## 3 Derivations of the Solutions
### Semi-Analytical Solutions
\begin{array}{l}\widetilde{u}_{\mathrm{w}}(z,s)=\frac{1}{C_{\mathrm{w}}^{ \prime}(\xi^{2}-\eta^{2})}\Big{\{}-\frac{\mathrm{ch}[\xi(H-z)]}{\xi^{2} \mathrm{ch}(\xi H)}\Big{[}\frac{s\widetilde{C}_{\mathrm{w}}}{\zeta_{\mathrm{a }}^{2}}\Big{(}u_{\mathrm{a}}^{0}+\widetilde{C}_{\mathrm{a}}u_{\mathrm{w}}^{0}+ \frac{\widetilde{D}_{\mathrm{a}}}{s+b}\Big{)}+\Big{(}\eta^{2}+\frac{s}{C_{ \mathrm{w}}^{\prime}}\Big{)}\Big{(}\widetilde{C}_{\mathrm{w}}u_{\mathrm{a}}^{0} +u_{\mathrm{w}}^{0}+\frac{\widetilde{D}_{\mathrm{w}}}{s+b}\Big{)}\Big{]}\\ +\frac{\mathrm{ch}[\eta(H-z)]}{\eta^{2}\mathrm{ch}(\eta H)}\Big{[}\frac{s \widetilde{C}_{\mathrm{w}}^{\prime}}{\zeta_{\mathrm{w}}^{\prime}}\Big{(}u_{ \mathrm{a}}^{0}+\widetilde{C}_{\mathrm{a}}u_{\mathrm{w}}^{0}+\frac{\widetilde{ D}_{\mathrm{a}}}{s+b}\Big{)}+\Big{(}\widetilde{C}_{\mathrm{w}}u_{\mathrm{a}}^{0}+u_{ \mathrm{w}}^{0}+\frac{\widetilde{D}_{\mathrm{w}}}{s+b}\Big{)}\Big{(}\xi^{2}+ \frac{s}{C_{\mathrm{w}}^{\prime}}\Big{)}\Big{]}\Big{\}}+\frac{u_{\mathrm{w}}^{ 0}}{s}+\frac{\widetilde{D}_{\mathrm{a}}-\widetilde{C}_{\mathrm{a}}\widetilde{D}_{ \mathrm{w}}}{s(1-\widetilde{C}_{\mathrm{a}}\widetilde{C}_{\mathrm{w}})(s+b)} \end{array}  \tag{14}\]  
\[ \begin{array}{l}\widetilde{w}(s)=-\frac{\mathrm{sh}(\xi H)}{\xi^{3}(\xi^{2}- \eta^{2})\mathrm{ch}(\xi H)}\Big{\{}\frac{1}{C_{\mathrm{w}}^{\prime}}\Big{[}( \widetilde{m}_{2}-\widetilde{m}_{1\mathrm{k}})\Big{(}\eta^{2}+\frac{s}{C_{ \mathrm{w}}^{\prime}}\Big{)}-\widetilde{m}_{2}\frac{s\widetilde{C}_{\mathrm{w}} }{C_{\mathrm{w}}^{\prime}}\Big{]}\Big{(}u_{\mathrm{a}}^{0}+\widetilde{C}_{ \mathrm{a}}u_{\mathrm{w}}^{0}+\frac{\widetilde{D}_{\mathrm{a}}}{s+b}\Big{)}\\ +\frac{1}{C_{\mathrm{w}}^{\prime}}\Big{[}(\widetilde{m}_{2}-\widetilde{m}_{1 \mathrm{k}})\frac{s\widetilde{C}_{\mathrm{a}}}{\widetilde{C}_{\mathrm{w}}^{ \prime}}-\widetilde{m}_{2}\Big{(}\eta^{2}+\frac{s}{C_{\mathrm{w}}^{\prime}} \Big{)}\Big{]}\Big{(}\widetilde{C}_{\mathrm{w}}u_{\mathrm{a}}^{0}+u_{\mathrm{w}}^{ 0}+\frac{\widetilde{D}_{\mathrm{w}}}{s+b}\Big{)}\Big{\}}+\frac{(\widetilde{m}_{2}- \widetilde{m}_{1\mathrm{k}})(\widetilde{D}_{\mathrm{a}}-\widetilde{C}_{\mathrm{a}} \widetilde{D}_{\mathrm{w}})H}{s(1-\widetilde{C}_{\mathrm{a}}\widetilde{C}_{ \mathrm{w}})(s+b)}\\ +\frac{\mathrm{sh}(\eta H)}{\eta^{3}(\xi^{2}-\eta^{2})\mathrm{ch}(\eta H)}\Big{\{} \frac{1}{C_{\mathrm{w}}^{\prime}}\Big{[}(\widetilde{m}_{2}-\widetilde{m}_{1 \mathrm{k}})\Big{(}\xi^{2}+\frac{s}{C_{\mathrm{w}}^{\prime}}\Big{)}- \widetilde{m}_{2}\frac{\widetilde{C}_{\mathrm{w}}}{C_{\mathrm{w}}^{\prime}} \Big{]}\Big{(}u_{\mathrm{a}}^{0}+\widetilde{C}_{\mathrm{a}}u_{\mathrm{w}}^{0}+ \frac{\widetilde{D}_{\mathrm{a}}}{s+b}\Big{)}\\ +\frac{1}{C_{\mathrm{w}}^{\prime}}\Big{[}(\widetilde{m}_{2}-\widetilde{m}_{1 \mathrm{k}})\frac{s\widetilde{C}_{\mathrm{a}}}{\widetilde{C}_{\mathrm{w}}^{ \prime}}-\widetilde{m}_{2}\Big{(}\xi^{2}+\frac{s}{C_{\mathrm{w}}^{\prime}} \Big{)}\Big{]}\left(\widetilde{C}_{\mathrm{w}}u_{\mathrm{a}}^{0}+u_{\mathrm{w}}^{ 0}+\frac{\widetilde{D}_{\mathrm{w}}}{s+b}\right)\Big{\}}-\frac{\widetilde{m}_{2} (\widetilde{D}_{\mathrm{w}}-\widetilde{C}_{\mathrm{w}}\widetilde{D}_{\mathrm{a}} )H}{s(1-\widetilde{C}_{\mathrm{a}}\widetilde{C}_{\mathrm{w}})(s+b)}+\frac{ \widetilde{m}_{1\mathrm{k}}q_{\mathrm{w}}q_{\mathrm{w}}pH}{s(s+b)}\\ \end{array}  \tag{15}\]  
where, \(\widetilde{m}_{1\mathrm{k}}=\widetilde{m}_{1\mathrm{k}}^{\mathrm{w}}+\widetilde{m}_{1 \mathrm{k}}^{\mathrm{a}}\), \(\widetilde{m}_{2}=\widetilde{m}_{2}^{\mathrm{w}}+\widetilde{m}_{2}^{\mathrm{z}}\);  
\[\xi=\begin{array}{l}\Bigg{[}\frac{\