WAVE FRONT SETS OF
SOLUTIONS
TO ELASTICWAVE
PROPAGATION PROBLEMS
INSTRATIFIED MEDIA
SENJO
SHIMIZU
(清水 扇丈)Faculty of
Engineering, Shizuoka
University,Hamamatsu 432-8561
1. Introduction
We considerelasticwavepropagationproblems in plane-stratified media$\mathrm{R}^{3}$with
the planer interface $x_{3}=0$. This problem is formulated as
an
elastic mixed orinitial-interfece problem in a stratified media.
An elastic equation has two speeds. Pressure or Primary wave (for simplicity
called $\mathrm{P}$ wave) and Share or Secondary wave
($\mathrm{S}$ wave). $\mathrm{P}$ wave is a longitudinal
wave and $\mathrm{S}$ wave is a transversal
wave. In general the speed of $\mathrm{P}$ wave is
grater
than that of $\mathrm{S}$
wave. In plane-stratified media problem, a lower half-space $\mathrm{R}_{-}^{3}$
called Medium $I$has $P_{1}$ and $S_{1}$ waves and an upper half-space $\mathrm{R}_{+}^{3}$ called Medium
II has $P_{2}$ and $S_{2}$ waves. The speed of $P_{1}$ (resp. $P_{2}$) wave is grater than that of$S_{1}$
(resp. $S_{2}$ ) wave. So the order relation of the speeds of
$P_{1},$ $P_{2},$ $S_{1}$, and $S_{2}$
waves
are six
cases.
Herewe
assume
$P_{2},$ $S_{2,1}P,$ $S_{1}$ waves in order of speedsince it is themost complex
case.
We put unit impulse Dirac’s delta in the lower half-space Medium I. Then $P_{1}$
incident wave which speed is faster than $S_{1}$ incident wave bumps against the
in-terface andcauses $P_{1}$ and $S_{1}$ reflected waves in Medium I and $P_{2}$ and $S_{2}$ refracted
waves in Medium II as in Figure 1.
$X\mathrm{a}$
Figure 1 Reflected waves and refracted waves
Moreover when time goes on, lateral waves in other words glancing waves or
total
reflected
(or refracted)waves
arise. In Figure 2, dottedarrows
show $P_{2^{-}}P_{1}$and $P_{2^{-}}S_{1}$ lateral waves in Medium I, and $P_{2^{-}}S_{2}$ lateral wave in
Medium
II for1991 Mathematics Subject Classification. Primary $35\mathrm{L}67$; Secondary $73\mathrm{C}3535\mathrm{E}9935\mathrm{L}20$
.
This work was supported in part by $\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{S}^{- \mathrm{i}}\mathrm{n}$ Aid for Encouragement
of Young Scientists
(grantA-09740098) from the Ministry of Education, Science and Culture of Japan.
$P_{1}$ incident
wave.
$P_{2^{-}}P_{1}$ lateral wave means that the wave originally should havebeen $P_{2}$ reflected
wave
tends to total reflection, then becomessource
andcauses
$P_{1}$reflected wave.
Figure 2 Lateral waves
We have 11th kind of lateral waves in all. It is a characteristic of our elastic
wave propagation problems in stratified media. If half-space problem which has
two speeds, there exist only one kind of lateral wave. If plane-stratified media
problem that each medium has one speed, there exist onlyonekind oflateral wave.
Thus this elastic wave propagation problems in plane-stratified media has many
lateral waves.
In this paper we prove the above physical situation mathematically by using a
expression of an inner estimate of singularities. The main technical tool of our
analysis is alocalization method.
We
gave
an inner estimate of the location ofsingularities of the reflected andre-fracted Riemann functions by making use of the localization method $[7,8]$. This
method is first studied by M. F. Atiyah, R. Bott, and L. $\mathrm{G}^{\mathrm{Q}}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}[1]$ and L.
H\"ormander [2] for initial value problem, then studied by M. Matsumura [5], M.
Tsuji [9], and S. Wakabayashi $[10,11]$ for half-space mixed problem.
In this paper, we give an outer estimate of wave front sets of the incident,
reflected and refracted Riemann functionsbymaking use ofthelocalization method.
$\mathrm{A}\mathrm{t}\mathrm{i}\mathrm{y}\mathrm{a}\mathrm{h}- \mathrm{B}\mathrm{o}\mathrm{t}\mathrm{t}- \mathrm{G}^{\mathrm{o}}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}[1]$ studied theouter estimate ofwavefront sets of solutions to
initial value problem. Wakabayashi studied for half-space mixed problem [12], and
for more general case [13]. We analysis an outer estimate of wave front sets ofthe
Riemann functions to the elastic mixed problem based on Wakabayashi’s theorem
[12, Theorem 4.2]. Combiningthe innerestimate and the outer estimate, weobtain
the exact wave front sets of the elastic mixed problem in stratified media.
I would like to express my gratitude to Professor Seiichiro Wakabayashi for his
invaluable suggestions.
2. Formulation ofProblems
We consider elastic wave propagation problems in the following plane-stratified
media $\mathrm{R}^{3}$ with the planar interface $x_{3}=0$:
$(\lambda(x_{3}), \mu(X_{3}),$$\rho(_{X))=}3\{$
$(\lambda_{1}, \mu_{1}, \rho_{\mathrm{l}})$ for $x_{3}<0$,
($\lambda_{2,\mu_{2},\rho_{2})}$ for $x_{3}>0$
.
Here the constants $\lambda_{1},$ $\lambda_{2},$
$\rho_{1},$ $p_{2}$ are densities.
We
shalldenote the lower half-space $\mathrm{R}_{-}^{3}$ byMedium
$I$and the upper half-space $\mathrm{R}_{+}^{3}$ by Medium II, respectively, as in Figure 3.Figure 3 Stratified media
We
assume
that(1.1) $\lambda_{i}+\mu_{i}>0$, $\mu_{i}>0$, $\rho_{i}>0$, $i=1,2$.
(1.1) is the natural assumption in practicalsituation. From the ro$o\mathrm{t}\mathrm{s}$ofthe
charac-teristic equationsof$P^{I}(D)$ and$P^{II}(D)$ which aredefined below $3\cross 3$ matrix valued
hyperbolic partial differential operators in Medium I and Medium II, respectively,
weobtain two speeds correspondto Pressure or Primary wave (for simplicity called
$\mathrm{P}$ wave) and Share or Secondary wave ($\mathrm{S}$ wave) on
each medium. $\mathrm{P}$
wave
is alon-gitudinal wave and $\mathrm{S}$ wave is a transversal wave.
$c_{\mathrm{P}1}$ denotes the speed of
$\mathrm{P}$ wave
in Medium I and $c_{s_{1}}$ denotes the speed of
$\mathrm{S}$ wave
in Medium I. $c_{p_{2}}$ and $c_{s_{2}}$ denote
the speed of$\mathrm{P}$ and $\mathrm{S}$
wave in Medium II, respectively. They are given by $c_{p_{i}}^{2}= \frac{\lambda_{i}+2\mu_{i}}{p_{i}}$, $c_{s_{i}}^{2}=$
.
$\frac{\mu_{i}}{p_{i}}$, $i=1,2$
.
By assumption $(1.\perp)$, the speed of $\mathrm{P}$ wave is greater than that of $\mathrm{S}$ wave
in each
medium. On account of this, these are six cases of the order relation of the speeds
of$\{cc, cc\}p_{1}’ s1p_{2’ S2}$
.
Herewe assume that$c_{s_{1}}<c_{\mathrm{P}1}\leq c_{s_{2}}<c_{p_{2}}$,
since ifwe put an unit impulse Dirac’s delta in Medium I, it is the
case
that themost number of lateral waves are appeared. The other cases can be treated in a
similar
manner
(cf. [6, Section 3]).Let $x=(x_{0,1,2,3}xxx)=(x’, x\mathrm{s})=(x_{0}, x^{\prime/})=(x_{0,3}x^{\prime//}, x)$ in $\mathrm{R}^{4}$. The variable
$x_{0}$ will play a role of time, and $x^{\prime/}=(x_{1,}.x_{2}, X_{3})$ will play that of space. $\xi$ is
a
realdual variable of$x$ andis equal to $(\xi_{0}, \xi_{1}, \xi 2, \xi_{3})=(\xi’, \xi_{3})=(\xi_{0}, \xi/’)=(\xi_{0}, \xi^{\prime/}/, \xi_{3})$ in
$\mathrm{R}_{\xi}^{4}$. We use the differential symbol $D_{j}=i^{-1}\partial/\partial X_{j}(j=0,1,2,3)$, where $i=\sqrt{-1}$
.
We shall denote by $\mathrm{R}_{-}^{n}$ the half-space $\{x=(x_{1}, \cdots, x_{n})\in \mathrm{R}^{n}|x_{n}<0\}$ and by
$\mathrm{R}_{+}^{n}$ the half-space $\{x=(x_{1}, \cdots, x_{n})\in \mathrm{R}^{n}|x_{n}>0\}$, and also
use
the notation $|x|$$=\sqrt{x_{1}^{2}+x_{n}^{2}}$
.
Let $u(x)={}^{t}(u_{1}(x), u2(x),$$u_{3}(x))\in \mathrm{R}^{3}$ be the displacement vector at time $x_{0}$
and position $x^{\prime/}$
.
Thepropagation
is formulated as mixed (initial-interface value) problem:
Here
$P^{I}(D)u=-D^{2}Eu0+ \frac{\lambda_{1}+\mu_{1}}{p_{1}}\nabla_{x’’}(\nabla_{x^{;\prime}}\cdot u)+\frac{\mu_{1}}{\rho_{1}}\Delta_{x’’}u$,
is a $3\cross 3$ matrix valued secondorder hyperbolic differential operator with constant
coefficients where $E$ is a 3 $\mathrm{x}3$ identity matrix,
$(B^{I}(D)u)k=i\lambda_{1}(\nabla x’’\cdot u)\delta_{k3}+2\mu_{1}\epsilon k3(u)$, $k=1,2,3$ ,
are the k-th component of symmetric stress tensors of$B^{I}(D)u$ where
$\epsilon_{k3}(u)=i/2(D_{3}u_{k}+D_{k^{\prime u}}3)$ , $k=1,2,3$,
are strain tensors. The $P^{II}(D)u$ and $B^{II}(D)u$ are defined by replacing $\lambda_{1},$
$\mu_{1},$ $\rho_{1}$ by $\lambda_{2},$
$\mu_{2},$ $p_{2}$
,
respectively.If
we
put unit impulse Dirac’s delta $\delta(x-y)$ with $x_{0}\geq y_{0}$ and $y_{3}<0$, thatis, put it in Medium I, then the Riemann function ofthis elastic mixed problem is
given by the following:
$G(x, y)=\{$
$E^{I}(x-y)-F^{I}(x, y)$ for $x_{3}<0$,
$F^{II}(x, y)$ for $x_{3}>0$.
Wecall $E^{I}(x-y),$ $FI(x, y)$, and $F^{II}(x, y)$ the incident, reflected, and refracted
Rie-mann functions, respectively, becausethesearecorresponding to incident, reflected, and refracted waves, respectively. $E^{I}(x)$ is the fundamental solution in Medium I
describing an incident wave defined by
$E^{I}(x)=(2 \pi)^{-4}\int_{\mathrm{R}_{\xi}^{4}}e^{ix}.P(\xi+i\eta)I(\xi+i\eta)-1d\xi$, $\eta\in-\gamma 0\theta-\Gamma(\det P^{I}, \theta)$,
where $\gamma_{0}$ is a positive real number,
$\theta$ and $\Gamma(\det P^{I}, \theta)$ are defined Definition 1.4
below, and $P^{I}(\xi+i\eta)^{-1}$ is a $3\cross 3$ inverse matrix. Taking partial Fourier-Laplace
transform with respect to $x’$ for the mixed problem, we obtain a interface value
problem for ordinary differential equation with parameters. Then taking partial
inverse Fourier-Laplace transform for the solution, we obtain explicit expressions
of reflected and refracted Riemann functions $F^{I}(x, y)$ and $F^{II}(x, y)$.
We define a wave front set $WF(u)$ and a analytic
wave
front set $WF_{A}(u)$ (cf.Definition
1.1.
Let $u(x)\in D’(X)$. Then the wave front set $WF(u)$ is defined asthe complement in $X\cross(\mathrm{R}^{n}\backslash \{0\})$ of the collection of the points $(x^{0}, \xi 0)$ such that
there exist a conic neighborhood $\triangle$ of$\xi^{0}$ in $\mathrm{R}^{n}\backslash \{0\}$ and $\phi\in C_{0}^{\infty}(X)$ such that
$\phi(x^{0})\neq 0$ and
$|\mathcal{F}[\phi u](\xi)|\leq C_{N}(1+|\xi|)^{-N}$ when $\xi\in\triangle \mathrm{a}\mathrm{n}\mathrm{d}$ $N=0,1,2,$ $\cdots$
Here $\mathcal{F}$ denotes the Fourier transform.
For the definition of a analytic wave front set $WF_{A}(u)$, we prepare that there
exist a bounded sequence $\{\phi_{N}\}$ in $C_{0}^{\infty}$ such that $\phi_{N}=1$ on a fixed neighborhood
of$x^{0’}$ in $X$
, independent of$N$, and
$|D^{\alpha}\phi_{N}|\leq C(CN)^{1}\alpha|$ for $|\alpha|\leq N$.
Definition 1.2. Let $u(x)\in D^{/}(X)$. Then the analytic wave front set $WF_{A}(u)$ is defined as the complement in $X\cross(\mathrm{R}^{n}\backslash \{0\})$ of the collction of the points
$(x^{0}, \xi 0)$ such that for some sequence $\{\phi_{N}\}$ of the above type there exists a conic
neighborhood $\triangle$ of$\xi^{0}$ in $\mathrm{R}^{n}\backslash \{0\}$ with
$|\mathcal{F}[\phi_{N}u](\xi)|\leq C(CN)^{N}(1+|\xi|)^{-N}$ when $\xi\in\triangle$, $N=0,1,2,$$\cdots$
By Definition 1.1 and Definition 1.2, we obtain
$WF(u)\subset WF_{A}(u)\backslash$,
and $WF(u)$ and $WF_{A}(u)$ are closed subsets of$X\cross(\mathrm{R}^{n}\backslash \{0\})$.
We define a localization of polynomials according to $\mathrm{A}\mathrm{t}\mathrm{i}\mathrm{y}\mathrm{a}\mathrm{h}- \mathrm{B}\mathrm{o}\mathrm{t}\mathrm{t}_{- \mathrm{G}^{\circ}\mathrm{r}\mathrm{d}}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{g}$ (cf.
[1]$)$:
Definition 1.3. Let $P(\xi)$ be a polynomial ofdegree $m\geq 0$ and develop
$l\text{ノ^{}m}P(l\text{ノ}-1\xi+\eta)$ in ascending power of $\nu$:
$\nu^{m}P(\nu^{-1}\xi+\eta)=\nu^{p}P_{\xi}(\eta)+o(_{\mathcal{U}}p+1)$ as J ノ $arrow 0$,
where $P_{\xi}(\eta)$ is the first coefficient that does not vanish identically in
$\eta$
.
Thepoly-nomial $P_{\xi}(\eta)$ is the localization of $P$ at $\xi$
,
the number$p$ is the multiplicity of $\xi$
relative to $P$
.
Moreover we introduce the following:
Definition 1.4. $\Gamma=\Gamma(P, \theta)$ is the component of$\mathrm{R}_{\eta}^{n}\backslash \{\eta\in \mathrm{R}_{\eta}^{n}, P(\eta)=0\}$ which
contains $\theta=(1,0, \cdots, 0)\in \mathrm{R}^{n}$. Moreover $\Gamma’=\mathrm{r}^{l}(P, \theta)=\{x\in \mathrm{R}^{n}|x\cdot\eta\geq 0,$
$\eta\in$
$\Gamma\}$ is the dual cone of $\Gamma$ and is called the propagation cone.
3.
Results
We obtain the exact wave front sets of the elastic mixed problem in stratified
media by combining an inner estimate and
an
outer estimate. First we mentionabout the results of the incident Riemann function, namely
fundamental
solutionofMediumI $E^{I}(x)$
.
This proposition is aversionofthe theorem provedbyProposition. For$\xi^{0}\in \mathrm{R}_{\xi}^{4}\backslash \{0\}$ satisfying $(\det P_{j}^{I})(\xi^{0})=0(j\in\{p_{1}, s_{1}\})$, that is, $(\det P_{p_{1}1}^{I})(\xi 0)=\xi_{0}^{0^{2}}-C_{\mathrm{p}}2|\xi 0//|^{2}=0$,
$or$
$(\det P_{s_{1}}^{I})(\xi^{002})=\xi 0^{2}-C|s_{1}\xi 0//|^{2}=0$,
then we have
$\lim_{\nuarrow\infty}\nu e^{-}i\nu x\cdot\xi 0IE(X)=E_{j}I(_{X}\xi 0)$, $j\in\{p_{1}, s_{1}\}$,
in the distribution
sense
with respect to $x\in \mathrm{R}^{4}f$ where$E_{j\xi^{0}}I(X)=(2 \pi)-4\int \mathrm{R}_{\xi}4eix\cdot(\xi+i\eta)\frac{(_{\mathrm{C}\mathrm{o}\mathrm{f}}P^{I})_{\xi^{0}}(\xi+i\eta)}{(\det P^{I}j)_{\xi}\mathrm{o}(\xi+i\eta)}d\xi$
for $\eta\in-\theta-\Gamma(\det PI\theta)j$
’ and $j\in\{p_{1}, s_{1}\}$
with a positive real $s$ large enough. Moreover we have
$WF(E^{I}(_{X)})= \bigcup_{\xi^{0}\neq 0}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}^{E}p_{1}\xi^{0}(_{X})\cup \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}E0\xi(s1)X)II\cross\{\xi^{0}\}$,
and
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}E_{j\xi}^{I}0(x)=(\Gamma_{j\xi^{\mathrm{O}}})’=\{x\in \mathrm{R}^{4}$ : $x\cdot\eta\geq 0$ for any $\eta\in \mathrm{r}_{\mathrm{i}\xi^{0}}$
$u=\Gamma((\det P_{j}^{I})_{\xi}\mathrm{o}(\eta), \theta)\}$, $\theta=(1,0,0,0)$, $j\in\{p_{1}, s_{1}\}$.
In general, $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}^{E_{j\xi}^{I}}\mathrm{o}(x)\subset(\Gamma_{j\xi^{0}})^{/}(j=\{p_{1\mathcal{J}}.s_{1}\})$, more precisely $ch[\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}E_{j\xi}^{I}0(x-$
$y)]=(\mathrm{r}_{j\xi^{0}})/$, where $ch$ denotes
convex
hull. However in our problem we obtain$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}E_{j\xi^{0}}^{I}(x)=(\Gamma\xi^{0})j/$ .
Secondly we mention about main result.
Since
we take a partial Fourier-Laplacetransform with respect to $x’$ of $\delta(x-y)$ regarding $y’$ as a parameter, $y’\mathrm{a}_{\mathrm{P}\mathrm{P}^{\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}}}$
only in the form $x’-y’$. So we put
$\tilde{F}^{\iota}(x, y_{3})=F^{L}(_{X,0,)}y3,$ $\iota=\{I, II\}$.
The following Main Theorem shows the exact wave front sets of $\overline{F}^{I}(x, y_{3})$ and
$\tilde{F}^{II}(x, y_{3})$.
Main Theorem. For$\xi^{0}\in \mathrm{R}_{\xi}^{4}\backslash \{0\}$ satisfying $(detP_{j}^{I})(\xi^{0})=0(j\in\{p_{1}, S1\})$ we
have the following:
(1) For the
reflected
Riemannfunction
$\tilde{F}^{I}(x, y\mathrm{s})$, we have(3.1) $\lim_{\nuarrow\infty}\nu e^{-i\{}k)-y3\xi^{0}3\}\tilde{F}\nu x\xi^{0-(\xi}’\cdot\prime 0’(+x3\mathcal{T}l)X,$$y_{3}=\tilde{F}^{I}0k(j\xi X, y_{3})$,
and
if
$\xi^{0’}$ arezeros
of
$\tau_{m}^{+}(\zeta’)$, that $i\mathit{8}_{y}\xi^{0}/sati_{\mathit{8}}fy|\xi^{0^{\prime//}}|=\xi_{0}^{0}/cm(m\in\{p_{1},p_{2,2}s\})$,then we have
(3.2)
$\lim_{\nuarrow\infty}\{\nu^{\frac{3}{2}}e^{-}(-\xi^{0})-y3\xi_{3}0_{\}I}\tilde{F}x\cdot+x_{3}\tau_{k}(Xi\nu\{;_{\xi}0’;, y3)-\nu^{\frac{1}{2}}\tilde{F}^{I}0k(j\xi x, y3)\}=\tilde{F}^{I}j\xi 0km(x, y3)$,
$(j, k, m)=\{(p_{1},p1,p_{2}),$ $(p_{1},p\mathrm{l}, S2),$$(p1, S1,p_{2}),$ $(p_{1}, S_{1}, S2),$ $(s1,p1,p2)$,
$(s_{1},p_{1,2}s),$ $(s1, s_{1},p_{2}),$$(S1, s_{1,2}S),$ $(s_{1}, s1,p_{1})\}$
in the dist$7’ ibution$
sense
with respect to $(x, y_{3})\in \mathrm{R}_{-}^{4}\cross \mathrm{R}_{-}$.
Moreover we have$WF( \tilde{F}^{I}(x’, x_{3}, y3))=WFA(\tilde{F}^{I}(_{X}/, x_{3},y_{3}))=\xi\neq 0\bigcup_{0}$
$[_{(}j,k)=p1,’ \mathrm{P}1’ p1,s_{1}),/\cup(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\tilde{F}I(j\xi 0kx, X_{3},y_{3})\cross\{(\xi 0’, \mathcal{T}^{-(\xi}0’k), -\xi_{3}^{0})\})$
$\cup$
$(j,k^{\bigcup_{m}},)=$
$(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\tilde{F}_{j\xi km}^{I}0(X,$$x/3,$
$y_{3})\cross\{(\xi^{0^{//}}, \mathcal{T}_{k^{-}}(\xi \mathrm{o}), -\xi 30)\})]$
$\{(p_{1},p1,p2),(p_{1p_{12}),(},,Sp1,s1,p_{2})$,
$(p_{1,1S}s,2),(s_{1,p1},\mathrm{P}2),(s_{1},p1,s_{2})$,
$(s_{1},s_{1},p2),(_{S1s_{1},s_{2})},,(s1,s1,p1)\}$
and
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\tilde{F}_{j\xi^{0}}^{I}(kX, y)=(\Gamma_{j\xi}0)k\equiv I\{(_{X}, y_{3})\in \mathrm{R}^{4}-\cross \mathrm{R}_{-}$ :
$(X’+X_{3\mathrm{g}\mathrm{a}}\mathrm{r}\mathrm{d}\xi\tau_{k}^{-}(\xi 0^{/}))\cdot\eta-/y3\eta_{3}\geq 0$ for any
$\eta\in\Gamma_{j\xi^{0}}\}$,
$(j, k)=\{(p1,p_{1}), (p1, S1), (S1,p1), (S_{1}, s_{1})\}$
for
$\xi^{0}$ satisfying$\tilde{F}_{j\xi^{0}k}^{I}(x, y3)\neq 0$,
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\overline{F}_{j\xi^{0}k}^{I}(mx, y)=(\Gamma_{j\xi^{0_{m}}})_{k}^{I}\equiv\{(x, y_{3})\in \mathrm{R}^{4}-\cross \mathrm{R}_{-}$:
$(x’+x3\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\xi\tau^{-}k(\xi 0’))\cdot\eta-/y_{3}\eta_{\mathrm{s}}\geq 0$ for any
$\eta\in\Gamma_{j\xi^{0}m}\}$,
$(j, k, m)=\{(p_{1}, p1,p_{2}),$ $(p1,p1, S2),$ $(p17S_{1},p2),$$(p1, s1, s2),$ $(_{S_{1,p_{1}}},p2)$,
$(s_{1},p_{1,2}S),$$(s_{1}, s1,p2),$ $(s_{1,1}s, s2),$ $(S1, S1,p_{1})\}$
for
$\xi^{0}$ satisfying $\tilde{F}_{j\xi^{0}km}^{I}(x, y3)\neq 0$.(2) For the
refracted
Riemannfunction
$\tilde{F}^{II}(x, y\mathrm{s})$, we have$(j, k)=\{(p_{1},p_{2}), (p_{12}, s), (s_{1p2},), (S_{1}, S_{2})\}$
and
if
$\xi^{0’}$ arezeros
of
$\tau_{m}^{+}(\zeta’)(m\in\{p_{2}\})$, then we have(3.4)
$\lim_{\nuarrow\infty}\{\nu^{\frac{3}{2}-}ek\xi 0’)-y3\xi 3\}\overline{F}i\nu\{x’\cdot\xi 0’x+3\tau^{\dagger}(0II(x, y_{3})-\nu^{\frac{1}{2}}\tilde{F}_{j\xi k(}II0X, y_{3})\}=\tilde{F}^{I}Imj\xi 0k(x, y_{3})$ ,
$(j, k, m)=\{(p1_{\text{ノ}}.s_{2,p}2), (_{S_{1}}, S2,p2)\}$
in the distribution
sense
with respect to $(x, y_{3})\in \mathrm{R}_{+^{\mathrm{X}}}^{4}\mathrm{R}_{-}$.Moreover we have
$WF( \tilde{F}^{II/}(X, x3, y3))=WFA(\tilde{F}^{II}(Xx\mathrm{s}, y/,3))=\bigcup_{\xi 0\neq 0}$
$[_{(j,k)\mathrm{t}(}(_{\mathit{8}1,p_{2}}),’(s1,s_{2}) \}x=\mathrm{P}1p\bigcup_{22),(p_{1},s)},$$(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{P}^{\tilde{F}(}j\xi 0kx3,$$y_{3})\mathrm{X}\{(II/,\xi 0’.(J\tau k^{+}\xi^{00}/), -\xi_{3})\})$
$\cup(j,k,m)=\{(p1,s2,p_{2}),(\bigcup_{2s1,s,p_{2})\}}(I/,)\cross\{(\xi^{0}\tau_{k}(\xi), -\xi \mathrm{s})\})(^{\sup \mathrm{P}^{\tilde{F}_{j}^{I}(}}\xi 0kmXx_{3}, y3/,+0’0]$
and
$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\tilde{F}_{j\xi k}^{lI}0(_{X}, y_{3})=(\mathrm{r}_{j\epsilon^{0}})^{II}k\equiv\{(x, y3)\in \mathrm{R}_{+}^{4}\cross \mathrm{R}_{-:}$
$(x’+x\mathrm{s}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}_{\xi}\mathcal{T}(k\xi^{0}+/))\cdot\eta’-y3\eta 3\geq 0$ for any $\eta\in\Gamma_{j\xi^{0}}\}$,
$(j, k)=\{(p_{1},p2), (p1, s2), (s1,p_{2}), (s1, s_{2})\}$
for
$\xi^{0}$ satisfying $\tilde{F}_{j\xi^{0}k}^{II}(X, y_{3})\neq 0_{f}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\tilde{F}_{j\xi km}^{lI}\mathrm{o}(_{X}, y_{3})=(\Gamma_{j\xi m}0)_{k}II\equiv\{(x, y_{3})\in \mathrm{R}_{-}^{4}\mathrm{x}\mathrm{R}_{-}$
:
$(x’+x_{3}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}_{\xi}\mathcal{T}_{k}+(\xi^{0}’))\cdot\eta-/y_{3}\eta 3\geq 0$for any $\eta\in\Gamma_{j\xi^{0}m}\}$,
$(j, k, m)=\{(p1, S2,p_{2}), (s1, S2,p_{2})\}$
for
$\xi^{0}$ satisfying $\overline{F}_{j\xi^{0}}^{II}(kmx, y3)\neq 0$.
Here$F_{j\xi^{0}k}^{I}(X, y_{3}),$ $F_{j\xi^{0}km}^{I}(x, y_{3})_{\mathrm{z}}F_{j\xi^{0}k}^{II}(x, y_{3})$, and$F_{j\xi^{0}km}^{lI}(x, y\mathrm{s})$ are localizations
in the
sense
of
(3.1), $(\mathit{3}.\mathit{2})_{f}(\mathit{3}.\mathit{3})_{y}$ and $(\mathit{3},\mathit{4})_{f}re\mathit{8}pectively_{f}$ andmore
$preC?\mathit{8}e$expres-sions are given in $([7]y[8])$. Moreover
$\Gamma_{j\xi^{0}}=\Gamma((\det P^{I})_{\xi^{0}}(\eta), \theta j)$, $\theta=(1,0,0, \mathrm{o})$, $j\in\{p_{1}, s_{1}\}$,
$\theta’=(1,0, \mathrm{o})$, $j\in \mathrm{f}^{p_{1}},$ $s_{1}\}$
,
$m\in\{p_{2}\}$,$\tau_{p_{1}}^{\pm}(\xi’)=\mathrm{S}\mathrm{g}\mathrm{n}(\mp\xi_{0})\sqrt{\frac{\xi_{0}^{2}}{c_{p_{1}}^{2}}-|\xi^{0^{\prime//}}|^{2}}$, if
$\frac{\xi_{0}^{2}}{c_{p_{1}}^{2}}-|\xi 0^{\prime//}|2\geq 0$,
and $\tau_{p_{1}}^{\pm}(\xi’)$ is taken a branch
of
$\sqrt{\frac{\xi_{0}^{2}}{c_{\mathrm{p}_{1}}^{2}}-|\xi^{0^{\prime//}}|^{2}}\mathit{8}uchthat\pm{\rm Im} \mathcal{T}_{p_{1}}^{\pm}(\xi’)>0$if
$\frac{\xi_{0}^{2}}{c_{p\mathrm{J}}^{2}}$-$|\xi^{0^{;J/}2}|<0$ . $\tau_{s_{1}}^{\pm}(\xi’)_{f}\tau_{p_{2}}^{\pm}(\xi’)$, and $\tau_{s_{2}}^{\pm}(\xi’)$ are
defined
as thesame
$a\mathit{8}\tau_{p_{1}}^{\pm}(\xi^{l})$
substi-tuting$c_{p_{1}}$
for
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