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\title{Energy propagation of the Rayleigh waves
in 3-dimensional half space}
\author{Mishio Kawashita and Wakako
Kawashita(n\'ee Dan)${}^{\dag}$ \\
Faculty of education, Ibaraki University, \\
${}^{\dag}$ Institute of Mathematics, University of Tsukuba
}

\begin{document}
\maketitle
\par
We study the energy behaviour
of the Rayleigh surface waves in a homogeneous
isotropic 3-dimensional half space with
the Neumann boundary condition.
It is known that the Rayleigh waves
with finite energy are asymptotically
concentrated along the boundary.
Our aim of this study is to clarify
the domain where the energy
of the Rayleigh waves concentrate
and to obtain the decay
order of the energy distributed
in the complement of the domain.
\par
First, we estimate local energy of the Rayleigh component
in the flat domains
$G_L=\{ {\bf x} \in {\bf R}^3_+ ; \ x_3 >L \}$ and
obtain an estimate of energy in $G_L$
with respect to $L$ but not time $t$.
This estimate explains that energy
of the Rayleigh waves concentrates
in the complement of the region $ G_L $ in some sense.
On the other hand, we can not
expect decay properties of energy in $ G_L $
as $ t \to \infty $.
Indeed, we obtain an example pointing out
this fact, which means the energy
of the Rayleigh waves may spread to $ G_L $ generally.
Guillot \cite{G} obtained
energy of the Rayleigh component in non-flat domain
$C_\epsilon= \{ ({\bf x}', x_3)
\in {\bf R}^3_+ ; \ x_3>\epsilon\ \vert
{\bf x}'\vert \ \}$
tends to $0$ as $ t \to \infty$.
Hence, our results show different feature of
the energy behaviour of the Rayleigh waves.
\par
Second, to make the energy behaviour more clear,
we also obtain decay order of energy with respect to $t$
in flat and annular domains.
The decay property of the Rayleigh component in \cite{G}
is showed by constructing asymptotic wave functions
introduced by Wilcox \cite{W}, however, it seems to be hard
to obtain decay estimates by the approach using
the asymptotic wave functions.
In stead of that,
we rewrite the Rayleigh component by a kernel representation,
which plays an important role in our approach.
As a result, we conclude
that the energy of the Rayleigh waves
concentrates in the domain
$\{ ({\bf x}',x_3)\in {\bf R}^3_+; \
\vert \vert {\bf x}' \vert -c_R t \vert < L', \ 0< x_3 < L\}$,
by obtaining various estimates in the complement of
the domain above, where $c_R$ is a velocity of the Rayleigh waves.
\begin{thebibliography}{99}
%
\bibitem{G}
J. C. Guillot,
{\it \lq\lq
Existence and uniqueness of a Rayleigh surface wave
propagating along the free boundary of a transversely
isotropic elastic half space"},
\rm Math. Meth. in the Appl. Sci.
{\bf 8} \rm (1986), 289--310
%
\bibitem{W}
C. H. Wilcox,
{\it \lq\lq Scattering Theory for the d'Alembert Equation in
Exterior Domains"},
\rm Lect. Notes in Math., no. 442, Springer, Berlin
\rm (1975)
\end{thebibliography}
\end{document}
