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\title{Article Title}
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\author{First Author\textsuperscript{1,2*} \and Second Author\textsuperscript{2}}
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	\textsuperscript{1,2}Department, Organization, Street, City, 100190, State, Country\\
	\textsuperscript{2}Department, Organization, Street, City, 10587, State, Country\\
	\textsuperscript{*}{Communicating Author, E-Mail: iiiauthor@gmail.com}\\ % Email ID of the communicating author
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\fancyhead[R]{\textit{2nd International Conference on Applied \\ Mathematics and Mechanics (ICAMM 2026)}, \\
	03-05 December 2026, IIT Indore, India. \\
	Abstract ID: XXXXX }
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\begin{abstract}
	The abstract serves both as a general introduction to the topic and as a brief, non-technical summary of the main results and their implications. Authors are advised to check the author instructions for the journal to which they are submitting for word limits and to determine whether structural elements such as subheadings, citations, or equations are permitted.
\keywords{Keyword1, Keyword2, Keyword3, Keyword4}
\end{abstract}
\linenumbers
\section{Introduction}

The Introduction section, of referenced text Manna et al. \cite{manna2022love} expands on the background of the work (some overlap with the Abstract is acceptable). The introduction should not include subheadings (cf. Belfield et al. \cite{belfield1983stress} ).

Springer Nature does not impose a strict layout as standard however authors are advised to check the individual requirements for the journal they are planning to submit to as there may be journal-level preferences. When preparing your text please also be aware that some stylistic choices are not supported in full text XML (publication version), including coloured font. These will not be replicated in the typeset article if it is accepted. 


\section{Theoretical background}
Consider a semi-infinite medium covered by a FRV layer of thickness $(H)$, as illustrated in Figure 1. The material particles are polarized in ${x}$-axis direction, `$O$' is the intersecting point of topmost layer $(\rm{m}_1)$ and a substrate $(\rm{m}_2)$. Love wave propagation is assumed to be horizontally polarized along the $x$-axis, while the $z$-axis is vertically downward. Therefore, the total out-of-plane displacement components $(u_1 ,u_2 ,u_3 )$ along the $(x ,y ,z)$ direction, respectively, can be expressed as
\begin{eqnarray*}
u_1  = u_3  = 0,\,\,u_2  = u_2 (x ,z ,t ),\; \text{and}\;\ \frac{\partial }{{\partial {y}}}\equiv0.
\end{eqnarray*}
\begin{figure}[h]
	\centering
	\includegraphics[scale=0.5]{Fig1}
	\caption{Geometry of the present model}
	\label{fig:x cubed graph}
\end{figure}
\section{Equations}
\subsection{Solution }
The governing equation of motion for the medium $(\rm{m}_1)$ are given by Belfield et al. \cite{belfield1983stress}
\begin{equation}
	\frac{{\partial \sigma _{21}^{\rm{m}_1 } }}{{\partial x }} + \frac{{\partial \sigma _{23}^{\rm{m}_1 } }}{{\partial z }} = \rho _1^{\rm{m}_1 } \frac{{\partial ^2 u_2^{\rm{m}_1 } }}{{\partial t^2 }}{\rm{ }},
\end{equation}
where, ${\sigma _{21}^{\rm{m}_1 } }$, ${\sigma _{23}^{\rm{m}_1 } }$ are the stress tensor components, $\rho _1^{\rm{m}_1 } $ represents the density of medium $\rm{m}_1$.

\subsection{Results}
The governing equation of motion of the medium $(\rm{m}_2)$ are given by \cite{belfield1983stress}
\begin{equation}
	\left( {\lambda _2^{\rm{m}_2 }  + \mu _2^{\rm{m}_2 }  + \eta _2^{\rm{m}_2 } \nabla ^2 } \right)\nabla \left( {\nabla .\bm{u_r}^{\rm{m}_2} } \right) + \left( {\mu _2^{\rm{m}_2 }  - \eta _2^{\rm{m}_2 } \nabla ^2 } \right)\nabla ^2 \bm{u_r}^{\rm{m}_2}  = \rho _2^{\rm{m}_2 } \frac{{\partial ^2 \bm{u_r}^{\rm{m}_2} }}{{\partial t^2 }},
\end{equation}
where, $\lambda _2^{\rm{m}_2 }$ and $\mu _2^{\rm{m}_2 }$ are the Lamé constants and $\nabla ^2  \equiv \left(\frac{{\partial ^2 }}{{\partial x ^2 }} + \frac{{\partial ^2 }}{{\partial z ^2 }}\right)$ is the two-dimensional
	Laplacian operator, ${\eta _2^{\rm{m}_2}} = {\mu _2^{\rm{m}_2}}{l^2}$ is the coupled stress coefficient, $l$ is the characteristic length, $\bm{u_r}^{{\rm{m}_2}}$ is the displacement vector and ${\rho _2^{\rm{m}_2}}$ is the mass density of the intermediate layer. With the help of the aforementioned conditions, we obtain $\nabla .\bm{u_r}^{{\rm{m}_2}} = 0$, and Eq. (9) can be  written as
\begin{equation*}
	\begin{array}{l}
		\left( {{\mu _2^{\rm{m}_2}} - {\eta _2^{\rm{m}_2}}{\nabla ^2}} \right){\nabla ^2}\bm{u_r}^{{\rm{m}_2}} = {\rho _2^{\rm{m}_2}}\frac{{{\partial^2}\bm{u_r}^{{\rm{m}_2}}}}{{\partial{t^2}}} 
	\end{array}, 
\end{equation*}
\begin{equation}
	\left( {\frac{{\partial ^2 u_2^{\rm{m}_2 } }}{{\partial x^2 }} + \frac{{\partial ^2 u_2^{\rm{m}_2 } }}{{\partial z^2 }}} \right) - l^2 \left( {\frac{{\partial ^4 u_2^{\rm{m}_2 } }}{{\partial x^4 }} + \frac{{\partial ^4 u_2^{\rm{m}_2 } }}{{\partial z^4 }} + 2\frac{{\partial ^4 u_2^{\rm{m}_2 } }}{{\partial x^2 \partial z^2 }}} \right) = \frac{1}{{c_2 ^2 }}\frac{{\partial ^2 u_2^{\rm{m}_2 } }}{{\partial t^2 }},
\end{equation}
where, ${c _2} = \sqrt {\frac{{{\mu _2^{\rm{m}_2}}}}{{{\rho _2^{\rm{m}_2}}}}}$. Assuming that ${u_2^{\rm{m}_2 }}$ be the solution of Eq. (10) is 
\begin{equation}
	{u_2^{\rm{m}_2 }}(x ,z ,t) = {U_2^{\rm{m}_2 }}(z){e^{i(k{x} - \omega t)}}, 
\end{equation}
where, $\omega$ and $k$ describe angular frequency and wavenumber, respectively.
Substituting  $u_2^{\rm{m}_2 }$ in Eq. (10), we obtain
	\begin{equation}
		\frac{{d^4 U_2^{\rm{m}_2 } }}{{dz^4 }} - P\frac{{d^2 U_2^{\rm{m}_2 } }}{{dz^2 }} + QU_2^{\rm{m}_2 }  = 0{\rm{ }},
\end{equation}
where, $P = 2{k^2} + \frac{1}{{{l^2}}}$ \;\text{and}\;\ $Q = {k^4} - \frac{{{k^2}}}{{{l^2}}}\left( {\frac{{{c^2}}}{{{c _2}}} - 1} \right)$.\\
Thus, ${U_2^{\rm{m}_2 }}(z)$ can be written as a solution of Eq. (12):
\begin{equation}
	U_2^{\rm{m}_2 } (z ) = T_3 e^{ - a_1 z }  + T_4 e^{ - a_2 z }, 
\end{equation}
where, $a_1  = \sqrt {\frac{{P + \sqrt {P^2  - 4Q} }}{2}} $, \;\ $
a_2  = \sqrt {\frac{{P - \sqrt {P^2  - 4Q} }}{2}} $ \;\text{and} $T_3$, $T_4$ are arbitrary constants.\\
We obtain the final solution for the half-space using Eq. (13) in Eq. (11). 
\begin{equation}
	u_2^{\rm{m}_2 } (x ,z ,t) = \left( {T_3 e^{ - a_1 z }  + T_4 e^{ - a_2 z } } \right)e^{ i(kx  - \omega t)}. 
\end{equation}
For a coupled stress layer, the constitutive equations are as follows:\\
\begin{align}
	{\sigma _{jr}^{\rm{m}_2}} &= \lambda_2^{\rm{m}_2} {e_{kk}}{\delta _{rj}} + 2{\mu _2^{\rm{m}_2}}{e_{rj}} - 2{\eta _2^{\rm{m}_2}}{\nabla ^2}{\omega _{rj}},\nonumber\\
	&= \lambda_2^{\rm{m}_2} u_{_{k,k}}^{{\rm{m}_2}}{\delta _{rj}} + {\mu _2^{\rm{m}_2}}(u_{_{r,j}}^{{\rm{m}_2}} + u_{_{j,r}}^{{\rm{m}_2}}) - {\eta _2^{\rm{m}_2}}{\nabla ^2}(u_{_{r,j}}^{{\rm{m}_2}} - u_{_{j,r}}^{{\rm{m}_2}}), 
\end{align}
where, ${\sigma _{jr}^{\rm{m}_2}}$ represents non-symmetric force stress-tensor, $u_{_{rj}}^{{\rm{m}_2}}$ represents the displacement component, ${\delta _{rj}}$ represents the Kronecker's delta, ${e_{rj}} = \frac{1}{2}(u_{_{r,j}}^{{\rm{m}_2}} + u_{_{j,r}}^{{\rm{m}_2}})$ represents the symmetric stress tensor, ${\omega _{rj}} = \frac{1}{2}(u_{_{r,j}}^{{\rm{m}_2}} - u_{_{j,r}}^{{\rm{m}_2}})$  (skew-symmetric rotation stress tensor). 

\section*{Acknowledgment}
Acknowledgements are not compulsory. Where included they should be brief. Grant or contribution numbers may be acknowledged.
Please refer to Journal-level guidance for any specific requirements.

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\begin{itemize}
	\item Funding
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	\item Ethics approval and consent to participate
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\noindent
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\section*{Appendix}
\begin{equation*}
	\begin{array}{l}
		Y_3  = \left( { - \frac{{a_1 }}{2}\overline {\mu _r }  + \left( {\overline {\mu _s }  - \overline {\mu _r } } \right)\left( {\ell _1 \ell _3 ik - \frac{{a_1 }}{2}\ell _3^2 } \right)} \right),\,\,Y_4  = \left( {\theta _1 \overline {\mu _r }  + \left( {\overline {\mu _s }  - \overline {\mu _r } } \right)\theta _1 \ell _3^2 } \right), \\ 
		Y_5  = \left( {a_2 \mu _2^{\rm{m}_2 }  - a_2 \mu _2^{\rm{m}_2 } k^2 l^2  + a_2^3 \mu _2^{\rm{m}_2 } l^2 } \right),\,\,Y_6  = \left( {a_2 \mu _2^{\rm{m}_2 }  - a_2 \mu _2^{\rm{m}_2 } k^2 l^2  + a_2^3 \mu _2^{\rm{m}_2 } l^2 } \right), \\ 
		Y_7  = \left( { - \frac{{a_1 }}{2}\overline {\mu _r }  + \left( {\overline {\mu _s }  - \overline {\mu _r } } \right)\left( {\ell _1 \ell _3 ik - \frac{{a_1 }}{2}\ell _3^2 } \right) + \kappa_n } \right),\,\,\,\,Y_8  = \left( {\theta _1 \overline {\mu _r }  + \left( {\overline {\mu _s }  - \overline {\mu _r } } \right)\theta _1 \ell _3^2 } \right). \\ 
	\end{array}
\end{equation*}
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\bibitem {kundu2014propagation}
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\end{thebibliography}
\end{document}
