{\left[ 1+\eta c(t) \right] ^{2}}\partial _{t}^{2}u_{+}^{n}+\dfrac{c(t)}{1+\eta c(t)}\partial _{\eta }u_{+}^{n}+\partial _{\eta }^{2}u_{+}^{n}+k_{+}^{2}u_{+}^{n}=0. \end{aligned}$$

The partial derivative with respect to $\eta$ leads to

$$\begin{aligned}&\dfrac{-c^{\prime }(t)\left[ 1+\eta c(t)\right] ^{3}+3c(t)\left[ 1+\eta c(t)\right] ^{2}\eta c^{\prime }(t)}{\left[ 1+\eta c(t)\right] ^{6}} \partial _{t}u_{+}^{n}+\dfrac{\eta c^{\prime }(t)}{\left[ 1+\eta c(t)\right] ^{3}}\partial _{\eta }\partial _{t}u_{+}^{n}\\&\quad -\frac{2c(t)}{\left[ 1+\eta c(t)\right] ^{3}}\partial _{t}^{2}u_{+}^{n} +\frac{1}{\left[ 1+\eta c(t)\right] ^{2}}\partial _{\eta }\partial _{t} ^{2}u_{+}^{n}-\dfrac{c^{2}(t)}{\left[ 1+\eta c(t)\right] ^{2}}\partial _{\eta }u_{+}^{n}\\&\quad +\dfrac{c(t)}{1+\eta c(t)}\partial _{\eta }^{2}u_{+}^{n}+\partial _{\eta } ^{3}u_{+}^{n}+k_{+}^{2}\partial _{\eta }u_{+}^{n} =0. \end{aligned}$$

Taking the limit $\eta \rightarrow 0$, we obtain

$$\begin{aligned}&-c^{\prime }(t)\partial _{t}u_{+}^{n}|_{\Gamma }-2c(t)\partial _{t}^{2}u_{+} ^{n}|_{\Gamma }+\partial _{{\textbf{n}}}\partial _{t}^{2}u_{+}^{n}|_{\Gamma } -c^{2}(t)\partial _{{\textbf{n}}}u_{+}^{n}|_{\Gamma }\\&+c(t)\partial _{{\textbf{n}}}^{2}u_{+}^{n}|_{\Gamma }+\partial _{{\textbf{n}}} ^{3}u_{+}^{n}|_{\Gamma }+k_{+}^{2}\partial _{{\textbf{n}}}u_{+}^{n}|_{\Gamma } =0, \end{aligned}$$

hence

$$\begin{aligned} \partial _{{\textbf{n}}}^{3}u_{+}^{n}|_{\Gamma }&=c^{\prime }(t)\partial _{t}u_{+}^{n}|_{\Gamma }+2c(t)\partial _{t}^{2}u_{+}^{n}|_{\Gamma }-\partial _{n}\partial _{t}^{2}u_{+}^{n}|_{\Gamma }\nonumber \\&\quad +c^{2}(t)\partial _{{\textbf{n}}}u_{+}^{n}|_{\Gamma }-c(t)\partial _{{\textbf{n}} }^{2}u_{+}^{n}|_{\Gamma }-k^{2}\partial _{{\textbf{n}}}u_{+}^{n}|_{\Gamma }. \end{aligned}$$

(A17)

Using Identity (A7), Relation (A17) becomes

$$\begin{aligned} \partial _{{\textbf{n}}}^{3}u_{+}^{n}|_{\Gamma }= & {} 3c(t)\partial _{t}^{2}u_{+}^{n}|_{\Gamma }+\left( 2c^{2}(t)-k_{+}^{2}\right) \partial _{ {\textbf{n}}}u_{+}^{n}|_{\Gamma }+c^{\prime }(t)\partial _{t}u_{+}^{n}|_{\Gamma } \nonumber \\{} & {} -\partial _{{\textbf{n}}}\partial _{t}^{2}u_{+}^{n}|_{\Gamma }+c(t)k_{+}^{2}u_{+}^{n}|_{\Gamma },\ \forall n\ge 0. \end{aligned}$$

(A18)

Therefore, using (A18) for $n=0$ and (A7) for $n\in \left\{ 0,1\right\}$, Relation (A16) leads to

$$\begin{aligned} \frac{1}{f(t)}\partial _{s}U_{m,2}^{3}(t,1)&=\left( f^{\prime }(t)\right) ^{2}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+2c(t)f^{\prime }(t)(d+f(t))\left( \alpha _{+}-1\right) \partial _{t}u_{+}^{0}|_{\Gamma }\nonumber \\&\quad +\frac{(d+f(t))^{2}}{2}\alpha _{+}c^{\prime }(t)\partial _{t}u_{+} ^{0}|_{\Gamma }+\alpha _{+}\partial _{n}u_{+}^{2}|_{\Gamma }+\left( 1-\alpha _{+}\right) f^{\prime }(t)\partial _{t}u_{+}^{1}|_{\Gamma }\nonumber \\&\quad +f^{\prime }(t)\left( d+f(t)\right) \left( 1-\alpha _{+}\right) \partial _{t}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-(d+f(t))\alpha _{+}\partial _{t}^{2}u_{+}^{1}|_{\Gamma }\nonumber \\&\quad -(d+f(t))\alpha _{+}c(t)\partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma }-(d+f(t))\alpha _{+}k_{+}^{2}u_{+}^{1}|_{\Gamma }\nonumber \\&\quad +\frac{(d+f(t))^{2}}{2}\alpha _{+}3c(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\frac{(d+f(t))^{2}}{2}\alpha _{+}\left( 2c^{2}(t)-k_{+}^{2}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\nonumber \\&\quad -\frac{(d+f(t))^{2}}{2}\alpha _{+}\partial _{{\textbf{n}}}\partial _{t}^{2} u_{+}^{0}|_{\Gamma }+\frac{(d+f(t))^{2}}{2}\alpha _{+}c(t)k_{+}^{2}u_{+} ^{0}|_{\Gamma } \end{aligned}$$

(A19)

So (A15) becomes

$$\begin{aligned}&\alpha _{+}\partial _{n}u_{+}^{2}|_{\Gamma }\nonumber \\&\quad =\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{2}|_{\Gamma }-\frac{1}{f(t)} c(t)\alpha _{+}\partial _{n}u_{+}^{1}|_{\Gamma }+(d+f(t))\alpha _{+} c(t)\partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma }-d\alpha _{-}c(t)\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }\nonumber \\&\qquad -\left( 1-\alpha _{+}\right) f^{\prime }(t)\partial _{t}u_{+}^{1}|_{\Gamma }-\frac{1}{f(t)}k_{m}^{2}u_{+}^{1}|_{\Gamma }+(d+f(t))\alpha _{+}k_{+}^{2} u_{+}^{1}|_{\Gamma }-d\alpha _{-}k_{-}^{2}u_{-}^{1}|_{\Gamma }\nonumber \\&\qquad -\frac{1}{f(t)}\partial _{t}^{2}u_{+}^{1}|_{\Gamma }+(d+f(t))\alpha _{+}\partial _{t}^{2}u_{+}^{1}|_{\Gamma }-d\alpha _{-}\partial _{t}^{2}u_{-} ^{1}|_{\Gamma }+\frac{d^{2}}{2}\alpha _{-}c^{2}(t)\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma }\nonumber \\&\qquad +\frac{d^{2}}{2}\alpha _{-}c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma } +\frac{d^{2}}{2}\alpha _{-}c(t)k_{-}^{2}u_{-}^{0}|_{\Gamma }-\frac{d^{2}}{2}\alpha _{-}k_{-}^{2}\left( \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) -\frac{d^{2}}{2}\alpha _{-}\partial _{t}^{2}\left( \partial _{{\textbf{n}}} u_{-}^{0}|_{\Gamma }\right) \nonumber \\&\qquad +\frac{d^{2}}{2}\alpha _{-}c^{2}(t)\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }+d^{2}\alpha _{-}c(t)\partial _{t}^{2}u_{-}^{0}|_{\Gamma }+\frac{d^{2}}{2} \alpha _{-}c^{\prime }(t)\partial _{t}u_{-}^{0}|_{\Gamma }-\frac{1}{2f^{2} (t)}c^{2}(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -\frac{c(t)}{2f^{2}(t)}\partial _{t}^{2}u_{+}^{0}|_{\Gamma }-\frac{c(t)}{2f^{2}(t)}k_{m}^{2}u_{-}^{0}|_{\Gamma }+\frac{1}{f^{2}(t)}f^{\prime }(t)\partial _{t}\left( \left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma }\right) -\frac{f^{\prime 2}(t)}{f^{2}(t)}\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -\frac{1}{2}\frac{1}{f(t)}\partial _{t}^{2}\left[ \left( f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right] +\frac{1}{2}\frac{f^{\prime \prime }(t)}{f(t)}\alpha _{+}\partial _{{\textbf{n}}}u_{+} ^{0}|_{\Gamma }+\frac{1}{2}c^{2}(t)\alpha _{+}\partial _{{\textbf{n}}}u_{+} ^{0}|_{\Gamma }\nonumber \\&\qquad +c(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }-\frac{1}{2}c^{\prime } (t)\partial _{t}u_{+}^{0}|_{\Gamma }-\frac{1}{2}k_{m}^{2}\alpha _{+} \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+dc^{2}(t)\frac{1}{f(t)}\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -\frac{1}{f(t)}\partial _{t}^{2}\left[ \left( d+f(t)-f(t)\alpha _{+}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }\right] +\frac{1}{f(t)}c(t)\left( d+f(t)\right) \alpha _{+}\partial _{t}^{2}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad +\frac{1}{f(t)}c^{2}(t)\left( d+f(t)\right) \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\frac{1}{f(t)}c(t)\left( d+f(t)\right) \alpha _{+}k_{+}^{2}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad +\frac{1}{f(t)}c(t)f^{\prime }(t)\alpha _{+}\partial _{t}u_{+}^{0}|_{\Gamma }-\frac{1}{f(t)}c(t)f^{\prime }(t)\partial _{t}u_{+}^{0}|_{\Gamma }+\frac{1}{f(t)}2dc(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -k_{m}^{2}\left( d+f(t)-f(t)\alpha _{+}\right) \frac{1}{f(t)} \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\frac{1}{f(t)}dc^{\prime } (t)\partial _{t}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -\left( f^{\prime }(t)\right) ^{2}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-2c(t)f^{\prime }(t)(d+f(t))\left( \alpha _{+}-1\right) \partial _{t}u_{+} ^{0}|_{\Gamma }-\frac{(d+f(t))^{2}}{2}\alpha _{+}c^{\prime }(t)\partial _{t} u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -f^{\prime }(t)\left( d+f(t)\right) \left( 1-\alpha _{+}\right) \partial _{t}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\frac{(d+f(t))^{2}}{2}\alpha _{+}3c(t)\partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\frac{(d+f(t))^{2}}{2}\alpha _{+}\partial _{{\textbf{n}}}\partial _{t}^{2}u_{+}^{0}|_{\Gamma }\nonumber \\&\qquad -\frac{(d+f(t))^{2}}{2}\alpha _{+}\left( 2c^{2}(t)-k_{+}^{2}\right) \partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }-\frac{(d+f(t))^{2}}{2}\alpha _{+}c(t)k_{+}^{2}u_{+}^{0}|_{\Gamma } \end{aligned}$$

(A20)

Using again Transmission conditions (A1), (A2), (A3) and (A6), we obtain

$$\begin{aligned}&\alpha _{+}\partial _{n}u_{+}^{2}|_{\Gamma }=\alpha _{-}\partial _{{\textbf{n}} }u_{-}^{2}|_{\Gamma }-{\mathcal {M}}-\frac{1}{2}\left[ \frac{d^{2}}{2}k_{-} ^{2}-d^{2}c^{2}(t)+\frac{1}{f^{2}(t)}f^{\prime 2}(t)-\frac{1}{2}c^{2} (t)\right. \\&+\left. \frac{1}{2}k_{m}^{2}\frac{1}{f(t)}\left( \frac{\left( f^{\prime \prime }(t)-f^{\prime \prime }(t)\alpha _{+}\right) }{\alpha _{+} }\right) -dc^{2}(t)\frac{1}{f(t)}+\frac{\left( f^{\prime }(t)\right) ^{2} }{\alpha _{+}}\right. \\&+\frac{(d+f(t))^{2}}{2}\left( 2c^{2}(t)-k_{+}^{2}\right) +k_{m}^{2}\left( \frac{d+f(t)-f(t)\alpha _{+}}{\alpha _{+}}\right) \frac{1}{f(t)}\\&+\left. -\frac{1}{f(t)}c^{2}(t)\left( d+f(t)\right) +\frac{c^{2} (t)}{2f^{2}(t)}-\frac{\left( f^{\prime }(t)\right) ^{2}}{f^{2}(t)}\right] \left( \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\alpha _{-} \partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) \\&+\frac{1}{2}\left[ \frac{d^{2}}{2}\alpha _{-}c(t)k_{-}^{2}-\frac{c(t)}{2f^{2}(t) }k_{m}^{2}+\frac{1}{f(t)}c(t)\left( d+f(t)\right) \alpha _{+}k_{+}^{2} -\frac{(d+f(t))^{2}}{2}\alpha _{+}c(t)k_{+}^{2}\right] \\&\left( u_{+} ^{0}|_{\Gamma }+u_{-}^{0}|_{\Gamma }\right) \\&+\left[ \frac{d^{2}}{4}\alpha _{-}c^{\prime }(t)-\frac{1}{4}c^{\prime }(t)-\frac{1}{2f(t)}c(t)f^{\prime }(t)-c(t)f^{\prime }(t)(d+f(t))\left( \alpha _{+}-1\right) \right. \\&+\left. \frac{1}{2f(t)}c(t)f^{\prime }(t)\alpha _{+}-\frac{(d+f(t))^{2}}{4}\alpha _{+}c^{\prime }(t)-\frac{1}{2}\frac{1}{f(t)}dc^{\prime }(t)\right] \left( \partial _{t}u_{+}^{0}|_{\Gamma }+\partial _{t}u_{-}^{0}|_{\Gamma }\right) \\&+\frac{1}{2}\left[ \frac{c(t)}{2f^{2}(t)}-\frac{1}{f(t)}c(t)\left( d+f(t)\right) \alpha _{+}-c(t)-\frac{1}{f(t)}2dc(t)\right. \\&+\left. \frac{(d+f(t))^{2}}{2}\alpha _{+}3c(t)-\frac{3d^{2}}{2}\alpha _{-}c(t)\right] \left( \partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\partial _{t} ^{2}u_{-}^{0}|_{\Gamma }\right) \\&+\frac{1}{2}\left[ \frac{(d+f(t))^{2}}{2}-\frac{d^{2}}{2}-\frac{1}{2} -\frac{1}{f(t)}\left( \frac{d+f(t)-f(t)\alpha _{+}}{\alpha _{+}}\right) \right] \\&\left( \alpha _{+}\partial _{t}^{2}\partial _{{\textbf{n}}}u_{+} ^{0}|_{\Gamma }+\partial _{t}^{2}\alpha _{-}\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma }\right) \\&+\frac{1}{2}\left[ -2\frac{1}{f(t)}\left( \frac{\left( f^{\prime }(t)-f^{\prime }(t)\alpha _{+}\right) }{\alpha _{+}}\right) -\frac{f^{\prime }(t)\left( d+f(t)\right) \left( 1-\alpha _{+}\right) }{\alpha _{+}}\right] \\&\left( \alpha _{+}\partial _{t}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }++\partial _{t}\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) , \end{aligned}$$

where

$$\begin{aligned}&{\mathcal {M}}=\frac{1}{2}\left( \frac{1}{f(t)}-f(t)\right) c(t)\left( \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{1}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{1}|_{\Gamma }\right) \\&+\frac{1}{2}\left( \frac{1}{f(t)}k_{m}^{2}-(d+f(t))\alpha _{+}k_{+} ^{2}+d\alpha _{-}k_{-}^{2}\right) \left( u_{+}^{1}|_{\Gamma }+u_{-} ^{1}|_{\Gamma }\right) \\&+\frac{1}{2}\left( 1-\alpha _{+}\right) f^{\prime }(t)\left( \partial _{t}u_{+}^{1}|_{\Gamma }+\partial _{t}u_{-}^{1}|_{\Gamma }\right) +\frac{1}{2}\left( \frac{1}{f(t)}-(d+f(t))\alpha _{+}+d\alpha _{-}\right) \\&\left( \partial _{t}^{2}u_{+}^{1}|_{\Gamma }+\partial _{t}^{2}u_{-}^{1}|_{\Gamma }\right) \\&+\frac{1}{2}\left( \frac{1}{f(t)}-f(t)\right) c(t)\left[ \dfrac{(d+f(t))f(t)\alpha _{+}-df(t)\alpha _{-}-1}{2f(t)}\left( \partial _{t}^{2} u_{+}^{0}|_{\Gamma }+\partial _{t}^{2}u_{-}^{0}|_{\Gamma }\right) \right. \\&+\left. \dfrac{(d+f(t))f(t)k_{+}^{2}\alpha _{+}-df(t)k_{-}^{2}\alpha _{-}-k_{m}^{2}}{2f(t)}\left( u_{+}^{0}|_{\Gamma }+u_{-}^{0}|_{\Gamma }\right) \right. \\&+\left. \dfrac{c(t)(f^{2}(t)-1)}{2f(t)}\left( \alpha _{+}\partial _{n} u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{n}u_{-}^{0}|_{\Gamma }\right) +\dfrac{f^{\prime }(t)(\alpha _{+}-1)}{2}\left( \partial _{t}u_{+}^{0}|_{\Gamma }+\partial _{t}u_{-}^{0}|_{\Gamma }\right) \right] \\&-dc(t)\dfrac{(d+f(t))f(t)\alpha _{+}-df(t)\alpha _{-}-1}{2f(t)}\left( \partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\partial _{t}^{2}u_{-}^{0}|_{\Gamma }\right) \\&-dc(t)\dfrac{(d+f(t))f(t)k_{+}^{2}\alpha _{+}-df(t)k_{-}^{2}\alpha _{-} -k_{m}^{2}}{2f(t)}\left( u_{+}^{0}|_{\Gamma }+u_{-}^{0}|_{\Gamma }\right) \\&-dc^{2}(t)\dfrac{(f^{2}(t)-1)}{2f(t)}\left( \alpha _{+}\partial _{n}u_{+} ^{0}|_{\Gamma }+\alpha _{-}\partial _{n}u_{-}^{0}|_{\Gamma }\right) -dc(t)\dfrac{f^{\prime }(t)(\alpha _{+}-1)}{2}\left( \partial _{t}u_{+} ^{0}|_{\Gamma }+\partial _{t}u_{-}^{0}|_{\Gamma }\right) \\&+\left( \frac{k_{m}^{2}-(d+f(t))f(t)\alpha _{+}k_{+}^{2}-f(t)d\alpha _{-}k_{-}^{2}}{f(t)}\right) \left( \dfrac{f(t)\alpha _{+}\alpha _{-}-(d+f(t))\alpha _{-}+d\alpha _{+} }{4\alpha _{+}\alpha _{-}}\right) \\ {}&(\alpha _{+}\partial _{{\textbf{n}}}u_{+} ^{0}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }) +\left( 1-\alpha _{+}\right) f^{\prime }(t)\left( \dfrac{f^{\prime }(t)\alpha _{+}\alpha _{-}-f^{\prime }(t)\alpha _{-}}{4\alpha _{+}\alpha _{-} }\right) (\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma })\\&+\left( 1-\alpha _{+}\right) f^{\prime }(t)\dfrac{f(t)\alpha _{+}\alpha _{-}-(d+f(t))\alpha _{-}+d\alpha _{+}}{4\alpha _{+}\alpha _{-}}(\alpha _{+} \partial _{t}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{t}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma })\\&+\left( \frac{1}{f(t)}-(d+f(t))\alpha _{+}-d\alpha _{-}\right) \left[ \left( \dfrac{f^{\prime \prime }(t)\alpha _{+}\alpha _{-}-(f^{\prime \prime }(t))\alpha _{-}}{4\alpha _{+}\alpha _{-}}\right) (\alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma })\right. \\&+\left( \dfrac{f^{\prime }(t)\alpha _{+}\alpha _{-}-f^{\prime }(t)\alpha _{-} }{2\alpha _{+}\alpha _{-}}\right) (\alpha _{+}\partial _{t}\partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{t}\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma })\\&+\left. \dfrac{f(t)\alpha _{+}\alpha _{-}-(d+f(t))\alpha _{-}+d\alpha _{+} }{4\alpha _{+}\alpha _{-}}(\alpha _{+}\partial _{t}^{2}\partial _{{\textbf{n}}} u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{t}^{2}\partial _{{\textbf{n}}}u_{-} ^{0}|_{\Gamma })\right] . \end{aligned}$$

Then

$$\begin{aligned} \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{2}|_{\Gamma }-\alpha _{-}\partial _{n}u_{-}^{2}|_{\Gamma }= & {} \vartheta _{1}\left( \alpha _{+}\partial _{ {\textbf{n}}}u_{+}^{1}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}} }u_{-}^{1}\right) +\vartheta _{2}\left( u_{+}^{1}|_{\Gamma }+u_{-}^{1}|_{\Gamma }\right) \\{} & {} +\vartheta _{3}\left( \partial _{t}u_{+}^{1}|_{\Gamma }+\partial _{t}u_{-}^{1}|_{\Gamma }\right) +\vartheta _{4}\left( \partial _{t}^{2}u_{+}^{1}|_{\Gamma }+\partial _{t}^{2}u_{-}^{1}|_{\Gamma }\right) \\{} & {} +\vartheta _{5}\left( \partial _{t}^{2}u_{+}^{0}|_{\Gamma }+\partial _{t}^{2}u_{-}^{0}|_{\Gamma }\right) +\vartheta _{6}\left( \partial _{t}u_{+}^{0}|_{\Gamma }+\partial _{t}u_{-}^{0}|_{\Gamma }\right) \\{} & {} +\vartheta _{7}\left( u_{+}^{0}|_{\Gamma }+u_{-}^{0}|_{\Gamma }\right) +\vartheta _{8}\left( \alpha _{+}\partial _{t}^{2}\partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma }+\partial _{t}^{2}\alpha _{-}\partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }\right) \\{} & {} +\vartheta _{9}\left( \alpha _{+}\partial _{t}\partial _{{\textbf{n}} }u_{+}^{0}|_{\Gamma }+\partial _{t}\alpha _{-}\partial _{{\textbf{n}} }u_{-}^{0}|_{\Gamma }\right) \\{} & {} +\vartheta _{10}\left( \alpha _{+}\partial _{{\textbf{n}}}u_{+}^{0}|_{\Gamma }+\alpha _{-}\partial _{{\textbf{n}}}u_{-}^{0}|_{\Gamma }\right) , \end{aligned}$$

where $\left( \vartheta _{i}\right) _{1\le i\le 10}$ are defined in Sect. 4.2.