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\title{THEORY OF ELASTIC VIBRATIONS IN ISOTROPIC INHOMOGENEOUS LAYERED MEDIA}
\author{S.G.Kiselev, A.N.Kuznetsov, V.M.Markushevitch}
{International Institute of Earthquake Prediction Theory \\
 and
Mathematical Geophysics, Russian Academy of Sciences}


 We consider isotropic elastic media which are plane, spherically or
cylindrically layered. It means that their parameters Lame $\lambda $,
$\mu $ and density $\rho$ in plane case depend only on a depth, and in the
other two cases, after transition to appropriate coordinates, only on
radius. The small vibrations of such media are subject to linear equations
of elasticity
\begin{eqnarray}
\rho \ddot v_i = ( \lambda v_ { k, k } ) _ {, i } + ( \mu v_ { i, j } ) _ {,
                     j } + ( \mu v_ { j, i } ) _ {, j },
\label{mark1}
\end{eqnarray}
where $v_i$, $i=1, ~2, ~3$ - components of a vector of displacement.

The conditions on a boundary and fields of external forces are not
considered.

 After separation of variables and decomposition of a general solution
into axially symme\-tri\-cal ones the system is transformed into a system
of ordinary equations with two parameters: frequency $\omega$ and wave
number $\xi$, which consists of equation for SH vibrations and
two-dimensional system for P-SV vibrations. The latter has the form:
\begin{eqnarray}
Pu=0, \quad where \;\; P=\partial ( B\partial + C) - C^T \partial + E,
\label{mark10}
\end{eqnarray}
where
$\partial = { \displaystyle \frac { d } { dx }} $\,; the products are
everywhere understood as compositions of the operators, that is, $\partial
A = A\partial + A^\prime$,\, $ { \displaystyle ( \; ) " \equiv \frac { d (
\; ) } { dx }} $.
This system is just a subject of research.

{ \it In plane layered case } $B = diag ( \nu,\mu ) $, \\
$\;\;$ $C=\xi\left( \begin{array}{rc} 0 \;& \;\lambda \\-\mu \;& \;0
\end{array}
\right ) $, $E = diag ( \omega^2 \rho- \xi^2 \mu,\;\;
\omega^2 \rho- \xi^2 \nu ) $, \\ where $\nu=\lambda + 2\mu$, $\omega$ -
frequency, $\xi$ - wave number.

Our results cover all three cases; however, the formulas will be presented
only for the plane one. The noted three types of layered media are
distinguishable from other layered media, for instance, from toroidal and
ellipsoidal, since complete separation of variables in equation (\ref{mark1})
can be achieved for them and only for them.

 The main result of this work consists in that the system (\ref{mark10}) can
be transformed into the matrix Sturm-Liouville's form, in which only
eigenvalue $\xi^2$ depends on $\xi$; then the part of operator of the
system which is independent from $\xi$ explicitly decomposes in product of
the differential operators of the first order, free terms of which have
matrices with zero traces.




 Let Int$_M (L) \stackrel {{ \rm def }} { = } M^ { -1 } LM$, where $M$ and
$L$ are matrices or the matrix differential operators with nonsingular $M$.

{ \it Proposition 1. In all three cases of layered structure there are
matrices $M_1$, $M_2$, $K$, $L$ and operator $Q$, which are explicitly
expressed through parameters of media, and such that the identity holds:

\begin{eqnarray}
   && P_1 \stackrel {{ \rm def }} { = } { \rm Int } _ { M_1 } \bigl ( M_2^ {
-1 } { \rm Int } _Q (P) \bigr ) = ( \partial -L + K ( \partial -L-K ) - \xi^2
\label{11}
\end{eqnarray}


Here in plane case: \\
$M_1=
\left( \begin{array}{cc} \kappa \mu^{-1} &   \; \;  0 \\    0 &
   \; \; \mu \nu^{-1}
\end{array} \right),
\; M_2 = \left(
\begin{array}{cc}
      \mu
   &   \; \;  0
   \\    - 2\mu^{\prime}
   &   \; \;  \nu
\end{array}
\right), \;
Q = \left(
\begin{array}{cc}
-\xi \;\; & \;\;0\\
\partial \;\;& \;\;1
\end{array}\right),$
\begin{eqnarray}
K= &&
\left(\begin{array}{cc}
      \mu^{-1}\mu^{\prime} &   \; \;  - \frac {1}{2}
   \kappa^{-1}\mu\nu^{-1}(\lambda+3\mu) \\ \kappa(\mu^{-2}\omega^2 \rho+
   (\mu^{-1})^{\prime\prime}) &   \; \;     - \mu^{-1}\mu^{\prime}
\end{array} \right) ,
\nonumber
\end{eqnarray}
\begin{eqnarray}
L= &&
\left(\begin{array}{cc} 0 &
   \; \;   \frac {1}{2} \kappa^{-1}\mu\nu^{-1}(\lambda+\mu) \\ -\kappa
   (\mu^{-1})^{\prime\prime} &   \; \;     0 \end{array} \right).
\nonumber
\end{eqnarray}

Here $\kappa $ is an arbitrary constant, its own for each case,
$\nu=\lambda + 2\mu$.  }

 Depth plays the role of independent variable $x$ in plane case; in other
cases $x$ is connected with $r$ by the formula $r=r_0 {\bf e}^{-x/x_0}$.

 The matrix $M_1$ is chosen to annihilate all members in the matrix
$L$ , which can be nullified by an explicit transformation.

 The following result, which can be called the theorem about
interpretation of the operators of a form (\ref{11}) with arbitrary
matrices $K$, $L$ (subject to the condition, that $ { \rm sp } (K) =0$ )
by means of the operators of elastic vibrations, opens up an opportunity
of the solution of Earth-flattening problem and is necessary for the
solution of inverse problems. The well known in seismology problem of
Earth-flattening  is  stated as follows: to reduce the solution of the
equation (\ref{mark10}) for the spherical body to the solution of the same
equation for a layered half space which is specially constructed to match
the body.

  Consider the operator $\widehat { P } = ( \partial -\widehat { L } +
\widehat { K } ) ( \partial -\widehat { L } -\widehat { K } ) - \xi^2$, in
which $\widehat { K } $ and $\widehat { L } $ are some matrix functions of
$x$.

{ \it Proposition 2. Let $ { \rm sp } ( \widehat { K } ) =0$. Then in some
vicinity of each point $x=x_0$ at fixed value of
frequency $\omega$ there exists the matrix transformation $H (x) $,
parameters of media $\lambda (x) $, $\mu (x) $, $\rho (x) $ and defined by
them in accordance with (\ref{11}) operator $P_1$,, such that $ { \rm Int
 } _H ( \widehat { P } ) =P_1$.  The solution depends on four arbitrary
 constants ( $\kappa$ is among them). }

 It should be pointed out, that without additional considerations it is
impossible to expect, that the determined $\lambda (x) $, $\mu (x) $ and
$\rho (x) $ will be positive and, especially, appropriate to some physical
media.  However, if the operator $P_1$ (\ref{11}) derived from $P$ in
spherically layered case is taken as $\widehat { P } $, the following
solution of Earth-flattening problem can be suggested:

{ \it Proposition 3. For any point of a sphere which is reasonably far
from a center there is a spherical layer containing this point and
admitting plane layered model with physically allowable parameters. For
any point of radius such layer exists also, but the density $\rho_f$ of
plane layered model can be outside a physically reasonable range. }

 This flattening solution can be expanded into a series with respect to
$r_0^ { -1 } $ and as a first term one can choose any initial
approximation, which were considered by Biswas and Chapmen:

\begin{eqnarray*}
\mu=r_0^ { -m } r^m\mu_s,\;\lambda=r_0^ { -m }r^m \lambda_s,\;
\rho=r_0^{-m-2}r^{m+2}\rho_s
\end{eqnarray*}

 The second term of the asymptotic series is received in an explicit form,
it depends on the solution of some four-dimensional  linear system.

 We studied the uniqueness of the representations. It was found that
the operator $P_1$ can be explicitly transformed to a perturbed wave
operator form:

\begin{eqnarray} P_2= ( \partial -
\tilde { L } + \tilde { K } ) ( \partial
-\tilde { L } -\tilde { K } ) + \omega^2 F-\xi^2,
\label{22}
\end{eqnarray}
 Here the matrix $F$ is diagonal, and matrices $\tilde { K } $ and $\tilde
{ L } $  do not depend on frequency.

 This result allows to drew an analogy of our theory and classical method
of elastic potentials, where the equations of elasticity in homogeneous
media are directly reduced  to a wave equation.

 The form (\ref{22}) is almost canonical, it admits of only the
transformations by subgroup of constant diagonal matrixes. It can be used
for study of dependence of the solutions and Earth-flattening
transformation on frequency, including derivation of high-frequency
asymptotics.




\begin{thebibliography}{99}

\bibitem{???-}\hspace{-6pt}.
{ \it S.G.\, Kiselev, A.N.\, Kuznetsov, V.M.\, Markushevitch,
A.S.\, Tsemachman. }
On equations of elastic waves in layered media allowing separation of
 variables .
// Due to appear in Computational Seismology, issue 29.

\bibitem{???'} { \it S.G.\, Kiselev, A.N.\, Kuznetsov, V.M.\,
Markushevitch. } Earth-flattening problem : an origin,
methods of the exact solution and expansion into a series.
// Due to appear in Computational Seismology, issue 29.

\end{thebibliography} 












\end{document}
