\documentstyle[twoside,psfig]{article}
\newcommand{\nazv}{}
\newcommand{\avtor}{}
\newcommand{\soavtor}{}

\def\refname{\underline{\normalsize References}}
\def\title#1{\begin{center}{\large\bf #1}\end{center}
             \renewcommand{\nazv}{#1}}

\renewcommand{\author}[2]{\renewcommand{\avtor}{#1}
\noindent\underbar{#1}\\#2\\}
\newcommand{\coauthor}[2]{\renewcommand{\soavtor}{ and #1}
\noindent\underbar{#1}\\#2\\}

\vsize 28.5cm
\hsize 20.5cm
\textheight 25cm
\textwidth  18cm
\hoffset -2.5cm
\voffset -2.5cm

\pagestyle{myheadings}
\markboth{DAY on DIFFRACTION'96}{DAY on DIFFRACTION'96}
\begin{document}

\title{INTERACTION OF STRUCTURAL AND ACOUSTICAL WAVES AT A
WEDGE-SHAPED JUNCTION OF TWO FLUID LOADED PLATES}
\author{A.N.Norris}{
Department of Mechanical and Aerospace Engineering, \\  Rutgers University,
Piscataway,  NJ \ 08855-0909, USA \\ email: norris@norris.rutgers.edu}

Two semi-infinite elastic plates are joined along a line forming a wedge
structure with unilateral fluid loading in the sector $-\Phi < \phi < \Phi$. The
structure is modeled using thin plate theory, allowing freely propagating
flexural and longitudinal waves. The junction is mechanically connected with
an applied force and moment acting there to simulate a possible internal
connection. The general 2D solution is described for incidence of time
harmonic structural or acoustical waves and numerical results are presented.

The general procedure for solving acoustic
diffraction from wedges with higher order boundary conditions
has only recently been developed \cite{Osi-94,Abr-95}, and has been
applied successfully to membrane boundaries \cite{Abr-95,ON}, and
also to plates.   Thus,   Osipov \cite{Osi-95} showed that
writing the acoustic field  as a  Sommerfeld integral transform
reduces the plate boundary conditions  to
functional difference equations for the pressure transform $S(\alpha)$.
He also provided an explicit procedure for solving these equations
using  Malyuzhinets functions and Fourier transforms.
The general solution has the form
\begin{equation}  S(\alpha)~=~\Psi(\alpha)
\, \left( P_0\, \frac{ \sigma(\alpha,\phi_0)}{\Psi(\phi_0)}\,
+ \Lambda(\alpha)\right) \, .  \label{1}\end{equation}
$P_0$ is the amplitude of an  acoustic of flexural wave incident at angle
$\phi_0$, where $\phi_0 \in (-\Phi,\Phi) $ for acoustic incidence and
$\phi_0$  is complex-valued for flexural incidence.
$\Psi(\alpha)$ is a combination of Malyuzhinets functions uniquely defined by the
material constants and the frequency,
and $\sigma(\alpha,\phi_0)$ contains a simple pole  which reproduces the wave
incident from $\phi = \phi_0$.  The additional function
$\Lambda(\alpha)$ is determined by the functional difference equations
and   involves eight unknown  coefficients.  Using the
inversion integral  of the Sommerfeld transform \cite{Mal-58b} it can be
shown that the coefficients  are  directly related to the
transverse displacement and its derivatives at the vertex.
A system of linear equations for the coefficients is
obtained by  applying (i) regularity conditions on $S(\alpha)$ yielding
four equations, and (ii)
the appropriate  junction conditions yielding four more, and thus completing
the solution.

\begin{figure}[htb]
\centerline{\psfig{figure=fig-1.eps,width=4.0in,height=4.0in}}

\caption{Diffracted structural energy for longitudinal incidence.  RL means
reflected longitudinal, TF means transmitted flexural, etc., and the total
radiated structural energy is F+L.} \label{fig1}
\end{figure}

In this paper we discuss the general solution for welded contact
at the point where the plates meet.  This case is of most
practical significance, but unfortunately, was not considered by
Osipov \cite{Osi-95}.   The  problem
involves six continuity conditions at the junction: two
for the two velocity components, two more for the force components,
and one each for the bending moment and the rotation.   It can be shown
that the six  continuity conditions can be
expressed solely in terms of the transverse plate displacements,
 $w_\pm (r) $ on $\phi = \pm \Phi$,   as a set of  four conditions \cite{Nor-95}:
\begin{eqnarray}
w_+^{\prime }(0)  + w_-^{\prime }(0)  = 0\, ,  ~~~~~~
B_+ w_+^{\prime \prime }(0)  - B_- w_-^{\prime \prime }(0)  &=&  M_0 \, ,
 \nonumber \\ &&  \nonumber \\
\frac{B_\pm}{i\omega} w_\pm^{\prime \prime \prime}(0)
  + Z_\pm  w_\pm (0) - \frac{ (Z_++Z_-) }{\sin^2 2\Phi }
\left( w_\pm (0) +w_\mp (0) \cos 2\Phi \right)
 &=&  \frac{F_r^\mp}{i\omega \sin 2\Phi}.     ~ \label{7}
\end{eqnarray}
Here,  $B_\pm$ is the bending stiffness, $Z_\pm $ is the longitudinal wave
impedance, and $e^{-i\omega t}$ is assumed.  The vertex can be
subject to an applied force $\bf F$ and moment $M_0$, and
 $F_r^\pm$ are the radial components of $\bf F$ along
$\phi = \pm \Phi$.   For acoustic or flexural wave incidence we have
$M_0= 0$ and $F_r^\pm = 0$.
Longitudinal waves traveling  inwards towards the junction on the
$\pm$plate can be considered by setting
$M_0= 0$ and
\begin{equation}
F_r^\pm = -2i\omega\, \left(
 Z_\pm  U_0^\pm +  Z_\mp  U_0^\mp \, \cos (2\Phi) \right) \, ,
\end{equation}
where  $U_0^\pm $ is the particle displacement of the incident
longitudinal wave  at $r=0$ in the radial direction.

\begin{figure}[htb]
\centerline{\psfig{figure=fig-2.eps,width=4.0in,height=4.0in}}

\caption{Diffracted structural energy for flexural incidence.} \label{fig2}
\end{figure}

It is interesting to note that
$P_0 = 0$  in (\ref{1}) if the excitation is an incident longitudinal wave,
but the four junction conditions   involve an
effective point force at the vertex, the amplitude of which is
related to the incident wave amplitude.   Thus, longitudinal
incidence is equivalent to the radiation problem with a source at the vertex.

\begin{figure}[htb]
\centerline{\psfig{figure=fig-3.eps,width=3.0in,height=3.0in}}
\caption{Diffracted structural energy for acoustic  incidence.} \label{fig3}
\end{figure}

The Figures  show some numerically computed results for the
particular case of identical steel plates in water
joined at the angle $\Phi =157.5^{\circ }$.   The
frequency $\omega_c$ is the flexural/acoustic  coincidence frequency.
The curves in Figures \ref{fig1}  and \ref{fig2} show the fraction of incident energy
that is converted into the different structural modes.
At low frequency the  total structural energy is conserved but is
significantly redistributed among the modes.   Figure \ref{fig3} shows the
fractional distribution of the diffracted structural energy
for acoustic plane wave incidence.
Further numerical computations indicate the coupling between the modes
for different wedge angles.  For example, we find that
acoustic plane wave
incidence on flat junctions ($\Phi \approx 90^{\circ }$) is converted almost
equally, in terms of energy, among diffracted flexural and longitudinal
waves. The coupling to flexural energy increases with the wedge angle, at
the expense of the longitudinal energy which vanishes as $\Phi \rightarrow
180^{\circ }$. An incident longitudinal wave generates relatively little
acoustic sound for all values of $\Phi $ considered, with most of its energy
redistributed among structural modes. The acoustical diffraction is
generally greater for flexural incidence.

Some of these trends can be understood by consideration of the special
cases (i): $\Phi = \pi$ for which a  Wiener-Hopf analysis leads to
relatively simple expressions for the acoustical and flexural diffraction
coefficients, and (ii) $\Phi \approx  \pi /2$, which can be analyzed by
perturbation methods, with the flat plate $ \Phi =  \pi /2$ solution as the
leading order term.

\begin{thebibliography}{10}
\bibitem{Osi-94} A.V. Osipov, ``General solution for a class of diffraction
problems", {\sl J. Phys. A: Math. Gen.} (UK), {\bf 27}, L27-L32 (1994).
\bibitem{Abr-95} I.D. Abrahams and J.B. Lawrie,
``Travelling waves on a membrane: reflection and transmission at a corner of
arbitrary angle. I'', {\sl Proc. R. Soc. Lond.} {\bf A 451}, 657-683 (1995).
\bibitem{ON} A. V. Osipov and A. N. Norris,
``Sound diffraction by a sharp ridge on a fluid-loaded membrane'',
 submitted to {\it Proc. R. Soc. Lond.}
\bibitem{Osi-95} A.V. Osipov, ``Sound diffraction by an angular joint
of thin elastic plates", submitted to {\sl Prikl. Matem. Mekh.} (Russia),
(1995).
\bibitem{Mal-58b} G.D. Malyuzhinets, ``Inversion formula for the Sommerfeld
integral", {\sl Sov. Phys. - Dokl.} (USA), {\bf 3}, 52-56 (1958).
\bibitem{Nor-95} A.N. Norris and A.V. Osipov,
``On acoustic interaction between two thin elastic plates through an
angular welded joint'',  submitted to  {\sl J. Sound Vib.} (1995).
\end{thebibliography}


\end{document}
