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\begin{document}



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\title{TRANSFORMATION MATRIX OF VIBRATIONAL WAVES
ON A NODAL JUNCTION OF PLATES}
\author{D.P. Kouzov, G.L. Nikitin, O.B. Agranova, V.G. Yakovleva
}{Marine Technical University}

We consider a rigid  nodal  junction  of $N$  semi-infinite  plates
linked along a straight line. On each plate we choose the  cartesian
coordinates ($x_{q},y)$, $q=1,2,\ldots
,N$ with common axis $Oy$. The coordinate $x_{q}$
in each plate is increasing with distance from the edge of a plate.

In each plate we mark a rectangle
$G_{q}: 0\le x_{q}\le a_{q}$, $-b\le y\le b$, $q=1,2,\ldots
,N$. These rectangles are forming together some construction $G$.
On the assumption that  the  external  surface forces  are  absent,  a
reciprocity principle (equality of mutual work) for  this  construction
can be written in the form
\par
$$
-i\omega \sum^{N}_{q=1}\int^{}_{\partial G_{q}}
\hskip -3pt({\bf f}_{q}({\bf u}_{q}),
{\bf \widetilde{u}}_{q})dl-i\omega \sum^{N}_{q=1}\sum^{4}_{j=1}
f_{qj}(w_{q})\widetilde{w}_{q}=
$$
$$
=-i\omega \sum^{N}_{q=1}\int^{}_{\partial G_{q}}
\hskip -3pt({\bf f}_{q}({\bf \widetilde{u}}_{q}),
{\bf u}_{q})dl-i\omega \sum^{N}_{q=1}\sum^{4}_{j=1}
f_{qj}(\widetilde{w}_{q})w_{q}.
\eqnum{1}
$$
Here: ${\bf u}_{q}=\left(\matrix{u_{q}\hbox{,}&v_{q}\hbox{,}&w_{q}
\hbox{,}&{\partial w_{q}\over \partial n}}\right)^{t}$
is generalized vector of displacement  (the
angle of rotation is taken as the fourth component
$), {\bf f}_{q}= (f_{qx}$, $f_{qy}$, $f_{qz}$,
$m_{q})^{t}$ is generalized vector of forces operating on the contour
$\partial G_{q}$\  of
the $q$-th plate (the bending moment is taken  as the fourth  component$),
f_{qj}(w_{q})$ is a normal corner point force, $j$  is  a  number  of  a
corner
point.  The  coordinates  of  displacements $(u_{q},v_{q},w_{q})$   and
forces
$(f_{qx},f_{qy},f_{qz})$ are measured in the  same  directions  as  the
cartesian
coordinates $(x_{q},y,z_{q})$. The axis $Oz_{q}$ for each plate is
chosen  so  that
the system of coordinates $(x_{q},y,z_{q})$ be right-hand.

The expressions for differential operators ${\bf f}_{q}$ and $f_{qj}$
depend of course on the chosen model of plates oscillations. Further
it was assumed that flexural oscillations of the thin plates are
described by
Kirchhoff's equation  and  symmetric  oscillations  are  described   by
Fylone's equations of the generalized flat theory of elasticity. Forces
and displacements of the second state are marked by a tilde.  The  time
dependence of processes is given by a factor $\exp (-i\omega t)$, which is
omitted throughout.

Let the incident linear wave ${\bf U}^{-}_{qj}(x_{q})$ with number $j$ in
the $q$-th plate serves as a source of the first field
$$
{\bf u}_{m}(x_{m},y)=\alpha e^{i\mu y}(\delta _{mq}{\bf U}^{-}_{qj}(x_{q})+
\sum^{4}_{l=1}\tau_{mqlj}(\mu ){\bf U}^{+}_{ml}(x_{m})).
\eqnum{2}
$$
We suppose that the dual field is excited by the $j^\prime $-th wave
incident along the $q^\prime $-th plate
$$
{\bf \widetilde{u}}_{m}(x_{m},y)=\beta e^{-i\mu y}(\delta _{mq^\prime }
{\bf \widetilde{U}}^{-}_{q^\prime j^\prime }(x_{q^\prime })+
\sum^{4}_{l^\prime =1}\tau_{mq^\prime l^\prime j^\prime }(-\mu )
{\bf \widetilde{U}}^{+}_{ml^\prime }(x_{m}))
\eqnum{3}
$$
The $y$-component of the  wave  vector  of
this wave is equal in magnitude and opposite in sign by comparison with
corresponding quantity of the first  field.  The  longitudinal,  shear,
flexural and decaying flexural waves are designated by the first subscripts
of 1, 2, 3 and 4 respectively.

The factors $\tau_{qq^\prime jj^\prime }$ determine a relative  excitation
degree of the $p$-th wave in the $q$-th plate,  when  the $p^\prime $-th
wave is incident along the $q^\prime $-th plate. The basic vectors
${\bf U}^{\pm }_{qp}$ can be  defined  (up to complex factor) from the
equations of plate oscillations.

The arbitrariness mentioned above leads to  the  arbitrariness  of
matrix elements $\tau_{qq^\prime jj^\prime }$  and  can  be  eliminated
with the help  of  the
reciprocity principle. Applying the procedure suggested in [1] to the
formulae (1), (2) and (3) we receive the equality
$$
\tau_{q^\prime qj^\prime j}(\mu )\varphi ^{+-}_{qj^\prime j^\prime }+
\tau_{qq^\prime jj^\prime }(-\mu )\varphi ^{-+}_{qjj}=0.
$$
where $\varphi ^{+-}_{qj^\prime j^\prime }$ are some factors.

The substitution corresponding to normalization of the basic vectors
$$
t_{mqlj}(\mu )=\tau_{mqlj}(\mu )\sqrt{{\varphi ^{+-}_{mll}\over
\varphi ^{+-}_{qjj}}} .
$$
transforms the above equality into the following form
of the reciprocity principle
$$
t_{q^\prime qj^\prime j}(\mu )=t_{qq^\prime jj^\prime }(-\mu ),
$$
or in the matrix form
$$
T^{t}(\mu )=T(-\mu ).
$$
The matrix $T=\{t_{qq^\prime jj^\prime }\}$,  ($q,q^\prime =1,2,\ldots
,N$; $j,j^\prime =1,2,3,4)$  will  be
called the transformation matrix. The  factor $t_{qq^\prime jj^\prime }$
as  well  as  the
factor $\tau_{qq^\prime jj^\prime }$ specifies a relative excitation
degree  of  the $j$-th
wave in the $q$-th plate, if the $j'$-th wave is incident along the $q'$-th
plate.

This normalization of basic vectors is carried out in   particular
in such a manner that for the travelling waves the value
$\mid t_{qq^\prime jj^\prime }\mid ^{2}$  is
equal to the part of energy, transferred by the $j$-th wave along the $q$-th
plate.

If we omit those rows and columns in the matrix $T$ which correspond
to decaying waves, then (in the  absence  of  losses  in  materials  of
plates) we obtain the unitary matrix.

At last, in the case when the decaying wave serves as a source  of
field, we have found that the conservation of  energy  yields  one  new
property of the transformation matrix
$$
2\,{\rm Im}\,t_{ppss}=\sum^{N}_{m=1}\sum^{}_{j}\mid t_{mpjs}\mid ^{2}.
\eqnum{4}
$$
By analogy with well-known optical theorem it is natural to  call  this
equality ``acoustic theorem''.

In the right-hand side of  the  formula  (4)  the  summation
extends over all travelling waves.  Thus  the  left-hand  side  of
equality (4) represents an energy flow transferred in  result  of
interaction of the incident and reflected decaying waves.

\begin{thebibliography}{99}
\bibitem{kouzov1} Kouzov D.P., Yakovleva V.G. On the reciprocity
principle  for
the transformation of linear waves at a rectilinear  plate  edge.
Acustica, 81 (1995), No. 2, 167-172.
\bibitem{Kouzov2}
Kouzov D.P., Kravtsova T.S., Yakovleva V.G. On  scattering  of
vibrational  waves  at  a  node   joint   of   plates.   Soviet
Physics-Acoustics 35 (1989), No. 4.
\end{thebibliography}
\end{document}
