Title

WAVE FRONT SETS OF SOLUTIONS TO ELASTICWAVE PROPAGATION PROBLEMS IN STRATIFIEDMEDIA (Analytical Studies for Singularities to the NonlinearEvolution Equation Appearing in Mathematical Physics)

Author(s) Shimizu, Senjo

Citation 数理解析研究所講究録 (2000), 1123: 83-91

Issue Date 2000-01

URL http://hdl.handle.net/2433/63551

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

WAVE FRONT SETS OF SOLUTIONS TO ELASTIC WAVEPROPAGATION PROBLEMS IN STRATIFIED MEDIA

SENJO SHIMIZU (清水 扇丈)

Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561

1. Introduction

We consider elastic wave propagation problems in plane-stratified media $\mathrm{R}^{3}$ withthe 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 simplicitycalled $\mathrm{P}$ wave) and Share or Secondary wave ( $\mathrm{S}$ wave). $\mathrm{P}$ wave is a longitudinalwave and $\mathrm{S}$ wave is a transversal wave. In general the speed of $\mathrm{P}$ wave is graterthan 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 MediumII 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 $S_{2}$ wavesare six cases. Here we assume $P_{2},$ $S_{2,1}P,$ $S_{1}$ waves in order of speed since 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 and causes $P_{1}$ and $S_{1}$ reflected waves in Medium I and $P_{2}$ and $S_{2}$ refractedwaves in Medium II as in Figure 1.

$X\mathrm{a}$

Figure 1 Reflected waves and refracted wavesMoreover when time goes on, lateral waves in other words glancing waves or

total reflected (or refracted) waves arise. In Figure 2, dotted arrows 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 for

1991 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

(grant A-09740098) from the Ministry of Education, Science and Culture of Japan.

Typeset by $A_{\lambda 4}\text{\c{a}} \mathrm{T}ffi$

数理解析研究所講究録1123巻 2000年 83-91 83

$P_{1}$ incident wave. $P_{2^{-}}P_{1}$ lateral wave means that the wave originally should havebeen $P_{2}$ reflected wave tends to total reflection, then becomes source and causes $P_{1}$

reflected wave.

Figure 2 Lateral wavesWe 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 hastwo speeds, there exist only one kind of lateral wave. If plane-stratified mediaproblem that each medium has one speed, there exist only one kind of lateral wave.Thus this elastic wave propagation problems in plane-stratified media has manylateral waves.

In this paper we prove the above physical situation mathematically by using aexpression of an inner estimate of singularities. The main technical tool of ouranalysis is a localization method.

We gave an inner estimate of the location of singularities of the reflected and re-fracted Riemann functions by making use of the localization method $[7,8]$ . Thismethod 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 functions by making use of the localization 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 the outer estimate of wave front sets of solutions toinitial value problem. Wakabayashi studied for half-space mixed problem [12], andfor more general case [13]. We analysis an outer estimate of wave front sets of theRiemann functions to the elastic mixed problem based on Wakabayashi’s theorem[12, Theorem 4.2]. Combining the inner estimate and the outer estimate, we obtainthe exact wave front sets of the elastic mixed problem in stratified media.

I would like to express my gratitude to Professor Seiichiro Wakabayashi for hisinvaluable suggestions.

2. Formulation of Problems

We consider elastic wave propagation problems in the following plane-stratifiedmedia $\mathrm{R}^{3}$ 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},$$\mu_{1},$ $\mu_{2}$ are called the Lame’ constants and the constants

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$\rho_{1},$ $p_{2}$ are densities. We shall denote the lower half-space $\mathrm{R}_{-}^{3}$ by Medium $I$ and theupper half-space $\mathrm{R}_{+}^{3}$ by Medium II, respectively, as in Figure 3.

Figure 3 Stratified mediaWe 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 practical situation. From the ro$o\mathrm{t}\mathrm{s}$ of the charac-teristic equations of $P^{I}(D)$ and $P^{II}(D)$ which are defined below $3\cross 3$ matrix valuedhyperbolic partial differential operators in Medium I and Medium II, respectively,we obtain two speeds correspond to 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 a lon-gitudinal wave and $\mathrm{S}$ wave is a transversal wave. $c_{\mathrm{P}1}$ denotes the speed of $\mathrm{P}$ wavein Medium I and $c_{s_{1}}$ denotes the speed of $\mathrm{S}$ wave in Medium I. $c_{p_{2}}$ and $c_{s_{2}}$ denotethe 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 eachmedium. On account of this, these are six cases of the order relation of the speedsof $\{cc, cc\}p_{1}’ s1p_{2’ S2}$ . Here we assume that

$c_{s_{1}}<c_{\mathrm{P}1}\leq c_{s_{2}}<c_{p_{2}}$ ,

since if we 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 asimilar 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 $x$ and is 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 $x^{\prime/}$ . The propagation problems of elastic waves in the stratified media

85

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 second order hyperbolic differential operator with constantcoefficients 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 of this elastic mixed problem isgiven by the following:

$G(x, y)=\{$$E^{I}(x-y)-F^{I}(x, y)$ for $x_{3}<0$ ,$F^{II}(x, y)$ for $x_{3}>0$ .

We call $E^{I}(x-y),$ $FI(x, y)$ , and $F^{II}(x, y)$ the incident, reflected, and refracted Rie-mann functions, respectively, because these are corresponding to incident, reflected,and refracted waves, respectively. $E^{I}(x)$ is the fundamental solution in Medium Idescribing 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 1.4below, and $P^{I}(\xi+i\eta)^{-1}$ is a $3\cross 3$ inverse matrix. Taking partial Fourier-Laplacetransform with respect to $x’$ for the mixed problem, we obtain a interface valueproblem for ordinary differential equation with parameters. Then taking partialinverse Fourier-Laplace transform for the solution, we obtain explicit expressionsof 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.[3], [4], [11] $)$ .

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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 thatthere 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 thereexist a bounded sequence $\{\phi_{N}\}$ in $C_{0}^{\infty}$ such that $\phi_{N}=1$ on a fixed neighborhoodof $x^{0’}$ 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 conicneighborhood $\triangle$ 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 of degree $m\geq 0$ 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 $\eta$ . The poly-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 $\mathrm{R}_{\eta}^{n}\backslash \{\eta\in \mathrm{R}_{\eta}^{n}, P(\eta)=0\}$ whichcontains $\theta=(1,0, \cdots, 0)\in \mathrm{R}^{n}$ . 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. ResultsWe 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 solutionof Medium I $E^{I}(x)$ . This proposition is a version of the theorem proved by Atiyah-$\mathrm{B}\mathrm{o}\mathrm{t}\mathrm{t}-\mathrm{G}^{\circ}\mathrm{a}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}$ [ $1$ , Theorem 4.10] adopted the present context.

87

Proposition. 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 $(detP_{j}^{I})(\xi^{0})=0(j\in\{p_{1}, S1\})$ wehave the following:

(1) For the reflected Riemann function $\tilde{F}^{I}(x, y\mathrm{s})$ , 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})$ ,

$(j, k)=\{(p_{1},p1), (p_{1}, s1), (S\mathrm{l},p1), (S_{1}, S_{1})\}$

88

and if $\xi^{0’}$ are zeros 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 Riemann function $\tilde{F}^{II}(x, y\mathrm{s})$ , we have

(3.3) $\lim_{\nuarrow\infty}\nu e^{-i}\nu\{x\xi\prime 0l(+x\mathrm{s}\tau_{k}^{+0}\xi^{0})l-y3\xi \mathrm{s}\}\tilde{F}II(x, y_{3})=\tilde{F}_{j\xi^{0}k}II(X, y3)$,

89

$(j, k)=\{(p_{1},p_{2}), (p_{12}, s), (s_{1p2},), (S_{1}, S_{2})\}$

and if $\xi^{0’}$ are zeros 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 $(\mathit{3},\mathit{4})_{f}re\mathit{8}pectively_{f}$ and more $preC?\mathit{8}e$ expres-sions are given in $([7]y[8])$ . Moreover

$\Gamma_{j\xi^{0}}=\Gamma((\det P^{I})_{\xi^{0}}(\eta), \theta j)$ , $\theta=(1,0,0, \mathrm{o})$ , $j\in\{p_{1}, s_{1}\}$ ,

$(\det P_{j}^{I})\xi^{0}(\eta),$ $\theta)\cap\{\Gamma(\frac{\xi_{0}^{0}}{c_{m}^{2}}\eta 0-\xi^{0}1\eta_{1}-\xi_{2\eta}^{0}2,$ $\theta/)\mathrm{x}\mathrm{R}_{\eta}\}$ ,

90

$\theta’=(1,0, \mathrm{o})$ , $j\in \mathrm{f}^{p_{1}},$ $s_{1}\}$ , $m\in\{p_{2}\}$ ,

$\tau_{p_{1}}^{\pm}(\xi’)=\mathrm{S}\mathrm{g}\mathrm{n}(\mp\xi_{0})\sqrt{\frac{\xi_{0}^{2}}{c_{p_{1}}^{2}}-|\xi^{0^{\prime//}}|^{2}}$, 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 the same $a\mathit{8}\tau_{p_{1}}^{\pm}(\xi^{l})$ substi-tuting $c_{p_{1}}$ for $c_{s_{1}\mathrm{z}}c_{p_{2}}$ , and $c_{s_{2})}$ respectively.

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