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\begin{document}



\title{ON COMPUTATION OF FIELDS OF LOW-FREQUENCY INTERFERENCE WAVES}
\author{N.S.Zabavnikova, Yu.A.Surkov}{St.Petersburg
Branch of Steklov Mathematical Institute \\
Fontanka 27, 191011, St.Petersburg, Russia}

     We consider the problem on the excitation of a plane displacement field
$\boldmath u_q\,=\,u_q(x,z,t)\boldmath j$ of the type  SH  by a surface action of
the form
$$\mu_1\frac{\partial \boldmath u_1}{\partial z}\bigg|_{z=0}\,=\,-f(t)\delta(x)
\boldmath j,                                                        \eqno (1) $$
turned on at the moment $t\,=\,0$, in a medium involving an elastic (thin)
layer $0<z<h \ \ (q=1) $ and the elastic half-space $z>h \ \ (q=2) $
that are in welded contact with each other. \par
    In the cases of a low-frequency dependence of a source upon the time
$$f(t)\,=\,\hbox{Re} \frac1{\pi}\int_0^{\omega_0} F(i\omega)e^{i\omega t}
d\omega,                                                       \eqno(2) $$
at the points $x>0$  of the profile on the surface $z=h+H$ inside
the half-space $z>h \ \ (q=2)$, the displacement field is determined by
the Fourier integral
$$u_2(x,z,t)\,=\,\hbox{Re}\frac1{\pi\mu_1} \int_0^{\omega_0} V_2(x,z,\omega)
F(i\omega)e^{i\omega t} d\omega,                                  \eqno(3) $$                                  
where the integrand involves the function   $V_2$      represented by the contour
integral
$$V_2 \,=\,\frac1{2\pi}\int_{(\lambda)} \frac1{\Delta_0(\zeta,k)}e^{-k\varphi_0
(\zeta,\tilde x)} d\zeta.                                \eqno  (4)$$
Here we denote
$$\Delta_0(\zeta,k)\,=\,\alpha\hbox{sinh}k\alpha\,+\,\sigma\beta\hbox{cosh}k\alpha,$$
$$\alpha=\sqrt{\zeta^2-1},\,\beta=\sqrt{\zeta^2-\gamma^2},\,\gamma=v_1/v_2,\,
\sigma=\mu_2/\mu_1,$$
$$\varphi_0=i\tilde x\zeta+\tilde H\beta,\, \tilde x= x/h,\,\tilde z=
z/h,\,\tilde H=H/h,\,k=\frac{\omega}{v_1}h.\eqno (5)$$ \par
On the complex $(\zeta)$ plane, cuts are drawn from the branch points of 
the radicals along the real axis to infinity without crossing the point 
$\zeta=0$. The branches of the radicals on the principal sheet of the 
$(\zeta)$ plane are specified by the conditions $\hbox{arg}\alpha=
\hbox{arg}\beta=\pi/2$ for $\zeta=0$.\par 
    The integration path $\lambda$ in the inner integrals starts at the 
point $\zeta=-\infty -i\delta,\,(\delta\ge 0)$ of the third quadrant, 
passes through the origin of coordinate $\zeta=0$ into the first quadrant 
and terminates at the point $\zeta=\infty+i\delta$ at infinity.\par 
   The field of displacement $u_2(\tilde x,t)$ at the points $\tilde x>0$ 
of the profile $z=h+H$ in the half-space $z>h$ given by (1) and (2) is of 
interest for the problem associated with the elucidation of mechanisms of 
the passage of low-frequency waves (disturbances) through thin layers 
in the case where the ray description of the propagation process 
is inapplicable to them.  Furthermore, the thin layer in the vicinity of 
the point of action can be considered the simplest model of the "upper 
part of a seismic section" (UPS) of a layered structure.  Therefore, the 
study of the field $u_2(\tilde x,t)$ is expected to contribute to the 
understanding of the influence of the UPS on the initiation of body waves 
in seismic media.  This problem was also discussed in [\,1\,] and [\,2\,] .

\begin{thebibliography}{9}
\bibitem{z1} G.I.Petrashen, B.M.Kashtan, and Yu.V.Kiselev. Quantitative study of
   nonstationary interference wave fields in layered homogeneous elastic media
   with plane-parallel interfaces. I. Statements of the problems and efficient
   methods for their solution. Zap.nauch.semin.POMI, vol.214, 1994, 7-186.
\bibitem{z2} Yu.V.Kiselev. Algorithms for evaluating basic components of spectral
   functions of interference wave fields. Zap.nauch.semin.POMI, vol.214,
   1994, 187-199.

\end{thebibliography}
\end{document}
