Diffraction by an Immersed Elastic Wedge
This monograph presents the mathematical description and numerical computation of the high-frequency diffracted wave by an immersed elastic wave with normal incidence. The mathematical analysis is based on the explicit description of the principal symbol
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2.1
Notation
We introduce in this section the notation used in the sequel of the paper. The angle of the wedge is ~, and we suppose 0 < ~ < 7r. We call ~2s (resp. f2/) the angular section filled by the elastic medium (resp. the fluid)
12s={(x,y)=(rcosO, rsinO), 0 < 0 < ~ ,
0U~ Z~. :
ZJ, Z 2 are called generation number g, and are constructed by the following recurrence relation
Z~+ 1 = 2~+ 1U
ro(2y
n
~?o), e _> 0
Z y + i = 2 ~ + l U T O ( 2 ~ n ~ ? o ), g > O where the auxiliary sets Z~, Zy, are defined by
(2.35)
16
2. Notation and Results
(2y n
u
n
e _> 0
(2.36)
2L1 rL(2yn.%)u~r(2~n~r), e>_o. Lemma
2.4.
The sets Z 1, Z 2 are finite.
Proof. It is sufficient to prove t h a t an infinite sequence z~ of the form z0 C C, ze+l = T ~ ( z e ) , ze E ~2~, u~ E {UL, UT, u0} does not exist. If z = u c o s 0 = u(cos01 cosh02 - i sin01 sinh02), 0 C 7), 0 _< R e 0 < 7r - ~ then R e ( T , ( z ) ) = ucos(01 + p ) c o s h 0 2 _< u(cos0x - e0)cosh02 _< R e z - e0u, where e0 > 0 depends only of ~. Thus, if the n u m b e r of points z~ is infinite, then Re z~ -+ - o o and Izel --+ +oo. This insures t h a t [0~el ~ + o c and cosh0e2 .~ [ sinh02~] --. 89 ~ Consequently, I Arg zt I ~ 0~ E]0, 7r - ~[ and [ Arg ze+l I "~ O~ + ~ ~ 0~+ 1. This gives a contradiction since it implies 0~ --+ +oc. .. We refer to Sect.2.2.2. for the physical interpretation of the sets Z 1, Z 2.
2.5
Main
Theorems
T h e first result concerns the existence and unicity of outgoing solutions of the problem, as settled in Definition 2.3. We need to suppose t h a t the incidence angle Oi~ is such t h a t {UL,ur, u o } n Z J ( u o c o s O i ~ , u o C o s ( O i ~ + ~ ) )
=(~
j=l,2
(H)
T h e hypothesis (H) means t h a t the incident wave 9 is not an incomming grazing wave along one of the two faces. 9 does not generate t h r o u g h the recurrence formula (2.35) an incomming grazing wave of the fluid-solid coupling by a plane interface. Because of L e m m a 2.4, the hypothesis (H) is satisfied except for a finite set of incidence angles 0i~.
17
2.5 Main T h e o r e m s ~z
;=r ~e
R-"
......................... ~ ........ ;h~
02 :
F i g . 2.4 The complex cosine transformation between the domainsc ~ and ~ .
~v -vco6,~
F i g . 2.5. T h e domain ~2~
18
2. Notation and Results
T h e o r e m 1. ( E x i s t e n c e a n d u n i q u e n e s s o f a n o u t g o i n g s o l u t i o n )
Under the hypothesis (H), the system (2.12)-(2.15) does have an unique outgoing solution (v, h). If aj,/3j, 7j E .4 are the single layer potentials linked to the outgoing solution (v, h) by (2.28), we define the spectral function of the system (5) by
~j(~) :
//~j(~)
,
j = 1,2.
(2.37)
The proof of Theorem 1 is postponed to Sect.4.2. We denote by U the domain U = C\] - oc,--~'L]
(2.38)
and by "H the space of the functions f , holomorphic on U and such that f ( x e i~) e L2(R, dx) for each a E]0, 7r[. Moreover, for j = 1, 2 we note
7)j = Z j (v0 cos 0i~, v0 cos(0in + ~)) cJ :
u zJ(-.r,--T)
(2.39)
u {-.0}.
(2.40)
The set 05 is such that Cy C [-u0,-
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