IV.l. Proof of existence in a general case. IV.2. An alternative construction when the obstacle cp is constant. V. Proof of Uniqueness: Infinite String. V.J. Results of trace. V.2. Proof of uniqueness. VI. The Finite String with Fixed Ends. VI.l. Existence. VI.2. Uniqueness. Appendix. A. 1. Elementary results about integration by substitution. A.2. The set t =--= a(x) in characteristic coordinates.

It is therefore natural to precribe the following conditions:

Ou~O in the sense of distributions in IR X IR1 = ~ X (0, oo ), (0.5) the support of Du is contained in the "set of contact," the set where u(x, t) = <p(x).

(0.6)

In this paper, we prescribe <p" ~ 0.

(0.7)

This convexity condition on the function <p expresses the fact that the n1aterial obstacle {(x, z)jz ~ cp(x)} is concave. Condition (0. 7) ensures that there will not be "too many" reflections on the obstacle, which would cause the method described hereafter to fail.

We need information on the nature of the reflection on the obstacle and we choose to impose that energy be conserved:; we shall, in fact, need a local energy conservation condition, which we will write as

~ (-2 ou ~) + ~ (lou 1 2 + lou 1 2 ) = 0
ox ox ot at ox at .

(0.8) in the sense of distributions in The aim of this paper is to prove that, if u 0 is in Hfoc( IR) and u 1 is in Lioc( ~), then there exists a unique function u in L~c(~+; Htoc(~)) n Wf~~(IR+; Lioc(IR))

with IR f• = [0, + oo) which satisfies conditions (0.1 )-(0.6) and (0.8). We prove an analogous result in the case of a finite vibrating string, with fixed ends.

Some results of continuity with respect to the data, of regularity, and the convergence of a numerical scheme are shown in the paper of A. Bamberger, and the author [START_REF] Bamberger | Etude d'un probleme hyperbolique avec contrainte unilaterale[END_REF].

Before Amerio and Prouse, who proved the first result on the vibrating string with obstacle in [START_REF] Amerio | Study of the motion of a string vibrating against an obstacle[END_REF], it seems that nobody had seriously progressed toward the solution of this problem. In the case with loss of energy, Citrini has given several results in [START_REF] Crtrini | Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un ostacolo[END_REF][START_REF] Citrini | Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un ostacolo[END_REF], along the line of Amerio and Prouse; the point obstacle has been studied by Amerio [1, 2], Citrini [START_REF] Crtrini | Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un ostacolo[END_REF], and the author [START_REF] Schatzman | Un probleme hyperbolique du 2eme ordre avec contrainte unilaterale: La corde vibrante avec obstacle ponctuel[END_REF].

The plan is the following: In what follows, we shall denote by u(x, t) the transverse displacement from its equilibrium position of the point of abscissa x of the string at the instant t, cp the obstacle, u 0 the initial position of the string, u 1 the initial velocity of the string.

We make the following hypotheses:

" > 0 where Lioc(IR) is the well-known space of locally square-integrable functions; u 0 (x) ~ g>(x) Yx,

(

(1.4)

(this makes sense because u 0 and cp are continuous by hypotheses (I. I) and (1.2)); almost everywhere on the set {xju 0 (x) = cp(x)}.

(1.5)

In the case of a finite string with fixed ends, we replace (I. J) by and <p(O) < 0, g1(L) < 0,

(1. [START_REF] Crtrini | Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un ostacolo[END_REF] and (1.2), (1.3) by Uo E Ho'(O,L) = lv EL 2 (0,L);;; EL"(O,L), v(O) = v(L) = oj ' (1. 7) Ut EL 2 (0,L). (I.8)

We shall prove the following results :

THEOREM IV.l (Infinite String). Under hypotheses (1.1 )-(!.5), there exists a unique function u such that u E Loo (iR+• 1! 1 (IR)) n WL<XJ(iR+• L 2 (IR)) loc ' loc loc ' loc ' u(x, 0) = u 0 (x) ou Bt (x, 0) = u 1 (x) u(x, t) ~ tp(x)

Ou~O 'Vx, almost everywhere,

V(x, t) E iR X !R_+,
in the sense of distributions in IR X IR; , supp D u C {(x, t) fu(x, t) = tp(x)}, ~ (-2 ou ~) + ~ (lou 12 lou 1 2 ) = 0 ox ox ot 8t ot + ox in the sense of distributions in ~ X ~; .

(1.9)

(1.1 0)

(1.11)

(1.12)

(1.13)

(1.14)

(1.15) THEOREM V.I (Finite String with fixed Ends). Under hypotheses (1.4)-(1.8), there • exists a unique function u such that

u E L <XJ (iR+• H 1 (0 L)) (\ WLCXJ(!R+• L 2 (0 L)) loc ' o ' loc ' '
(1.16)

and u satisfies conditions (1.1 0)-(1.15) with the relevant modifications on the domain.

For simplicity, we shall denote problem (1.9)-(I. 1 5) by the symbol P oo , and problem (1.10)-(1.16) by the symbol Pr.

The functional spaceL~c(IR+ ; Hfoc(IR)) n Wt 0 ~(~+;Lioc(IR))is defined as the set of real functions v on IR X IR+ such that, for any finite positive L and T e~~,~p

(( [1 u(x. t)l 2 + 1 :: (x. t) 1• + 1 !~ (x. t) n dt) < + oo. (1.17)
The functional space L~c(IR+ ; H 0 1 (0, L)) n Wf 0 ~(R+ ; L 2 (0, L)) is defined to be the set of real functions v on [0, L] X R + such that, for all finite positive T e~~,~~p (( [1 u(x, t)l 2 + 1 : :

(x, t) 1• + 1 a; (x. t) n dt) < + oo and (1.18) u(O, t) = u(L, t) = 0 'tit ~ 0.
It will be shown in the course of the paper that condition (1.12) implies that

(1.1 1 ) has a meaning.

I.2. Justification of the Mathematical Model

Conditions (I. I 0) to (1.12) are quite natural. Condition (1.14) implies that, when the string does not touch the obstacle, it satisfies the wave equation.

If at a time t and a point x, the string touches the obstacle with a nonzero velocity, there must necessarily be a discontinuity in the time derivative. Hence, generally, Du cannot be a function. As the string must leave the obstacle after the shock, Ou must be positive, i.e., (1.13).

Condition (1. 1 5) expresses that the divergence of the vector field (1.19)

.

IS zero.

The first component of this vector field is an energy flux density; it is similar to the Poynting vector which is well known in electromagnetism; see, for example, Landau and Lifschitz [I 1, p. 98].

The second component of Su is but a total energy density: kinetic plus poten- tial energy. Condition (I.l5) is satisfied on any open set where D u is null: to convince oneself it is enough to multiply Du by oufot, and rewrite the expression in I divergence form. This classical manipulation is done in Courant and Hilbert [9, p.

660].

It is not enough-for example, in the case of the finite string with fixed ends-to impose an integral condition of the type ( [\ ~ (x, t) 1 2 + I ~= (x , t) n dx = r (I ~: ,. + I u, 1 2 ) dx almost everywhere in t.

(1.20)

Condition (1.20) is not local, and does not ensure that the velocity is reversed everywhere when a shock happens.

Condition (1.15) expresses the conservation of energy across any curve, and especially the discontinuity curves of oujox and oufot. Proposition V.3 lays down precise! y this property.

Idea of the Proof of Existence

Role of the concavity hypothesis. Let us first consider the "free solution'' w, or solution of the problem without obstacle, with the initial data u 0 , u 1 of the problem with obstacle:

Dw = 0, w(x, 0) = u 0 (x), 8w 7ft (x, 0) = u 1 (x).
(1.21)

It is well known that there exists a unique w in the functional class L~0(1R+; Hfoc(IR)} 11 W{~~(IR+;Lroc(IR)) which satisfies (1.21). Such a w is a continuous function.

Let us now consider the set E:

E = {(x, t) E IR X IR+jw(x, t) < <p(x)}.
As w is continuous, E is unambiguously defined. Define the domain of influence to be the union, I, of all forward wave-cones with vortex in E. I is hatched in Of course, E is included in I. By definition

w(x, t) ~ <p(x) if (x, t) f# I.
We can easily see that if w(x, t) > <p(x) for all (x, t) in a backward cone, then the solution u must be equal to w in that cone.

We take u to be equal to w in all the complement of I, and we must now extend u across the boundary of I.

The boundary of I, called line of influence, is the graph t = r(x) of a Lipschitzcontinuous function r:

I r( X) -r(y) I ~ I X -Y I Vx, yE IR.
Section 11 is devoted to studying the line of influence, and proving the following properties:

w(x, T(x)) = <p(x) if I T'(x)l < 1, if w(x, r(x)) = <p(x).
In Section Ill, we consider the following problem:

V E W 1 • 00 (~+ • L2 (!R)) n L 00 (~+ -fll (!R))• loc ' Ioc loc ' loc ' Ov! {(x,t) / t;ea(x)} := 0, v(x, 0) = v 0 (x), ov ot (x, 0) = v 1 (x), (1.22) o+v o-v
Tt (x, a(x)) = -Tt (x, a(x)) a.e. on {x/l a'(x)l < 1 and a(x) > 0}.

Here a is a given nonnegative Lipschitz-continuous function, with Lipschitz constant 1, and v 0 , v 1 are given, respectively, in Hfoc(IR) and Lioc(IR).

We can prove that (I.22) possesses a unique solution v given explicitly in terms of a and w, the free solution with data v 0 and v 1 by

V = W + tf * fL( W ),
where <f is the elementary solution of the wave equation, and p.( w) is a measure given by

<~L(w), t/1) = -2 r (1 -r'(x) 2 ) ~w (x, r(x)) tfo(x, r(x)) dx, J cu t "T(x) > O} ut
To complete the proof of existence, it remains to check that conditions (1.12) and (1.13) are satisfied by u, solution of (1.22) with (]' = r, u 0 = v 0 and u 1 = v 1 • The latter condition is easily verified, but to prove the former, we rely heavily on hypothesis (1.1 ). We can infer indeed from (I. I) that, once the string has touched the obstacle, it does not touch it any more. From the similar hypothesis (1.6), we can deduce that once the finite string has touched the obstacle, it does not touch it again after a finite time, depending only on the geometry of the obstacle and on the initial total energy; thus we can iterate easily the above procedure and obtain a solution at a finite time after a finite number of steps.

If hypothesis (I.l) were left out, let us see what could happen: Let u 0 , u 1 , and q> be given by Then the free solution w is

u 0 (x) = q>(x) = -x 2 , u 1 (x) = 0. w(x, t) = -x"' -tZ so that E = I = 1R X fR+ and -r = 0. The solution v of (1.22) satisfies Ov = 0 if t > 0, v(x, 0) = v 0 (x) = -x2, ov ot (x, 0) = vl(x) = 0
and it is readily computed:

v(x, t) = w(x, t) = -x 2 -t 2 •
We see that condition (1.12) cannot be satisfied for any x and any positive tl Of course if the initial were nowhere zero, we could build a solution of (I.22) which would be positive in a ne~ghborhood of !R X {0}, and define a new line of influence, and perhaps do the same again a certain number of times. But in general nothing guarantees that there will be only a finite number of influence lines in a finite time. The lines of influence could accumulate, and a simple argument shows that in the limit, the velocity would be zero on the limit curve, or more precisely, on the noncharacteristic parts of the limit curve where the obstacle is convex. So we need a more powerful device-which is not yet known -to prove an existence theorem without hypothesis (I.l ).

Finally, the reader should note that the existence proof given here is not at all variational. Of course, conditions (I.12) to (1.14) "look variational" but it is definitely difficult to include condition (1.15) in an efficient variational formulation.

Comparison with the Results of Amerio and Prouse

In their first paper [START_REF] Amerio | Study of the motion of a string vibrating against an obstacle[END_REF], Amerio and Prouse constructed a solution taking a rather unusual functional class of initial data. Lately, they could relax their hypothesis, in a still unpublished paper [START_REF] Amerio | [END_REF]. They consider a constant obstacle ( g;' = 0), and they build a weak solution step by step, following the characteristics, and extending functions across the lines of influence by means of convenient formulas. They do not define a functional class in which they seek a solution of a partial differential problem, in order to have a genuine theorem of uniqueness. The fundamental idea of line of influence originated from Amerio and Prouse's paper [START_REF] Amerio | Study of the motion of a string vibrating against an obstacle[END_REF], and the present work uses this idea to a large extent.

Explicit Computation of Examples

EXAMPLE I. Infinite String. The simplest example one can devise is the following one:

Then u 0 (x) = c > 0, u 1 (x) = -1, g;(x) = 0. w(x, t) = c -t, E=l=~+x[c,+oo]
and the solution of problem P 00 is given by

u(x, t) = c-t u(x, t) = t -c if if tE [0, c], t ;); c.
EXAMPLE 2. Infinite String. This example shows how a wave "coming from minus infinity'' is reflected against a plane, oblique obstacle. Let us set

u 0 (x) = X if x~O 0 if X< 0, u 1 (x) = -1 if x;)=O 0 if X< 0, <p(x) = ax with a E (0, 1).
Then the free solution is given by

w(x, t) = (x -t)+ = sup(O, x -t).
The set E is defined by

E = {(x, t)j(x -t)+ < ax} = {(x, t)jx ~ 0 and t ~ x(I -a)}
and the set I is

I= {(x, t)ft ~ max(-x, (I -a) x)}.
Then the solution of problem P oo is given by

u(x, t) = (x-t)+ u( x, t) = (2a -1) x + t u(x, t) = a(x + t) I if if if t ~ max( -x, (1 -a) x), (1 -a) x ~ t ~ x, t ~I X'• X FIG. 2.
The sets E and I of Example 2.

u 0 (x) = 0, U 1 (x) = -1T COS 1TX, cp(x) = -!,
we can compute e~plicitly the solution of Pr for any time t.

The initial data correspond to the free solution w(x, t) = -sin 1rt sin 1TX, so that E = {(x, t)/-sin 1Tl sin 1TX < -!} and the (first) line:, of influence is given by (see Fig. 4).

r(x) = -~--x r(x) ~ (l/7T) arc sin(l/(2 sin 1rx)) r(x) = -l + x if if if 0 ~X~ t 1
In the set where t ~ T( x }, u = w; the velocity is reversed in those points where

t = r(x)
and Therefore, in the curvilinear triangle

T(x) ~ t ~ ! -I x -! I we have u = -1-w. I 1 ._ ____ .... ----: -•• 6 ' 1 : l __ _ Ll_ 0 1 1 6 4 .. . .1. --. •••-• _____ ___ I _ -+-----t-•------t>- 1 3 5 l X 2 4 6
Fie . 4. The sets E and I of Example 3 .

.

---•• / / -1 -.3L' IL--n;x: Si?t.-11..t ------. . .. .. --~ Si.,!\-U s[,~ n.t -h--• 0 ... -------1~-x-•-C> FIG. 5.
Values of function u of Example 3 in the different regions of the x, t plane. Next, we compute u as a free solution of the wave equation, with data on the characteristic segments

t = !-!x-ll t = li-xl if if I X -ll < !, l X -lJ . . . . . . . . . . . ~ 2
:;::::-4! as long as there is no new influence of the obstacle. The value of u is given, according to the regions, in Fig. 5.

There is a new influence of the obstable beyond t == t• More precisely, we shall have the influence line:

7" 1 (x) = 2 -T(x) = 2-x =I+x if if if O~x ~t i~x ~l.
We can see that here, the solution is periodic in time, with period 3, and moreover u(2 -t) = u(t).

Figures 6 and7 show the evolution of u for t comprised between 0 and 1, with a time step of 1/12. The computations were carried out explicitly using Fig. 5.

Jl. THE LINE OF INFLUENCE AND ITS PROPERTIES

II.l. Definition and First Properties of the Domain of Influence and of the Line of Influence

Let us recall that the backward cone (or cone of dependence) of a point (x, t) is the set

r-;.t = {(x', t') ~ IR X ~+;o ~ t' ~ t -l X -x' I} (11.1)
and that the forward cone (or c. one of influence) of a point (x, t) is the set

r;,t = {(x', t') E IR X !R+Jt' ~ t + l X -x' )}. (II.2)
A fundamental result of the. theory of hyperbolic equations is the following; Let v be a solution of the wave equation

Ov = 0, v(x, 0) = v 0 (x), ov ~ (x, 0) = v 1 (x). ut •
If v 0 and v 1 have their support in C, then the support of v is included in u r-:.o.

xec Let us define

E = {(x, t) E IR X !R+fw(x, t) < cp(x)}. (11.3) One must not mistake E for the set {(x, t) E lR X JR+ jzv(x, t) ~ cp(x)} =I= E.
We shall suppose that E is not empty; if it were, then the function w would be a solution of Pro and the existence problem would be solved.

We now proceed to define the domain of influence and the line of influence.

DEFINITION II.l. The domain of influence of the obstacle cp with respect to w, for t nonnegative is the set

1 = U r:.t. (11.4) (x,t)EE DEFINITION 1!.2.
The line of influence of the obstacle g; with respect to w, for t nonnegative, or more simply, the line of influence, is the boundary of the set I.

The reader is referred to Fig. 8, to visualize the sets I, E, and the line of influence.

/ ~-------"--+---------••••••••• • ••• • ••-••--•••x:-!>
Frc. 8. The sets r:,t T ;:t, E, and I; the line and the domain of influence.

The names "domain in influence" and "line of influence" are derived from the fact that outside the domain of influence, the obstacle does not alter the problem, and inside the domain of influence it does. These assertions will be a result of the existence and uniqueness theorem.

The boundary of I has the following properties.

I= {(x, t) E !R X ~+jt ~ r(x)}
and therefore, I is closed.

Proof. Define on ~ a function r by r(x) = inf{t' + [ x-x' 1/(x', t') E E}. (11.5) Obviously if (x, t) is in J, then t ~ r(x).
Conversely, let t be greater than or equal to r(x). Then, for any positive E, there exists (xE, t,J in E such that

t ~ t" + I X -XE I -€ (11.6)
and therefore

I x" I + t,. ~ t + I x I + €.
Extract from the bounded sequence (xe, t")" a convergent subsequence still denoted by ( x" , t")" . As E is closed, we shall have and using (11.6) (11.7) and that

t ~ 1 0 + [ x -x 0 I . This shows that r(x) = min{t' + I x-x' lf(x', t') E E}
I= {(x, t)/t ~ r(x)}.
Let us show now that r is Lipschitz-continuous, with Lipschitz constant 1. Given x, there exists (x 0 , t 0 ) in E such that

r( x) = t 0 + I x -x 0 I . (11.8)
On the other hand, for any x, the definition of r implies that -r( x) ~ to + I X -Xo I . (11.9) We deduce from (11.8) and (11.9)

r(.x) -T( x) ~ t 0 + I x -x 0 I -t 0 -I x -X 0 I ~ I x -x I .
By exchanging the roles of x and x, we obtain proposition 11.4.

I

II.2. Study of the Values of the Free Solution and of Its Derivatives on the Line of Influence

LEMMA II. 5. The following implication holds: (11.10) Proof (by contradiction). Suppose I -r'(x 0 )[ < 1 and I.e., By continuity of w and cp, there exists a positive number r, such that

I -r'(x)l < I =>-w(x, r(x)) = cp(x).
I x-x 0 1 2 + I t --r(x 0 )[ 2 ~ r 2 => (x, t) ~E.
In particular, we shall have r

I x -x 0 I ~ 2112 => (x, T(x)) Ef= E.

(II.ll)

There exists (x 1 , t') in E, by virtue of (11.7) such that

T( x 0 ) = t' + I x' -x 0 I
and, thanks to (!1.11 ),

x 0 #-x'. Suppose, for instance, x' > x 0 . Then, if x 0 < x < x 0 + rf2 1 1 2 , T(x) = ( T(x) -T(x 0 )) + -r(x 0 ) = T'(x 0 ) (x -x 0 ) + o(x -x 0 ) + t' + x' -x 0 = t' + x' -x + (x -x 0 ) (1 + T'(x 0 )) + o(x -x 0 ) ?;:: -r(x') + x' -x + (x -x 0 ) (1 + T'(x 0 )) + o(x -x 0 ).
We deduce from this last inequality that, for an x near enough to x 0 ,

1(x) > 1(x') + x' -x (II. I 2)
by the hypothesis I r'(x 0 )l < 1.

On the other hand, the Lipschitz constant of r is equal to I, which contradicts (11.12).

1

Our knowledge of the influence line is improved by the following result. (II.l4)

(II.l5) (II.l6)
Adding inequalities (11.15) and (II.16), and comparing the result to (II.I4), we obtain easily

1(X) -X = T(a) -a = r(b) -b,
i.e., relation (11. J 3).

Suppose next that (a, T(a)) and (b, 1(b)) do not both lie on one characteristic segment, and set

-T(x) = min(1(a) + x-a, r(b) + b-x). (I I. I 7) Clearly, on [a, b], we must have r(x) ~ r(x).
Suppose there exists x 0 E (a, b) such that (11.18) Therefore, by (II.7), there exists (x 1 , t 1 ) in E such that (II.I9)

As the Lipschitz constant ofT is 1, the following inequality holds: Hence (11.20) and, by definition of E (11.21)

The number x 1 cannot be equal to a or b. Suppose, for example, that x 1 ~ x 0 • Then, write I.e., which implies

T(a) -(x 1 -a) ~ r(x 1 ) = T(x 0 ) + x 1 -x 0 < f(x 0 ) + x 1 -x 0 ~ T( a) + x 0 -a + x 1 -x 0 = T( a) + x 1 -a, x 1 -a > -I x 1 -a I ,
x 1 >a.

We could show in a similar way that

x 1 <h.

Relation (II. 21) contradicts the hypothesis w(x, ' T' (x)) > <p(x), therefore relation (11.18) cannot be fulfilled.

I

Lemma II.6 shows that on a connected component of U, the line of influence is made either of a characteristic segment (of slope + 1 or -1) or of a characteris- tic segment of slope + 1, followed by a characteristic segment of slope -1.

We introduce now the characteristic coordinates whence In this section, we study a linear problem, whose solution will be a "candidate" for a solution of the nonlinear problem P oo • THEOREM III.l. Let a be a nonnegative Lipschitz-continuous function with Lipschitz constant 1 , and let v 0 be in H[ 00 ( IR), v 1 in L~0 0 ( IR). Then there exists a unique function u such that

-x + t 1]= 21/ 2
V E wl.oo(~+• L 2 (IR)) n L 00 (JR+• H 1 (~)) loe ' loc loc ' loc ' Ov \{(x.t)/t"'a(x).t>O} = 0 (in the sense of distributions), v(x, 0) = v 0 (x), av at (x, 0) = vl(x), o+v 8-v at (x, a(x)) = -8t (x, a(x))
a.e. on {xfa(x) > 0 and I a'(x)l < 1}.

(III.l)

(111.2) (111.3) (III.4) (111.5)
Condition (III.l) does not imply that Conditions (111.4) and (111.5) make sense. Thus, the first stage is to show (Lemma 111.2) that, with the help of (111.2), Conditions (111.4) and (111.5) do make sense.

Next we shall suppose that there exists a solution v of (Ill. I) to (Ill. 5), and we shall compute Ov in the open set IR X IRi . We shall find an explicit formula for ~-t( w) = Ov in terms of a and w, the free solution of the wave equation with initial data v 0 and v 1 (Proposition 111.3).

Then, it will only remain to show that w + C * ~-t(w) is a solution of (III.l)-(III.5), for any a, v 0 , and v 1 ; moreover w + ~ * J.L( w) fulfills condition (1.15) (Proposition 111.6). Here C is the elementary solution of the wave equation.

Computation of Ov in Terms of w and a

LEMMA 111.2. Let v satisfy conditions (Ill. I) and (111.2). Then (8vfot) (x, 0) is defined almost everywhere on IR, and (o+vfot) (x, a(x)) and (8-vfot) (x, o{x))

are defined almost everywhere on the set {xfa(x) > 0 and I a'(x)l < 1}.

Proof. Let V be a connected component of {(x, t) jt > 0, t =1= a(x)}. On any rectangle R included in V, with characteristic sides, and vertices (x + h, t + k), (x -1k, x + h) (x -h, t -k), and (x -k, x -h), a solution v of the wave equation is of the form v(x, t) = f(x + t) + g(x-t); moreover, f E Jll(x + t -I h + k I , x + t + I h + k j) and g E H 1 (x -t - I h -k I ' X + t + J h -k 1), by (III.l).

The functions f and g are well defined save for an additive constant.

We can join any points (x, t) and, (x, i) by a continuous arc in V, and this continuous arc can be covered by a finite number of open rectangles Ri with characteristic sides included in V. On each of these rectangles Ri we have

(xi+ t i -l hi+ ki I , xi+ ti + I hi + ki I) n (xi+! + ti+l -I hi+l + k i +l I , xi+ I + t i +l + I hi+l + ki+r I) =F 0 .
Therefore, we can choose the additive constant in such a way that fi = fH 1 on the intersection of these two intervals. Thus, because V is arcwise connected, we can define, save for an additive constant, two functions f and g such that

f E Hloc(inf(x + t j(x, t) E V), sup(x, tf(x, t) E V)), g E Hloc(inf(x -tf(x, t) E V), sup(x, t/(x, t) E V)), v(x, t) = f(x + t) + g(x -t), 'r/(x, t) E V.
More specifically, if V = {(x, t)/t > a(x)}, which is, of course, connected, we know from Corollary A.3, that almost everywhere on {x// a'(x)l < I}, j'(x + a(x)) and g'(x -a(x)) are defined. Then, in such a point 1 . v(x, a(x) + h) -v(x, a(x)) lffi _ . ; _ _..:._.:__ -: ----' ---.;........c_ hfO h = lim f(x + a(x) +h)-f(x + a(x)) + li f(x-a(x)-h) -f(x-a(x)) h~O h hw h exists, and defines the right time-derivative of vat (x, a(x)). We argue similarly for (o-vjot) (x, a(x)) and (ovfot) (x, 0).

I

We can now compute Ov if vis a solution of problem (III.l)-(111.5).

PROPOSITION 111.3. Let v be a solution of (III.l )-(III.5), then Ov is a measure p,( w) defined by <t-t(w), "') = -2 J (1a'(x) 2 ) t/J(x, a(x)) 8

; (x, a(x)) dx. Proof. To avoid notattonal ambiguities we shall write when f is a function of x and t

(III.7) whence -(X+ t -X+ t) f(x, t) = f 2112 ' • 2112 • (111.8)
Let 1/J be an infinitely differentiable test function with compact support in the open half-plane. We shall suppose that this support meets only one component

V of {(x, t)/0 < t < a(x)}. Then <ov, f> = <ov, {i> = <2v~11, ~> = -2<ve, {111>• Let us denote by Y the graph of a in characteristic coordinates ~ + 1] (~-' 1]) 1] E Y(~) ~ 2112 = a 2112 • By Lemma A.3, Y is a decreasing graph.
Y is the function defined on (111.12) from which, after having integrated in TJ and substituting '= (x + a(x))/2 1 1 2 , we obtain

{~/Y(~) is one-valued} with values Y(e) on this set. Let on V, v( X, t) = j (X + t) + g( X -t) v(x, t) = / 1 (x + t) + g 1 (x-t) on {(x, t)/t > a(x)}.
<Dv, f ) = -2 J [f'(x + a(x)) -f~(x + a(x))] if;(x, a(x)) (I + a'(x)) dx.
As u is continuous,

f(x + a(x)) + g(x-a(x)) = / 1 (x + a(x)) + g 1 (x-a(x)),
which we differentiate; according to Corollary A.2,

(1 + a'(x)) f'(x + a(x)) + (1 -a'(x)) g'(x -a(x)) (III.14) =(I + a'(x))f~(x + a(x)) + (1 -a'(x)) g~(x-a(x)) a.e.
On the other hand, (111.5) is written:

f'(x + a(x))-g'(x-a(x)) = -f~(x + a(x)) + g~(x-a(x)), (111.15)
a.e. on {xjj a'(x)j < I and a(x) > 0}.

By linear combination of (III.l4) and (III.15), we deduce

f ~ ( x + a( X)) = a' (X) f ' (X + a( X)) + ( I -a' (X)) g' ( x -a( X))
a.e. on {xfj a'(x)j < I and a(x) > 0} and from (III.14) (III.16) f~(x + a(x)) = f'(x + a(x)) a.e. on {x/a'(x) = 1 and a(x) > 0}. (III.l7)

Therefore f'(x + a(x))-f~(x + a(x)) = (f'(x + a(x))-g'(x-a(x))) (1 -a'(x)) (111.18)
a.e. on {x/a'(x) > -1 and a(x) > 0}.

Carrying (III.l8) into (III.13), we obtain (111.6).

I III.3. Partial Results of Existence

Let us recall that the elementary solution t! of the wave equation in one space dimension is where the distribution ~-t(w) is defined by (111.6).

t!(x, t) = ~ C(x, t) = 0 if t ~ I X I ,
LEMMA 111.4. Let a be a Lipschitz-continuous function, with Lipschitz constant J, and let v 0 , v 1 be given, respectively, in Htoc(IR) andLroc(IR). Then the linear form ~ H--2 J ifi(x, a(x)) : (x, a(x)) (1a' 2 (x)) dx defined for all"' in !2J(IR X IRt), is a measure. (Here w is the free solution of the wave equation with initial data v 0 , v 1 ).

Proof. It is enough to show that X H-( ow I ot) ( x, a( X)) ( 1 -a' 2 ( X)) is locally integrable. Classically w(x, t) = f(x + t) + g(x-t)
with f and g in Htoc(IR). Therefore

I ~~ (x, a(x)) (1 -a'(x) 2 ) I ~ 2 I f'(x + a(x)) (1 + a'(x))l + 2 I g'(x-a(x)) (1 -a'(x))l
and by Lemma A. I, the functions x f-+ (1 + a'(x)) f'(x + a(x)) and x ~---+ (Ia'(x)) g'(xu(x)) are locally integrable.

I

The convolution of this measure p.(w) with C can be defined thanks to the support condition of Schwartz [13, p. Condition (111.4) is satisfied almost everywhere on {xja(x) > 0}; it is also satisfied almost everywhere on {xja(x) = 0} as will be proved by next lemma.

LEMMA 111.5. Let A be the set {xfa(x) > 0}, and let B be the set of right and left Lebesgue points of the function

x ~---+ (1 -a' 2 (x)) ~ (x, a(x)) IA(x).
h.!.O • h -lim ) x -ha(h) • 1 (h) Jx I A(x') ~ (x', a(x')) (1 -a' 2 (x')) dx' h.j.O ~ x -a a(h) ut b(h)-X. 1 fb(h} 1 ( ') ow (I ('))(I '2( '))d '~ + h b( h) -X x A X Ot X ' 0' X - U X X )
and by the hypotheses we made, (o+ufot) (x, a(x)) exists, and o;;

I

Clearly, the complement of the set ' where ( o+vj ot) (x, a(x)) exists is null, so we obtain (III.4) and (111.5).

Let us compute now the derivatives of v in characteristic coordinates, in the sense of distributions.

We can write <fl, «/ J) = -2 1 12 J [we(x, a(x)) + ~71 (x, a(x))] (1 -a' 2 (x)) lA(x) «/J(x, a(x)) dx.

(III.27) By the substitutions we transform (111.27) into

. f _ ----2Y'(~) (~-Y(g)) <fl, «/J) = -2. we(~, Y(t)) t/1(~, Y(~}) • 1 _ Y'(t) lA 2112 dg f _ ---. -2X'('IJ) (X('IJ)-'IJ) -2 w71(X(7J}, 7J) t/J(X(7J), 7J) • 1 _ X'(7J) lA 2112 d'T],
(III.28

)
where Y is defined in (111.10), X= Y-I, Y is the "one-valued part" of Y, and X is the ''one-valued part'' of X. The functions Y and X are defined everywhere except on a qenumerable set.

If X is one-valued at 7J, and Y is one-valued at t, then, the value of vis given

h( ') = -2X'(71') • 1 (X(71') -7J' ) 7J 1 -X'('r]') A 2 1 1 2 • (111.33)
Let us compute oyfog in the sense of distributions, from formula (111.30)

Interchange the order of integrations in ~ and t:

(III.34) Formula (111.34) proves that we can identify ay --

{(~, TJ)/~ ~ X(TJ)},
0 ~ (g, TJ) = w€(~, Y(g)) g(g) a.e. on (III.35) =0

a.e. on {(~, TJ){g < X(TJ)} .

Similarly oz .

-

{(g, TJ)/g ~ X(TJ)}, ag (~, TJ) = w1)(g, Y(g)) g(~)
a.e. on (III.36) = 0

a.e. on {(g, TJ) /t < X(TJ)}• Now, formulas (111.35) and (111.36) allow us to compute ovfo~, using the fact that ow;ag does not depend on T):

~ = Wf(g, Y(g)) (I -g(~)) -w1)(g, Y(~)) g(t)
a. e. on {(t, ' YJ)fg ~ X(TJ)},

= wf(g, Y(g))
a.e. on {(g, ' YJ)/ g < X('Y))}.

(III.37) The goal of this section is to prove that the divergence of S;; (and therefore of Sv) is zero, and from that, to infer property (Ill. I).

PROPOSITION III.6. Let v be defined by (111.20). Then -I we(g, Y(g)) (1 -g(g)) -w 71 (g, Y(g)) g(g}l 2 ] ag.

V • Sv = 0 in the
In the same fashion, from formula (111.38), we get

f I av 1 2 of;_ dg d BYJ a~ 1J = f f(X(?J), YJ) [w7l(X(1]), 77)12
(III.43) -1 -u\~(X(?J), YJ) h(TJ) + w 71 (X(?J), .,) (1 -h(.,)}l 2 ] d"l.

Add (III.42) and (III.43), and go back to the x variable by the substitutions Then

g = x + u(x) 21/2 7] = -x + a(x) tn tn 2112 J [J ~~ r ~ + 1 ~:r ~ 1 df d~ (III.42), (III.43). = JA z/;(x, a(x)) {[I We / 2 -/ w~a' -w 11 (1 -a')l 2 ]
(1 + a')

+ [I W 11 ! 2 -/ wE(I + a') + W 11 a' 1 2 ]
(1 -a')} dx and an elementary computation proves (111.40).

I (111.44)

End of the Proof of Theorem (III.l ). It remains only to prove (III.I ). For almost all t 0 in ~R+,

mes{xfa(x) = t 0 } = 0. (III.45) Let D be the set D = {(x, t)/0 < t ~ min(A + t 0 -/ x I , t)},
where t 0 satisfies (111.45), and A is an arbitrary positive number.

Let n be the exterior normal to oD, the boundary of D. Condition (III.45) implies that ovfot and ovfox are defined almost everywhere on IR X {t 0 }; the characteristic derivative ovf at is defined almost everywhere on the g character- istic going through (-At 0 , 0), and likewise, the characteristic derivative ovf O' I' J is defined almost everywhere on the ' fJ characteristic through (A + t 0 , 0). Thanks to (III.40) and therefore f Su 'n = 0 laD L: ( / ~: (x, to) J' + 1 :~ (x, to) J' ) dx < {;~:. (1 v,(xw + 1 : • (x) n dx, which implies (Ill .1 ).

(IV.6) u(x, 0) = u 0 (x), 8u Vx E IR, (IV.?) ot (x, 0) = ul(x), a.e. 1n IR, (IV.8) u(x, t) ~ cp(x), V(x, t) E IR X IJl+, (IV.9)
Ou ~O in the sense of distributions in IR X IR! , (IV.lO)

supp Du C {(x, t) fu(x, t) = f~J(x)}, (IV.ll) ~ (-2 ou ~) + ~(I ou 1 2 + IOU 1 2 ) = 0 ox ox ot at ot ox (IV.12)
in the sense of distributions in 1R X Ill! .

Moreover, the solution u is explicitly known:

THEOREM IV.2. Let w be the free solution of the wave equation with initial data u 0 and u 1 Ow =0,

w(x, 0) = u 0 (x), ow Tt (x, 0) = u 1 (x).

Define a measure t-t(w) on IR

X ]0, + oo[ by (~-t(w), if;) = -2 J ~w (x, T(x)) (1 -T ' 2 (x)) if;(x, T(x)) dx, T(x)>o t (IV.13) (IV.14)
On the set {x/'r(x) = 0}, the relation holds (still Lemma 11.51), and condition (IV.5) implies 1 cp'(x) ft(x + T(x)) ~ -2-a.e. on {x/-r(x) = 0}.

(IV.23) Relation (111.17

) implies 0 ) 1/2 I ( ) cp' (X) 0 ~ [ft(x + t) + Ct(x -t ] !t=T<x> = 2 f t(x + -r x) = 2112
a.e. on {x/-r'(x) = 1, -r(x) > 0, w(x, -r(x)) = cp(x)}.

(IV.24)

We can deduce from relations (IV.22)-(IV.24) that az where C is defined as

~ (x, T(x)) ~ 0 a.e. on {x/-r'(x) > -1, w(x, T(x)) = cp(x)},
C = {~/2(~, Y(~)) = 0} (IV.27)
and thus, (IV.28)

Recall the definition of the set V,

U = {xjw(x, -r(x)) > cp(x)}.
Save for a null set, there is the obvious relation between U and C and besides, on U, the function -r is known very precisely.

Let U be the (at most) denumerable union of disjoint open intervals (ai, hi) (i E /), define ci by and define a subset J of I by Denote Then~ by Lemma II.6~ sa, ve for a null set,. the following relation holds:

cc = U (ai , Yi)• (IV.29) iEJ Moreover if (IV.30)
On the other hand, by relation (111.37), az aw -

1 I (g-Y(~)) a{(~, TJ) =a{(~, Y(~))-2112 cp 2112 if ai ~ ~ < Y i , TJ ~ Y (g), (IV.31)
we can now estimate using relations (IV.29)-(IV.31 ).

Let ~0 belong to the interval [ai, rd• Then (IV.32)

As we suppose that X is one-valued in 7J, we never have

Proof. If we suppose that v is smooth, we can write for arbitrary a, bin [a 0 , b 0 ] and c, din [c 0 , d 0 ). Then the conclusion is obvious.

In the general case, regularize v by convolution with a smooth nonnegative function, and let this function tend to a Dirac mass; the conclusion still holds.

I

Let u be a function in W{~~( ~ X !Rt), such that Du is nonnegative. As a result of Lemma V .1 , the functions

(V.l) (V.2)
are defined on the set{(~, 77)/g ~ N~, "fJ > -g}, with N( a null set, and increasing in "fJ• Similarly, the functions ~ r-+ lim ~u (g + h, 7J) = ij+(~, 7J),

1)~0 U'YJ t r-+ lim ~u (~ -h, 71) = ij_(~, 7J) h~O UT) (V.3) (V.4)
are defined on the set {(~, 7])/77 ~ N 17 and g > -71}, with N 71 a null set, and increasing in g.

LetL~oc(X; m dx) be the set of functions on a subset X of 1R which are locally integrable with respect to the measure m(x) dx; here m is a locally integrable function (with respect to the Lebesgue measure).

We have more precise information on P± and q± in the following proposition.

PROPOSITION V.2. Let u be in W{~~( 1R X IRi) and let Du be nonnegative. Let a be an arbitrary Lipschitz-continuous function, with Lipschitz constant 1, and let a be positive on IR.

Then P±(x, a(x)) E Lloc(IR; (1 + a') dx), q±(x, a(x)) E Lloc(IR; (1 -a') dx), (V.5) (V.6)
and, almost everywhere on {xfj a'(x)j < 1},

o+u ( ( )) _ 1 . u(x, a(x) + h) -u(x, a(x)) 8 x , a x -tm h t h~O 1 = 2112 (p+(x, a(x)) + q+(x, a(x))), (V.7) 8-u ( ( )) -l' u(x, a(x) -h) -u(x, a(x)) 8 x, a x -tm h t h-l-0 - I = 2112 (p_(x, a(x)) + q_(x, a(x)).
(V.8) a.e. on 1R X IR; .

0 < t 1 ~ min a(x) ~ max a(x) ~ t 2 [a,b] [a,h]
and

The functions p+ and p_ are increasing in ' r' J, so that (V.9) which proves (V.5). Here Y is the graph of a in characteristic coordinates. Relation (V.6) is proved in the same fashion.

On the other hand, the following equality holds for almost all A in (-rr1in~ a, +oo):

ou 1 [au ou ] 8 t (x, a(x) + A) = 2112 8 ~ (x, a(x) + A) + 07J (x, a(x) + A) • (V.lO)
Let us show that ' lfol ~~ (x, a(x) + .\) = 2 ~1, (P+(x, a(x)) + q+(x, a(x))) (V.ll) in the L{ 0 c(~: (1 -a' 2 ) dx) topology.

Let A =(a+ a(a))/2 1 1 2 , B = (b + a(b))/2 1 1 2 ; we shall estimate p+(x, a(x)) -uE(x, a(x) + A) by a computation in ~, 1' J coordinates. Substitute in the first integral of (V.l2), g + A/2 1 1 2 = ,, which supplies Estimate (V.9) and the Lebesgue dominated convergence theorem imply that expression (V.l3) tends to zero as A decreases to zero.

The same estimate (V.9) allows us to conclude that the second term of (V.l2) converges to zero.

Therefore, we have shown that l.im topology. Thus we get (V.ll).

To conclude, denote k(x, A) = u(x, a(x) + A);

(V.l5)

we have just proved that :~ ELfoc(-mjn a(x), +oo;Lloc(IR;

(1-a' 2

) dx))

(V.16)
and that okjoA has right and left limits at any A. This implies that 1 . k(

•, A + h) -k( •, A) = 1 . ~ (. \ + h) lffi h lffi ':I\ , 1\ ' h+O h+O Ul\
in the Lf 00 (1R; (1a' 2 ) dx) topology, i.e., for A = 0,

1 2112 (P+(x, u(x)) + q+(x, u(x))) _ 1 . u(x, u(x) + h) -u(x, u(x)) _ o+u ( ( )) - lffi h -'=~ X, a X htO ut a.e. on {x E IR/1 a'(x)l < 1}.
This completes the proof of Proposition V.2.

I

The next result accounts for the fact that the derivative (ofox) u(x, a(x) +A) has no jumps across the line t = a(x). PROPOSITION V.3. Let u be in Wf 0 ~(R X Rt) and let Du be nonnegative. Let a be an arbitrary Lipschitz-continuous function, with Lipschitz constant 1, and let a be positive on IR. Then

P+(x, a(x)) (1 + a'(x)) -q+(x, a(x)) (1 -a'(x)) = p_(x, a(x)) (1 + a'(x)) -q_(x, a(x)) (1 -a'(x))
a.e. on {xfl a'(x)l < 1}.

(V.17)

Proof. With notation (V.l5), the following relation holds for almost all A.

~= (' , . \ )

= (1 + u'(•)) ~; (•, u{•) + .\)-{1 -u'{•)) ~ {•, u{•) +A). (V.18)
From (V.15), we can infer that 0 ~2:x ELfoc(-mJn u(x), +oo; .@'(IR)).

Therefore ( ok I ox) (., A) is continuous to P)' ( IR) with the weak topology.

On the other hand, by (V. Similar relations hold when the limit is taken as A increases to zero. The (very weak) continuity of okf ox with respect to ,\ completes the proof. I

We need the weaker assumption that a is only nonnegative; to obtain conclusions analogous to those of Propositions V.2, and V.3, we shall make a stronger assumption on u.

CoROLLARY V.4. Suppose that a is nonnegative and

au ou L(XJ ( +.£ 1 ( )• ot , ox E loc IR , loc IR) ,
let the other assumptions of Proposition V.2 remain.

(V.19)

Then, the functional inclusion of Proposition V.2 on P+ and q+ as well as (V.7) still hold. On the other side, p_(x, a(x)) eLloc({xfa(x) > 0}, (1 + a') dx), q_(x, a(x)) eLloc({x/a(x) > 0}, (1 -a') dx), a-u 1 at (x, a-(x)) = 2112 (p_(x, a(x)) + q_(x, a(x)))

a.e. on {x/1 a'(x)l < 1 and a(x) > 0}, and the conclusion of Proposition V.3 is replaced by p+(x, a(x)

) (I + a'(x)) -q+(x, a(x)) (1 -a'(x)) = p_(x, a(x)) (1 + a'(x)) -q_(x, a(x)) (1 -a'(x)) (V.20)
a.e. on {xfl u'(x)l < 1 and a(x) > 0}~

Proof. Hypothesis (V.I9) implies that the function

t t-+ P+(. -t, t) 1 [a-t,b-t]
which takes values in L 1 (1R) and is increasing, tends to a certain function p+ 0 • I [a.b] when t decreases to zero.

This simple remark allows us to relieve ourselves of the condition t 1 > 0 for estimate (V.9). Then (V.ll) is still true.

Concerning p_ and q_ , one must consider a compact interval [a, b] included in {xfa(x) > 0}, and argue as before.

I V.2. Proof of Uniqueness

Note first that conditions (IV.lO) and (IV.6) allow us to apply Corollary V.4, and to take a trace of oufot on the curve a(x) = 0. Thus, condition (IV.8) makes sense.

With the help of subsection V.l, it is now possible to state how the timederivative of a function satisfying the local energy conservation condition is transformed across a space-like curve. This is the goal of next proposition . a.e. on {x/1 a'(x)l < 1 and a(x) > 0}.

(V.22)

Proof By Lemma V.l, the functions e ~ ! uTI(~, 7J)!

r = {(x, t) I a~ x ~band t = a(x)},
in terms of the functions P± and q± defined in (V.l)-(V.4). We assume that a(x) is positive on (a, b).

(b+a(l;l))/2 1 1 2 • p(T) = p(f) = 2 1 1 2 f [P+(g, Y(g)) 2 -p_(g, Y(0)2] dg (a+o(a)) /2 1 / 2 (-a+a(a))/2 1 / 2 + I [q+(X(T)), '7)2 -ij_(X(YJ), 7])2] d7j. (-b+a(b))/2112 Substitute ~ = (x + a(x))/2 1 1 2 in the first integral and 7J = (-x + a(x))/2 1 / 2 in the second one. Then b p(F) = J {[p+(x, a(x)) 2 -p_(x, (x))2] (1 + a'(x)) a + [q+(x, a(x)) 2 -q_(x, a(x)) 2 ] (1 -a'(x))} dx. (V.23)
By Corollary V.4, the following relation holds:

(p+(x, a(x)) (1 + a'(x)) -q+(x, a(x)) (1 -a'(x))) 2 -(p_(x, a(x)) (1 + a'(x)) -q_(x, a(x)) (1 -a'(x))) 2 = 0 (V.24)
a.e. on {x/1 a'(x)l < 1 and a(x) > 0}.

Thanks to the identities

2p± 2 (l + a') + 2q± 2 (1 -a') -(P±(I +a')-q±(l -a')) 2 = (P± +. q±)2 (1 -a'2)
we obtain by subtracting the half of (V.24) from (V.23

) b p(T) = ~• J [(p+(x, a(x)) + q+(x, a(x)))2 a -(p_(x, a(x)) + q_(x, a(x)))2] (1 -a' 2 (x)) dx
and by relations (V. 7) and (V.20),

J b (\ o+u \2 j o-u \2) p(F) = (L Tt (x, a(x)) -Tt (x, a(x)) (1 -a'2(x)) dx.
As a and b are arbitrary, we obtain (V.22), if (V.21) is fulfilled.

I

Let u be the solution of P oo defined in (IV.I5), with associated measure 1-L = Du and let v be another solution of P oo , with associated measure v === Ov.

Let F be the support of v, and define a set J, analogous to the set I defined in We may apply Proposition V.5, and obtain o+v ot (x, a(x)) = 0 a. e. on {x/0 < a(x) < T(x) and I a'(x)\ < 1 }. (V.30) On the other hand, if we argue as in Proposition 111.3, we can estimate v \ 1 c by f ( o+v

8-v ) (v ! 1 c, ljJ) = (I -a' 2 (x)) 8t (x, a(x)) -Bt (x, a(x)) if;(x, a(x)) dx.
Using (V.29) and (V.30), we conclude that v ! 1 c = 0, and therefore V n Jc is empty. This contradicts (V.28).

I

Let us now apply relation (V.22) to v on the line t = r(x).

l fJ+v I I aw I Tt (x, T(x)) = Bt (x, T(x))
a.e. on {xjT(x) > 0, \ T'(x)\ < l}.

We may infer from Lemma 11.27 and condition (IV.9) that a+v ow Tt (x, T(x)) = -Tt (x, T(x)) and we obtain a contradiction when we compute v with a formula analogous to (V.31).

o+u = Tt (x, T(x)) a.e. on {x(T(x) > 0 \ T'(x)] < 1}, Moreover, the function v -u satisfies v(x, T(x)) -u(x, T(x)) = 0 on D(v -u) IHx,t)/t>-
Therefore P is empty, and the proof of uniqueness is completed. I

VI. THE FINITE STRING WITH FIXED ENDS

The solution of problem P 00 will enable us to solve problem Pr, and to give existence and uniqueness theorems analogous to Theorem IV.l.

We are given a function ({J such that cp(O) < 0, and initial data in the sense of distributions on (0, L), q;(L) < 0,

U 0 E H 0 1 (0, L), UI E L 2 (0, L ),
with the compatibility conditions u 0 (x) ):: rp(x) u 1 (x) ):: 0

'Vx E [0, L],
a.e. on {x E (0, L]/u 0 (x) = cp(x)}.

Then, the following result holds: It is elementary to check that u 0 , 11 1 , and cp satisfy conditions (IV.l) to (IV.5). Therefore, by Theorem IV.l, problem P oo possesses a unique solution u. Our goal is to show that u, restricted to ]0, L[ X ]0, 0::( solves problem Pr if a is suitably chosen.

Let w be the free solution of the wave equation in all of !R, with periodic data u 0 and u 1 • This function w is classically known to be odd and 2L-periodic.

There exists a positive number Ci. such that or I Lx I < a => w(x, t) -ffl(x) > 0.

(VI.l6) (VI.l9) where E = r (I ~: r + I ul 1") dx.

Let w be the free solution of the wave equation with data u 0 , u 1 • Then, by (VI.16), W = W In TL; 2 ,L; 2 +a = {(x, t)/0 ~ t ::;; L/2 + a:-I Lj2-X I}• (VI.20)

Besides, the influence line of the obstacle cp, with respect to w, satisfies

f'(x) = -1 f'(x) = +1 if if X E (0, a)
and f(x) < x + a, x E [ La:, L + a:] and f(x) < L + ex -x.

To prove this, define the sets E = {(x', t')fw(x', t') < q>(x')} and E' = {(x', t') E E and x' ~a:}. Let v be a solution of (VI. 7)-(VI.13). We shall extend v to !R+ X [0, cx/2] as foilows.

Let u be the solution of problem P oc; with initial data (VI.l4), (VI.I5), and the obstacle f;. Then, using (VI.l6), we can check that v is a solution of P cc• with data u 0 , u 1 , and f;.

The theorem of uniqueness for P oo shows that necessarily v is equal to u in [0, L] X [0, ~J2]. This process can be iterated any number of times to prove uniqueness in [0, L J X !R+.

I

APPENDIX Throughout this appendix, we will denote by a a Lipschitz-continuous function, with Lipschitz constant 1, defined on ~.

A. I . Elementary Results about Integration by Substitution

T

  . lntroduct1:on. 1.1. Notations and general hypotheses_ Statement of the results. 1.2. justification of the mathematical model. 1.3. Idea of the proof of existence. Role of the concavity hypothesis. 1.4. Comparison with the results of L. Amerio and G. Prouse. 1.5. Explicit computations of examples. II. The Line of Influence and Its Properties. Il.l. De~ finition and first properties of the domain of influence and of the line of influence. II.2. Study of the values of the free solution and of its derivatives on the line of influence. Ill. Solution of a Linear Auxiliary Problem. !ILl. Statement of the result, idea of the proof. 111.2. Computation of Ov in terms of w and a . III.3. Partial results of existence. III.4. Energy condition. End of the proof of theorem III.l. IV. Proof of Existence : Infinite String. I. INTRODUCTION I. I. Notations and General Hypotheses. Statement of the Results

2 )jdv 2 Iu 0 E

 220 H10c(~) = /v ELioc(IR dx ELloc(IR) ,

Fig. 1 .FIG. 1 .

 11 Fig. 1.

XFIG. 3 .

 3 FIG. 3. Evolution of function u of Example 2.

FIG. 6 .FIG. 7 .

 67 FIG. 6. Evolution of function u of Example 3.

PROPOSITION 11 . 3 .

 113 The line of influence is the graph of a Lipschitz-continuous function r, with Lipschitz constant I,

LEMMA 11 . 6 .

 116 Let U = {xjw(x, r(x)) > <p(x)}, and let (a, b) be any connected component of U. Then r(x) = min(r(a) + xa, r(h) + bx), Vx E [a, b]. (II.I3) Proof. Suppose first that (a, r(a)) and (b, -r(b)) both lie on one characteristic segment. For instance, r(b) -b = r(a) -a. By Proposition 11.4, we know that r(x) -x .:s; T(a) -a, r(b) -b ~ r(x)x, if a~ x ~b.

  elsewhere.We shall study in this section the distributionv = w + C * ~-t(w), (111.19) (111.20)

  170]. According• to Lemma A.5; p.(w) * tff is a continuous function, because the parts of characteristics are null with respect to the measure IL• Obviously, the function v defined by (111.20) satisfies conditions (111.2) and (111.3).

  (x, a(x)) = ~~ (x, a(x)) -[ 1 _ 1 a'(x) + I + l~'(x)l (I -a' 2 (x) 1,.(~).(III.26) Relation (Ill.26). giv~s (111.21) and••(I1I.22).

  by v(~, 77 ) = w(~, TJ)-_y(~, 77 ) -z(~, TJ), (IIL29) where .Y(~, 7J) = J~ we(f, Y(f)) g(f) df ) = I -Y'(f). 1,. ( 2112 ) ,(111.31)z(~, 'TJ) = fn wn(X(1J'), 'TJ') h(1J') d1J'

  TJ)ITJ ~ Y(t)}, OTJ = w 11 (X(TJ), TJ) h(TJ) a.e. on = 0 a. e. on {(t, 'YJ)/7J < Y(~)}; ay -07J = w{(X(TJ), TJ) h(TJ) a.e. on {(~, TJ)ITJ ~ Y(g)}, = 0 a.e. on {(~, TJ)fTJ < Y(t)}; and therefore ~ --OTJ = -wf(X(TJ), TJ) h(TJ) + (I -h(7J)) w 11 (X(7J), TJ) a.e. on {(t, TJ)/TJ ~ Y(~)}, (IIL38) = wn(X(TJ), TJ) a.e. on {(~, TJ)/7J < Y(~)}. 111.4. Energy Condition. End of the Proof of Theorem Ill. I The vector field S v is given by In characteristic coordinates, Sv is transformed into (III.39)

I

  IV. PROOF OF EXISTENCE; INFINITE STRING IV.I. Proof of Existence in the General Case Let us recall the hypotheses and the statement we have in the Introduction The function cp represents the obstacle, and in the sense of distributions. The initial data are Uo E Hloc(IR), ui ELroc(IR) with the compatibility conditions u 0 (x) ~ cp(x) u 1 (x) ~ 0 Vx E IR, a.e. on {xfu 0 (x) = cp(x)}.

  l (Infinite String). Under hypotheses (IV.l)-(IV.5), there exists a unique function u such that

(

  IV.25) we know from (IV.l5) that DZ ~ 0, and we can now infer from (IV.25) that a. e. on {(g, TJ)/Y(e) ~ TJ and~ E C}, (IV.26)

.

  fl!-At•'l' I :~ ( ~, y ( ~-2~/2) + 2~/2) -P+(~, Yg)) Id~. A-~/21/2 S (V.l3) As Y is decreasing, and by definition of P+, a.e.

8 8 ~

 8 (x, a(x) + A) -P+(x, a(x)) = 0"-l-0 £ . . . (V.l4)in the Lloc(IR; (I -a') dx) topology; the result is still true in the Lf 0 c(IR;(1a' 2 ) dx) topology, which is weaker. In the same fashion we showthatlim ~u (x, u(x) + A)-q+(x, q(x)) = U AtO O'YJin the Lfoc(IR;(1 + a') dx) topology, and therefore in the Lf 0 c(IR; (1 -a' 2 ) dx)

  14) lim ~~ ( •, a(•) + ,\)p +( •, a(•)) = 0 A~O u~ tn Lloc([R, (1 + a') dx) and analogously lfi Lloc(IR, (1 -a') dx).

. PROPOSITION V. 5 . 2 )J

 52 Let u be in the space Wf~~(~;; Ltoc(IR+)) n L~c(~;; Htoc(R)), such that Du is nonnegative, and let a be a Lipschitz-continuous, nonnegative function, with Lipschitz constant 1. If u satisfies ~ ( _ 2 ou ou ) c ( lou 1 2 + I [}u 1 Tt (x, a(x)) = Tt (x, a(x))

(

  II.4) by I = U r:..t • (V.25) (z,t}EF hence OW Tt (x, a(x)) = 0 a.e. on {x/0 < a(x) < T(x) and I a'(x)l < 1}. (V.29)

  VI.J. Under hypotheses (VI.1 )-(VI.6) there existe a unique function u such that(VI.7) u(x, 0) = u 0 (x) ou ot (x, 0) = ul(x) u(x, t) ? cp(x) Vx E IR, 'VxE ~' \f(x, t) E ~ X !R+, Ou~O in the sense of distributions in ~ X ~R; , Supp Du C {(x, t)ju(x, t) = cp(x)}, ~ ~ (-2 ou ~) ~ ( ot + ot ot + oxin the sense of distrz"butions in iR X IR~ .VI.l. ExistenceExtend the function cp to all of 1R by settingcp(x) = cp(O) + xcp'(O+) cp(x) = cp(L) + (x-L) cp'(L-) if ifExtend the functions u 0 and u 1 to all of IR by setting fori= 0, 1.ui(x) = ui(-x), ui(x + 2L) = ui(x)Define initial data u 0 , z1 1 on IR by u 0 (x) = max(u 0 (x), cp(x)), x) > cp(x), u 0 (x) ~ ffl(x).

  Then f(x) = min{l x-x' I + t'f(x', t') E 2}. (VI.21) Taking as new initial data u(•, a) and ut(•, a), we can solve again a problem Pr in a bounded time interval [ ~, a' + a]. But the number ex' depends only on the energy of the data, and therefore can be taken to be equal to ex. By recurrence problem Pr can be solved on the whole infinite rectangle [0, L J X ~+; relation (VI. 7) holds, because (VI.23) is true almost everywhere on fR~. I VI.2. Uniqueness

  LEMMA A.l. The following equivalence holds:(i) f ELloc((l + a) R, dx), (ii) f o (I+ a) ELfoc(~, (1 +a') dx).Moreover, if one of conditions (i) or (ii) is satisfied, then the identity b b+a(b)J j(x + a(x)) (1 + a'(x)) dx = J f(x) dxa a+a(a) (A.l) £s true for arbitrary real numbers a and b.

  Under the hypothesis that E is nonvoid, M 7 (and therefore M 7 + and M 7 -) is not null (with respect to the Lebesgue measure); this is a result of Theorems IV.2 and III.l below; see the end of subsection IV.l.

				(IL24)
	a 1 (a a) ax = 2 112 8~ -81) ' IlL SoLUTION OF A LINEAR AuxiLIARY PRoBLEM
	III.l. Statement of the Result, Idea of the Proof		(II.25)
				then the
	following inequality holds:
	ow a~ (x, r(x)) ~ 2112 tp'(x+)		(II.26)
	(respectively
	aw	(<p' (x-)) 21 / 2	•	(II.27}
	Proof By hypothesis
		w(x, r(x)) = cp(x)		(11.29}
	and, by definition of r,
	w(x + h, r(x) +h) ~ <p(x +h), Vh ~ 0.	(II.22) (11.30}
		g-T}
		X= 2112 ,

(II.23) and therefore a 1 (a a )

of = 2 112 ox + ot ' a 1 ( a a ) O'YJ = 2 112 -ox + Tt ' LEMMA I I. 7. Let x be such that w(x, r(x)) = cp(x). If moreover (3wfo~) (x, r(x)) exists (respectively (owfih]) (x, T(x)) exists), OTJ (x, -r(x)) ~ -If (owjot) (x, -r(x)) exists, then ow Tt (x, T(x)) ~ 0.

(II.28)

Subtracting (11.29

) from (II.30) and passing to the limit, (II.26) obtains. The other two inequalities are similarly proved.

I

As a result of Corollary A.2 note that

(11.26)

holds almost everywhere on MT + = {x/T'(x) > -1 }, (II.27) holds almost everywhere on MT-= {xjT'(x) < 1}, and (11.28) holds almost everywhere on M 7 = {x/1 T'(x)! < 1}.

  Proof We can write formula (111.20) in the form v(x, t) = w(x, t) + f J dp.(w).

					(111.22)
					lx.t
	Suppose t is greater than r(x), and let a(x, t) be the greatest number such that
			a(x, t) -a(a(x, t)) = x -t,
	and let a' be any number such that
			a' -a(a') = x -t.
	Then a'(s) = 1 on (a', a(x, t)) and
	f a(x,t) ,	OW 1 A(x') Tt (x', a(x')) ( 1 -o.7(x') 2 ) dx' = 0.
	v(x; t) = w(x, t)-	f a(x,t) b(x.t)	OW IA(x') T (x', a(x')) (l -a'(x') 2 ) dx. t	(111.23)
	For simplicity, set a(x, a(x) + h) = a(h) and b(x, a(x) + h) = b(h). From the
	hypothesis I a'(x)l < I, we deduce
			lim x -a(h) = h~O h	1 1 -a'(x)'	(111.24)
			lim b( h) -. x = 1 h~O h 1 + a'(x)	.	(111.25)
	Using (111.23), we may write
	1 . v(x, a(x) + h) -v(x, a(x)) lffi _..;_____;:..._.:...__-=--~_,;..._:....:.... h.j.O h
	= lim w(x, a(x) +h)-w(x, a(x))

Then, if the point xis in B, zf (owfot) (x, a(x)) exists, if a'(x) exists, and if

I a'(x)l < 1, then ( ()+vfot) (x, a(x)) exists. Moreover, if a(x) > 0, then o+v ow . Tt (x, a(x)) = -ot (x, a(x)).

(III.21)

If a(x) = 0, then fJ+v Tt (x, a(x)) = 0. a If, similarly, b(x, t) is the smallest number s~ch that b(x, t) + a(h(~, >t)) = x + .t

then we may write, if t is greater than a(x),

  Proof. The divergence of S v , in the sense of distributions is given by

	sense of distributions.	(111.40)
	Now, using formula (111.37)
	Integrate by parts in ?J;
	f I av l2 ot/!_ dg d ag a?J 11
	= f {l(g, Y(g)) [I we(g, Y(g))l 2	(III.42)

< v . s .r.> _ 21, 2 [(I Bv 1 2 a{l) (I av 1 2 a{;)] v ' 'f --aTJ ' a g + a g ' a 11 •

(111.41)

  Proof. Let us show, for instance, that P+ and p_ are in L} 00 (!R, (1 + a') dx), and for this purpose, fix two arbitrary numbers a < b. By definition,

	ou .P+(x, t) = p_(x, t) = ag (x, t)	a.e. on 1R X ~R;
	and, in the same way
	ou q+(x, t) = q_(x, t) = 8 ' YJ (x, t)
	We can find real numbers t 1 and t 2 such that

  2 and 7J ~I iiE(~, 7])[ 2 are of bounded variation, respeqtively, for almost every ' rJ and almost every t.

	Therefore the distribution

a ( au ou ) o ( jou \ 2 jou j 2 ) P = ax -2 ax Tt + ot • ox + 8i '

which is written, in characteristic coordinates, is in fact the sum of two measures.

Let us compute the measure by p of the set

  I a'(x)l < 1, we obtain as in the proof of Lemma V.6 that

	cp(x) = v(x, a(x)) >-u((x, a(x)) >-cp(x);
	hence
	o+v Tt (x, a(x)) = 0	a.e. on {xja(x) > T(x) and I a(x)/' < 1},
					(V.31)
					(V.32)
					(V.33)
					(V.34)
	We deduce from (V.32)-(V.34) that
	v(x, t)-u(x, t) ~ 0	on	{(x, t)ft > T(x)}.	(V.35)
	Set
	V= V l{(x.t)/t>T(x)}'
	P = suppv,
	and

r(x}} ):: 0.

J = u r;t = {(x, t)Jt ~ a(x)}.

(:x:,t>eP Suppose P is nonvoid. Then if t is greater than a(x), v(x, t) = u(x, t) + lv( T;,t) > u(x, t). lf

  ~ w(O, t) -cp(O) -I x [ 112 0 ox (x, t) + cp'(x) dx) -I x 11 cp'(O+)! , mtn Evz + (f~ cp'(x)2 dx)112 + I cp'(O+)I Ll/2 '

					2	1 /2
					1
					(VI.l7)
	w(x, t) -cp(x)
	~ w(L, t)-cp(L)
	-	( rL low I L-x 1 112 Jo ox (x, t) + cp'(x) dx) -(L-x) I cp'(L-)1, lz 112
					(VI.18)
	which supplies a possible a,
		112 _	•	[	-<p(O)
					-cp(L)

Indeed w(x, t) -cp(x) (f t I ow a: -J Ellz + (f ~cp'(x)2 dx)vz + 1 cp'(L -)I £1 12 '

where x ~ r(x) is the line of influence of the obstacle 9' with respect tow, following Definition

11.2.

Then the junction u = w + C * tt(w), (IV.l5) with 8 the elementary solution of the wave equation solves problem (IV.6}-(IV.12).

Proof of Theorem IV.2. By Theorem III.l, we know that u given by (IV.15) satisfies conditions (IV.6)-(IV.8) and (IV.ll ). Condition (IV.lO) is seen to be satisfied by Lemma (II. 7) and the explicit expression (IV.I4) and condition (IV.12) results from Proposition 11!. [START_REF] Crtrini | Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un ostacolo[END_REF].

So it remains to check condition (IV.9), for t ~ r(x). Set if X is one-valued in TJ (same notations as in subsection III.2).

With notations (III.ll) and (11!.12), and thanks to Lemma II.5, we have

a.e. on M,. = {x/1 r'(x)l < 1 and -r(x) > 0}.

(IV.19)

Let us differentiate (IV.l9); then by a linear combination with (III.I5) which we write again as

a.e. on M,.

we obtain

Therefore, Lemma Ill. 7 implies a.e. on M ., , a.e. on M., .

a.e. on M ,. .

By the definition of the line of influence 2(X(?J), ?J) ~ 0. Uniqueness will be proved in next chapter.

We can infer from Theorems III.l and IV.2 that if E is nonvoid, then I T'(x)l l T'(x)l is not almost everywhere equal to one. Assume indeed that \ T'(x)l = 1 a.e. on {x/T(x) > 0}.

(IV.35)

Then, w is clearly the unique solution of problem (111.5)-(III.9) as conditions (111.6) and (111.9) are automatically fulfilled. We have proved by Theorem IV.2 that on IR X IR+.

Here, the measure /L(w) is equal to zero; therefore on IR X lfl+, (IV.36)

and E is void.

I

In the same fashion, if u 1 is almost everywhere nonnegative, E is void.

IV.2. An Alternate Construction when the Obstacle cp is Constant

A. Bamberger has the idea of a simpler construction, which is especially convenient when cp is a constant, which can be taken equal to zero.

For this purpose, we can deduce from the results established between (IV.21) and (IV.24) the formulas if ~ ?: X(?J), ~ + Y(e) > 0, Y'(e) <:: 0, and w(~, Y(~)) = 0;

(IV.37)

elsewhere.

If we define almost everywhere a kernel K by (IV.39) Formula (IV.39) can be converted into the still simpler expression, where r -= -inf(r, 0), (IV.40) Formula (IV.40) will be proved, and generalizations and consequences will be given, in a joint work with Bamberger [START_REF] Bamberger | Etude d'un probleme hyperbolique avec contrainte unilaterale[END_REF].

V. PROOF OF UNIQUENESS: INFINITE STRING

The proof of uniqueness relies on several trace results which, I believe, do not appear elsewhere in the literature. The main difficulty is to define cleanly traces of derivatives ux and Ut on a space-like curve, i.e., a curve x ~ r(x) with I -r' loo < 1, when u is a solution of P oo . Once these traces are defined, we can derive from (IV. J 2) how the derivatives are transformed across a discontinuity line. Then, by comparison with the solution built in Section I V, uniqueness is proved. Note that allowing the initial position u 0 to be equal to rp introduces an extra difficulty, as we have to give a meaning to (IV.8), using the other relations.

V .l. Results of Trace

We prove how condition (IV.IO) can be used to define traces. LEMMA V.l. Let il be a function in WI• 1 ([a 0 , b 0 ] X [c 0 , d 0 J), such that Then YJ ~ (Oil( o~) ( ~, YJ) is an -increasing function from ( a 0 , b 0 ] to L 1 ( a 0 , h 0 ) and e~ (oujor;) (t, 17) is an increasing junction from [a 0 , h 0 ] to L 1 (c 0 , d 0 ).

Arguing as in Section II, we can see that

where a is Lipschitz continuous, with Lipschitz constant l. Moreover, if

Let t be greater than a(x). Then, by the definition of the support of a measure v(x, t) = w(x, t) + }v(T;,t) > w(x, t).

As a first stage, we shall show, by contradiction, LEMMA V.6.

Proof. We need to show that

Let V= {(x, t)fv(x, t) > w(x, t)} and suppose that

(V.28)

and in particular V I Vrll(J = 0.

If a(x) is smaller than r(x), and if I a'(x)l is smaller than 1, then relation (V.26) holds, and moreover

If (x', t') is in /t, then by (V1.16) and (VI.20) or t' ~ x' + a:.

. By the assumption that f(x) < x + ex, we see that

and this proves (VL21 ). Relation (VI.22) is similarly proved.

Note that u satisfies the boundary conditions, for t ~ ex. Indeed

then, thanks to (VI.21) and (V1.22),

Then by (VI.20), and since w is odd and 2L-periodic, This classical result is proved for instance by Hewitt and Stromberg [10, 18.24, 18.25, and 20.5].

Applying this result to t/; . I + a,

the implication (i) ~ (ii) obtains, with identity (A.l ).

The inverse of I+ a=~ can be multivalued; in this lies the difficulty of the converse.

The first point to check is that if f o if; is measurable with respect to the measure if/ dx, then f is measurable with respect to dx.

For this purpose, let

Set M is measurable with respect to if/ dx. Therefore, there exists for any positive e and compact K, a compact set K' such that

As the function ~-~ defined almost everywhere is increasing, it is measurable. Let K 1 be any compact set in IR, and E 1 be any positive number. Then, there exists a compact K~ such that is continuous;

Then llb(M) = lM o l{;-I, restricted to the compact set K{ n if;(K'), is continu- ous. Besides, the measure of K 1 n ~(K)i(K~ n f(K')) is majorized by e + e 1 .

Therefore, i{l(M) is measurable. Now the proof of (ii) ::::? (i) is easy by contra- diction: Suppose that f is not locally integrable. Then there exists a sequence of integrable functions fn converging to I almost everywhere on [a + a( a), b + a( h)] such that J I fn(x + a(x))l dx--+-+ oo; a+a(a)

Vn, a.e. in x.

This gives a contradiction, using identity ( A.l ).

I

CoROLLARY A.2. Let I be in Wl;~(ll + a) IR), then f o (I + a) is absolutely continuous on any compact interval of IR and

Proof. By Lemma A.l we have

a if~ =I+ a, and [a, b] is an arbitrary compact interval.

The left-hand side of (A.5) can be expressed as tb(b)

since f is absolutely continuous. This proves Corollary A.2.

I

A.2. The Set t = a(x) in Characteristic Coordinates

The characteristic coordinates (~, ' YJ) are defined as

The graph of a in characteristic coordinates is the set We shall write Y-1 =X and

The set Y has the following property.

and this expression is non positive for all x and x'.

I

We shall denote by Y the function defined on the set

and likewise, we define X as the "one-valued part" of X = Y-1 • It is possible to show that Y is maximal monotone (for a definition see Brezis [START_REF] Brezis | Operateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert[END_REF]) in IR X IR. A.3. (i) u is continuous. (ii) The parts of characteristics are null with respect to Du.

Proof. Let Du = ,_,+ -,..,_be the decomposition of Du = ,..,_ in its positive and negative parts.

Let us prove first (i) => (ii). The following inequalities hold:

If u is continuous, we can pass to the limit in (A. 7), and which shows that f.J-(Roc) = 0. Conversely, to prove (ii) ==> (i), suppose that the parts of characteristics are null with respect to Du, and fix an arbitrary point ( x 0 , t 00 ) with t 00 > 0.

Let tn be a sequence decreasing to too , and t~ a sequence increasing to ! 00 • Then (A.8) n U T~. t~ = int T~. tctJ • (A.9) n As the parts of characteristics are Ou-null, Therefore, thanks to (A.8) and (A.9), This shows that ( tff * !L±) (x, t) is continuous with respect to t in any point.

To prove the continuity of u with respect to x and t, just note the inclusions T;;,.t-l x-x 0 1 C T;,t C T~.t+ lx-x 0 1 • We have thus the inequalities and this completes the proof of Lemma A.4.


On the article by S. I. Vavilov: "Remarks on the empirical accuracy of the optical superposition principle".1 K. Shaposhnikov.

Prof. V a v i l o v writes: "It is being said that the independence of the propagation of a bundle of light of the passage of other bundles on its path is explained by the wave theory of light, however, this is not so easily reconciled with the primitive corpuscular idea". It is extremely strange and not understandable why Prof. V a v il o v contrasts the developed wave theory with the "primitive corpuscular idea" and does not say anything about the more or less developed corpuscular theory.

In the following lines I will show that the superposition principle (the independence of the propagation of waves from the encounters of waves) is extremely easily reconciled with the mechanical (corpuscular) theory of light.

Indeed, let us assume that an encounter of two quanta of light occurs: one has the energy ℎ𝜈 and the momentum ℎ𝜈 𝑐 before the collision, the other ℎ𝜈 1 and ℎ𝜈 1 𝑐 , respectively. After the collision, the first one has got the energy ℎ𝜈 ′ and the momentum ℎ𝜈 ′ 𝑐 , the second ℎ𝜈 ′ 1 and ℎ𝜈 ′ 1 𝑐 , respectively. In the case of an elastic collision, the principle of energy conservation demands:

ℎ𝜈 + ℎ𝜈 1 = ℎ𝜈 ′ + ℎ𝜈 ′ 1 (1)
or:

𝜈 + 𝜈 1 = 𝜈 ′ + 𝜈 ′ 1 . . . . . . . . . . . . . . . (2)
The principle of momentum conservation, on its turn, gives after division by ℎ 2 𝑐 2 (cf. fig. 1).

𝜈 2 + 𝜈 2 1 -2𝜈𝜈 1 cos 𝜙 = 𝜈 ′ 2 + ℎ𝜈 ′ 2 1 -2𝜈 ′ 𝜈 ′ 1 cos 𝜙 ′ .
(3) 216 From the last equation and from (2) follows:

𝜈𝜈 1 sin 2 𝜙 2 = 𝜈 ′ 𝜈 ′ 1 sin 2 𝜙 ′ 2 . . . . . . . . . . . . . (4)
As shown in our figure in a clear manner, the two relations ( 2) and ( 4) are insufficient to determine the direction of movement of the quanta of light after the encounter where the total momentum 𝐴𝐵 is conserved, but the directions of movement of the quanta of light before and after the collision can differ from each other considerably.

As in the wave theory of light, we do not have an unique solution. However, together with Prof. V a v i l o v we accept the superposition principle as an empirical fact. Do the equations ( 2) and ( 4) contradict it? No! Let 𝜙 = 𝜙 ′ ; then we obtain from (4) :

𝜈𝜈 1 = 𝜈 ′ 𝜈 ′ 1 . . . . . . . . . . . . . . . . . (5)
After squaring both parts of (2), we subtract from it the product (5) multiplied by 4. After taking the root we obtain:

𝜈 -𝜈 1 = ± (𝜈 ′ -𝜈 ′ 1 ) . (6)
If we take the + sign, from it and equation ( 2) follows:

𝜈 = 𝜈 ′ ; 𝜈 1 = 𝜈 ′ 1 .
One sees easily, that this way a passage of one quantum of light through the other will occur what does, perhaps, not correspond to our idea of a collision. However, if we take the -sign, we obtain by the same way:

𝜈 = 𝜈 ′ 1 ; 𝜈 1 = 𝜈 ′ ,
i.e. in the collision of a quantum of light with another one a complete exchange of frequencies and directions occurs. Formally, this is precisely the superposition principle. This way we see, that the mechanical theory does not capitulate in view of this "marvellous property of light"... [START_REF] Ивановский | Ivanovskiȋ gosudarstvennyȋ ènergeticheskiȋ universitet[END_REF] We now direct attention to the following: As the above considerations are completely independent of the dimensions of the quanta of light, reasoning about the latter cannot be performed from out the superposition principle. And meanwhile, Prof.

V a v i l o v, apparently relying on "primitive ideas", is solving this whole question extremely quickly. He says at the beginning of § 2: "But if the effective size of the quanta was of the order of the light wave, we should observe a completely noticeable scattering of light, when two bundles intersect", i.e. a lack of superposition. From the above discussion it is clear, that the problem of the dimensions of quanta cannot be solved that easily. However, we will not delve into its closer considera-217 tion now but will do this, perhaps, at the appropriate point in time. We only say this: The mechanical theory demands that the linear extensions of the cross section of the quanta should be of the order of the wave length. As we have convinced ourselves, this idea spelled out by many sufficiently competent authors cannot, under no circumstances, be undermined by the critical remarks by Prof. V a v i l o v.

  Fig. 1.

2 [

 2 the Redaction on February 27, 1929) ПО ПОВОДУ СТАТЬИ С. И. ВАВИЛОВА: «ЗАМЕЧАНИЯ ОБ ЭМПИРИЧЕСКОЙ ТОЧНОСТИ ОПТИЧЕСКОГО ПРИНЦИПА СУПЕРПОЗИЦИИ».1К. Шапошников.Проф. В ави л ов пишет: «Независимость распространения светового пучка от прохождения других пучков на его пути объясняется, как сказано, волновой теорией света, но не столь легко примиряется она с примитив ными корпускулярными представлениями». Чрезвычайно странно и непо нятно, почему проф. Вавилов противополагает раз работанной волновой теории «примитивные корпуску лярные представления» и ничего не говорит о более или менее разработанной корпускулярной теории.В последующих строках я покажу, что принцип суперпозиции (независимость распространения волны от встречных волн) чрезвычайно легко примиряется с механической (корпускулярной) теорией света.Действительно, положим происходит встреча двух Рис. 1. квантов света: один до удара имеет энергию hν и ко личество движения hν/c, другой соответственно hν1 и hν1/c. После удара пер вый из них получает энергию hν' и количество движения h ν ' / c , второй соот ветственно hν1' и hν1'/c. Закон сохранения энергии в случае упругого = ν' + ν1' . . . . . . . . . . . . . . . ( 1 ) Закон сохранения количества движения дает в свою очередь после сокращения на h2/c2 (см. рис. 1). ν 2 + ν12 -2νν1 cosφ = ν2' + ν12' -2ν'ν1' cosφ' 1 C. И. Вавилов. Ж.Р.Ф.-Х.О. (ч. физич.) 60, 555, 1928. Ж . Р . Ф . -Х . О ., ч. физич., т. LXI. вып. 3 . Facsimile of Zh. Russk. Fiz.-Khim. Obshch., Ch. Fiz. 61(1929)215-217] К. ШАПОШНИКОВ Из этого последнего равенства и из (1) вытекает: 2 ν ν 1 s i n 2 φ / 2 = 2 ν ' ν 1 ' s i n 2 φ ' / 2 . . . . . . . . . . . . . (2 ). Двух соотношений (1) и (2) недостаточно для определения направле ния движения квантов света после встречи, что наглядно представлено на нашем рисунке, где полное количество движения AB сохраняется, но на правления движения световых квантов до удара и после могут сильно от личаться друг от друга. Как и в волновой теории, мы не имеем здесь одно значности в решении. Примем, однако, вместе с проф. В а в и л о в ы м принцип суперпозиции как эмпирический факт. Противоречат ли ему уравнения (1) и (2)? Нет! Пусть φ = φ'; тогда получаем из (2 ): ν1ν = ν'ν1' . . . . . . . . . . . . . . . . . ( 3 ) Возведя обе части равенства (1) в квадрат, вычтем из него учетве ренные произведения (3). После извлечения корня получаем: ν -ν1 = ± (ν' -ν1') Если взять зн ак + , то отсюда и из равенства (1) следует: ν = ν' ; ν1 = ν1'. Легко видеть, что при этом будет проникновение одного кванта света через другой, что, может быть, не соответствует нашим представлениям об ударе. Но если мы возьмем знак -, то таким же путем получим: ν = ν1' ; ν1 = ν', т. е. при ударе одного кванта света о другой происходит полный обмен частотами и направлениями.Формально это тот же принцип суперпозиции. Мы видим таким обра зом, что механическая теория не становится в тупик перед «удивительней шим свойством св е та »...1Обратим теперь внимание на следующее: так как предыдущие сооб ражения совершенно не связаны с размерами кванта света, то о последних нельзя судить с точки зрения принципа суперпозиции. А между тем проф. В а в и л о в , очевидно опираясь на «примитивные представления», решает чрезвычайно быстро весь этот вопрос. Он говорит в начале § 2: «Если бы эффективные размеры квантов были порядка световой волны, мы должны бы наблюдать вполне заметное рассеяние света при пересечении двух пучков», т. е. отсутствие суперпозиции. Из предыдущего ясно, что вопрос о размерах кванта не может быть так просто решен. Однако входить в его подробное 1 См. слова Г ю й ге н са , приводимые проф. В ави л овы м . обсуждение мы сейчас не будем, но, может быть, сделаем это в свое время. Скажем только следующее: механическая теория требует, чтобы линейные размеры поперечного сечения кванта были бы порядка длины волны. Эта идея, высказанная многими в достаточной мере компетентными авторами, ни в коем случае, как мы убедились, не может быть поколеблена крити ческими замечаниями проф. В а в и л о в а . Иваново-Вознесенск. Политехнический институт. (Поступило в Редакцию 27 февраля 1929 г.)

С. И. В а в и л о в [S. I. V a v i l o v]. Ж.Р.Ф.-Х.О. (ч. физич.) [Zh.R.F.-Kh.O. (ch. fizich.)] 60, 555, 1928.

Cf. the words by H u y g e n s, cited by Prof. V a v i l o v.

Acknowledgements

I am indebted to M. S. Aksent'eva (redaction of the journal Usp. Fiz. Nauk, Moscow) for their invaluable help in exploring the copyright situation of the original Russian article.

Contents

Supplementary information

Translator's notes:

1. Given the copyright situation in the Soviet Union at the time of publication (1929) of the original article it seems reasonable to assume that the article is in the public domain.

2. The numbers on the right margin refer to the page numbers of the original article. A facsimile of the original article is appended (see p. P-9).

3. The English translation of the quotes from the preceding article by Vavilov (Zh. Russk. Fiz.-Khim. Obshch., Ch. Fiz. 60(1928)555)) is taken from its English translation arXiv:1708.06817.

Concerning footnote 1 on p. 216 [T-2]:

The words of Huygens are quoted here after the English translation of Traité de la Lumière: [2], pp. 21-22.

Literature

[For references in Cyrillic letters, we apply the (new) Mathematical Reviews transliteration (transcription) scheme (to be found at the end of index issues of Mathematical Reviews).]

Full details of the cited literature:


Luc Verdure

Franck Schoefs

Pascal Casari

Humberto Yáñez-Godoy

Humberto Yanez-Godoy

Uncertainty updating of on-pile wharf after monitoring

Keywords: on-pile wharf, monitoring, uncertainty updating, manufactory uncertainties

INTRODUCTION

Reliability of complex structures such as bridges, offshore structures and wharves require some investigations for uncertainty and model calibration. Moreover, building uncertainties are high due to complexity of conditions during construction (effect of tide, duration of building...). Expert judgment or uncertainty analysis coming from other similar structures are very hard to transfer. This paper focuses on this point and focuses on the interest of conducting quasi-systematic monitoring.

An instrumentation has been achieved on an onpile wharf in order to monitor its behavior at short and long terms. A probabilistic approach of modeling which takes the variability of mechanical loadings, behavior and manufacturing into account is conducted. Usually, the instrumentation reports do not give priority to the use of mechanical modelling to explain the response of the structure. The focus is mostly put on the validation of a pre-existing model which has to be validated (Quirion and Ballivy 2000), on a basic health monitoring (Seim et al. 1999), or on the evaluation of monitoring devices (Maaskant et al. 1997).

Here the use of mechanical modeling is the way to identify some more features from measurements. The method is applied to the lateral behavior of the wharf as it conditiones noticeably the reliability of the wharf submitted to extreme events (storms). The second section aims to gives a global overview of monitoring strategy in the specific case of wharves. The third part is deoted to the design of the wharf and the presentation of the instrumentation. Then the fourth part explains the 3D finite element meshing to represent the mechanical behavior and the uncertainty and sensitivity studies. The fifth part presents a simplified 2D model based on Timoshenko's beam hypotheses and allows to perform Monte-Carlo simulations. The focus is put on anchoring rods because they have been considered as a priority in the experimental investigations after a report of severe damage on similar components on other structures. Finally, the sixth section aims to estimates parameters of selected input basic random variables from measured loading in rods an inverse analysis. It is performed by means of simplex algorithm. Several cases are studied and their pertinence in regard to measured loading is analyzed on the basis of Monte-Carlo simulations.

STATE OF THE ART FOR WHARF MONITORING

Only a few experimental investigations have been reported on wharf structures. This fact is probably due to the difficulty to select suitable sensors associated to a lack of real need of a feedback in design studies for further wharves design andconstruction. Five studies have been found in scientific references:

The Seaforth Dock in Liverpool (Uhf, 1969): moulded wall supported by vertical arches topped by a docking frame and a slab itself tied up to the bank by active injected anchoring rod placed in a vertical position.

The Bougainville dock at Le Havre [START_REF] Blivet | Comportement d'un quai en paroi moulée au port du Havre[END_REF]: moulded wall anchored with a layer of active injected rods.

The Osaka dock at Le Havre (Delattre et al., 1999) and a dock in Calais harbour [START_REF] Delattre | Etude expérimentale du comportement du quai en eaux profondes du port de Calais[END_REF]: moulded wall anchored with two layers of passive rods clamped to a line of sheet piles.

Docks of Hamburg harbour (Gatterman et al., 2001[START_REF] Rodatz | Back analysis of geotechnical measurements at the port of Hamburg based on FEM[END_REF][START_REF] Rodatz | Pressure and deformation measurements at two retaining walls at the port of Hamburg[END_REF]: hybrid technology with a retaining wall and slab on piles, anchored to the bank by slanting passive rods.

The most widespread monitoring strategy consists in instrumenting only one section of the structure with a set of sensors to measure mainly the stress state in the soil, the vertical deflection of the wall and anchoring loads like in classical experiments for retaining walls. On the other hand, seldom are measurements along the length of the structures.

This approach of investigation in a cross section brings a representation of the main loads and corresponding displacements and strains involved at the location of the sensors, but two problems are to be noticed: Firstly, measurements depend on the natural variability of the soil, which is impossible to quantify at this location compared to the others along the structure, and this leads to a lower confidence in the data recorded.

Secondly, the successive cross sections of the structure are not mechanically independent and events occuring in one section may affect significantly the behaviour of another one and especially the monitored one.

The last point to mention is that even if monitoring is conducted during a long time, only the first months of data collected are reported. Then scientists can not investigate long term phenomena like creep behaviour.

WHARF DESIGN AND INSTRUMENTATION

Technological description of the wharf

The wharf is built on piles driven in the ground. It has been designed for big tonnage ships, and is located near the sea, at the mouth of the river Loire. Two pictures of this wharf during its building are shown on picture in figure 1. The main design concept is to distinguish the ability of the structure to support vertical and transversal loads. It generally leads to the best compromise between technology and cost: as an example, rods are dedicated to bear a high part of transverse loading (ship mooring,...) and allows to keep pile diameter in a reasonable range in terms of beating power.

It is made up of a reinforced concrete deck (255 x 43 m) with a triangular network of reinforced concrete beams 0.75 m high and a reinforced dam 0.25 m thick. At each node of this network, a steel pile is filled up with concrete: They are altogether 332 with diameters varying from 0.711 to 0.914 m and steel thickness from 10.3 to 12 mm depending of the pile's location. They are beaten down to the rock deck at around 45 m deep. The function of this platform is to support vertical loading of containers, a mobile crane (335 tons) and lifting frame structures (1100 tons). The wharf is leaned against a vertical reinforced concrete wall ("back-wharf wall") 4.35 m high. It is dedicated to embankment loading: The embankment ballast is 0/180 mm. Under this wall, a vertical sheet pile 4.5 m high prevents small soil particles from leaking. The role of the platform is also to average horizontal loading due to embankment and ship mooring which can not totally pass through the radial capacity of piles. For this reason, the wharf is anchored inside the bank by 38 steel rods of diameter 8.5 mm and length 20 m. At their ends, rods are clamped between the back-wharf wall and a vertical reinforced concrete anchoring plate embedded in the bank. An explaining diagram shows the architecture of the structure in Fig. 2 Figure 1. Views of the wharf during its building.

Instrumentation

The wharf is instrumented with two kinds of sensors (shown in Fig. 2).

• 12 couples of vibrating wire sensors (WR) clamped on the anchoring rods: The use of two sensors in the same cross section, one on top and one at the bottom, allows the measurement of normal load and bending moment. Instrumented rods are the end ones (upstream and downstream), and 10 regularly distributed ones along the wharf.

They are classified with their distance x along the wharf, from upstream (x = 0) to downstream (x = 250 m). Rods at the ends are instrumented in three sections in order to analyze the effect of soil along the rod (see figure 5). • 1 piezometer, located inside the bank, in the middle of the wharf, used to measure the water level. Thus, knowing the actual sea level in the river Loire -provided by the port services-, the difference between the levels in the sea and in the bank can be deduced.

The measures are recorded every 3à minutes with a Campbell CR10X acquisition system. This period is relevant for the description of periodic processes like tide affecting the mechanical response of the structure (approximately 12 hours periodic). The same period of data storage was used at the Container Terminal Altenwerder, in the port of Hamburg, which is another instrumented on-pile wharf, build with quite the same technology [START_REF] Gattermann | Modified instrumentation and results of stress and deformation monitoring at the new quay wall construction -Container Terminal Altenwerder, Port of Hamburg[END_REF]. Data have been recorded since the 1 st of October 2002. The number of sensors has been determined for cost reasons and also to get a satisfactory confidence in the estimate of mean and standard deviation of measurements.

Data pre-processing.

A first processing is made on the initial data. The water level is deduced from a pressure measurement providing a proportional electric signal. For load measurements, the vibrating wires give specific frequency values transformed into strains by means of equation 1 :

) N K(N ε 2 0 2- = (1)
where K is the calibration coefficient given by the supplier, N 0 the frequency at a time t = t 0 and N the current frequency. Values of N 0 have been recorded for all sensors just once they have been clamped on the rods. They are called "zero-states". ε is then an absolute measure.

The tension load F is deduced from both strain values for the same couple of sensors with equation 2 :

2 ε ε ES F l u + = (2)
where E is the Young's modulus of the rod (steel), S the cross section of the rod, u ε and l ε are respec- tively the strain in the upper and lower vibrating wires.

SENSITIVITY AND UNCERTAINTY STUDIES

Finite element model as numerical support for sensitivity analysis

Reliability analysis suggests to carry out uncertainty and sensitivity studies before and during modelling in order to provide robust probabilistic models as input to limit state functions. Response Surface Methodology is devoted to this last point. It has been widely developed during the last decade and response function models, including non-linear ones are now tractable, especially for the specific requirements of structural reliability analysis (Labeyrie and Schoefs, 1995). This way is no more investigated in this paper as the subject is to focus on a better understanding of physical underlying mechanisms which are the source of main variations and affect reliability of wharves. The point of view selected here is to analyse effects of randomness around the main mechanical model which describes accurately the wharf behaviour by means of the use of a 3-D finite element model. It is described on figure 3. The following elements are selected to take into account the main constitutive elements of the structure: -triangular web on reinforced concrete beams is modelled with bar elements, assuming that their main loading case is a tensile one. e bending stiffness of the wharf.

piles are modelled by beams with Winkler model for taking the soil-pile interaction into account. Only a linear elastic behaviour is considered due to low measured and computed displacements (see [START_REF] Verdure | Cadre statistique du suivi en service des ouvrages de génie civil : application à un quai sur pieux[END_REF]). -rods are modelled by cable elements, which behaviour law is drawn in figure 4. F denotes axial load and u axial displacement. K t is the overall stiffness of rod-soil. δ 0 is the play in the balljoint. It allows to introduce a lack of prestressed loading and of wharf displacement and leads to unloaded rods. It will be used further in the paper. They are simply supported on the back-wharf wall, and the other end can be subjected to given displacements. e axis bending: 32% for the slab(deck), 10% for beam network, 58% for the berthing beam and back-wharf wall. Design hypotheses, where only the effect of slab is considered, leads to underestimate the stiffness with factor 3. It is conservative but not suitable for in-service behaviour modelling and reliability analysis.

complexity level in pile modeling: the complete 3D FE meshing allows to discuss the relative bending stiffness of axis

→ x
e for each row of pile and rods: the percentage is 51% for rod and 45, 20, 13, 8, 7, 3, 2, 2, 1 percent respectively for the 8 rows of piles from river to backwharf wall. The effect of bending stiffness of axis

→ y e has been discussed according to its contribution in case of wharf rotation and because of the width of wharf (42 m): it has been shown to be negligible (Verdure,204).

Design hypothesis without monitoring data

First we present main hypothesis which are generally assumed. They come from expert judgment and uncertainty studies performed during preparation of European semi-probabilistic code format called Eurocode 7 (de [START_REF] Grave | Validation partial safety factors from « probabilistic analysis sheet pile constructions (IGWR)[END_REF]. Judgment of expert of wharf design leads to several analyses: -rods are pre-stressed and are loaded by the platform depending of the platform deformation only. -immediately after construction, loading on the vertical reinforced concrete anchoring plate embedded inside the bank allows are sufficient to assume that passive earth pressure acts in totality: limit state is reached.

Benefit of monitoring data

The instrumentation of several wharves in Nantes need to stand back to analyses presented above. First, loading on reinforced concrete anchoring plate on which rods are clamped, is very fair. In fact two rods located at the ends have been instrumented in three points (see Fig. 5). It allows to analyze how loading varies along a rod. Figure 6 presents variations around 0 of loading during two days at this three sections: higher amplitudes are obtained for the section near the back-wharf, and lower for the section near the plate. This amplitude near the plate is about 10 times lower than the amplitude near the back wharf. Thus, it is very questionable to assume that displacement is sufficient to be at the source of passive earth pressure. In case of storm, the displacement could be higher and be the source of structural disorder. Second, the variations of normal load in rods along the length of the wharf are large. Figure 8 gives average values upon a tide, on September 20 th 2002, for these measured loads after embankment works. Moreover, some rods seems not to be loaded which makes questionable expert judgment presented upon. The following of this paper will focus on this anomaly to show how measurements and finite element model can be coupled for the identification of basic variables.

The evolution of rods loading with time are also of first interest and are widely commented in Verdure ( 2004) and Verdure et al. (2003).

Sensitivity analysis

Sensitivity analyses have been performed including several factors : -embankment loading: two selected values of passive earth pressure coefficient Ka are 0.23 obtained from limit state balance [START_REF] Soubra | Active nad Passive earth pressure coefficients by a kinematical approach[END_REF] and 0.6 from relaxed sol loading. The first one is selected because it allows to explain the upper bound of loading in rods. For embankment loading only, it leads to a soil load per meter of 134 kN/m or 896 kN every rod location: they are distant of 6.65 m. For a great tide loading, the load is of 14.1 kN/m (94 kN every 6.65 m). -soil-pile interaction with three configurations: design hypotheses and two realistic ones versus presence or absence of sludge. They respectively lead to values of equivalent stiffness for pile of 43.2, 36.3 and 94.9 MN/m. As the aim of this paper is to understand underlying mechanisms which are source of randomness, these studies which have led to second order effects, are not presented. Results are given in [START_REF] Verdure | Cadre statistique du suivi en service des ouvrages de génie civil : application à un quai sur pieux[END_REF]. Focus is put on the sensitivity analysis on rod and its interaction with soil. In fact soil-rod interaction model should include four levels: play in the clamping device, soil elasto-plastic behavior around the anchoring plate, adhesion of soil to rod and bending behavior of rod due to soil compression As these options are very hard to quantify for in-service behavior, a global soil-rod stiffness is considered in the following. Detailed investigations on this subject are available in [START_REF] Verdure | Cadre statistique du suivi en service des ouvrages de génie civil : application à un quai sur pieux[END_REF].

MONTE-CARLO SIMULATION

Simplified beam model for simulations

The model presented in section 3.1 is interesting as reference model but time cost is too high to perform sensitivity studies in a probabilistic way. For cost reason a more simple model, based on Timoshenko beam theory is introduced: -the deck is modeled by a Timoshenko beam with stiffness EI = 2.15 10 14 N.m² and KGS = 1.9 10 11 N. -each row of piles, in the rods remain the same as in the 3-D model, i.e. cable elements. It allows to take into account local shear effects which come here from mooring or transverse crane loading, due to wind. Figure 7 gives a sketch of this model. This model allows to keep the mechanical behaviour of the wharf and to perform simulations.

Simulation strategy.

Two behaviors of the wharf can be considered depending of loading: -behavior considering only embankment loading which allows to analyze the role of the technological play δ 0 and the soil-rod stiffness presented in figure 4. -behavior during an increasing tide which leads to introduce the equivalent stiffness of rod-soil. For this last hypothesis, it must be assumed that rods cannot change their state (loaded or unloaded) during a given tide. Ins this paper only the first one is considered. Monte-carlo simulations are performed according to techniques of equalization kernel presented by [START_REF] Akaike | An approximation to the density function[END_REF], [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF] and [START_REF] Parzen | On the estimation of a probability density function and the mode[END_REF]. Randomness of play and stiff-ness are not considered simultaneously as output statistics are too poor.

Simulation of the play.

This first study aims to analyze effect of play in mean loading profile. Variation of loading with length is presented on figure 8. Several rods have been put out of the data base. Reasons are given in [START_REF] Verdure | Cadre statistique du suivi en service des ouvrages de génie civil : application à un quai sur pieux[END_REF]. Some of loads are null or negative, others vary with a profile which cannot be understood by a classical mechanical deterministic analysis.

Embankment loading is 896 kN every 6.65 m (distance between rods). We draw probability densities from this profile according to the assumption that negative loads are considered as null (see figure 9). In fact, this first analysis is performed with the one-side-only rod model shown in figure 4. To analyze the effect of the play ! 0 only on the loading distribution in rods, numerical tests are performed through Monte-Carlo simulations. As the main statistical effects are seeked, a normal distribution is considered with three hypotheses for sensitivity study: The mean takes three values (0.5, 1 and 1.5 cm) when the standard deviation is set to 0.5 cm. Then sgtatistical samples of 38 rods are built. To remove bias in estimates due to small size of samples, the same basic sample is selected to compare the hypotheses.

Figure 10 gives results obtained for a given sample. We can notice that the mean value of play affects the number of unloaded rods. In fact, rods are loaded only if the wharf displacement exceeds the play. The most suitable statistical properties for ! 0 seem to be; m! 0 =10 and "! 0 =0.5.

Simulation of the overall soil-rod stiffness.

Let us consider now the effect randomness in soilrod stiffness only at given play, set here to zero. It allows to take into account the second order effect of friction forces between wharf-wall and instrumented section 1 meter apart (see figure 5). Several hypotheses for soil-rod stiffness are tested and presented on figure 11 Previous sections allow to understand main effects of distributions of couple of basic random variables (δ 0 , K t ). The objective is now to compute parameters of distributions which lead to output distributions. This inverse problem is solved here with a simplex optimization method [START_REF] Nelder | A simplex method for function minimization[END_REF]. The algorithm is detailed and illustrated in [START_REF] Verdure | Cadre statistique du suivi en service des ouvrages de génie civil : application à un quai sur pieux[END_REF]. The number of loops selected for simulations is 50 which results from a compromise between calculation costs and result stability. In what follows, only δ 0 is considered as a random variable. The full study is available in [START_REF] Verdure | Cadre statistique du suivi en service des ouvrages de génie civil : application à un quai sur pieux[END_REF]. δ 0 is assumed to be normally distributed. Thus, optimization problem can be written as a minimization of a cost function λ, which has here a quadratic form. Two optimization strategies are presented in the selection of optimization parameters. Equation 3 presents this cost function for two optimization parameters: mδ 0 and σδ 0 (mean and standard deviation of (δ 0 )). In equation 4, one more parameter called p is added. p is the percentage of unloaded rods. It allows to condition the problem. Due to the small data base, several values are selected for p. It is to notice that in this case, statistics of rod loads are estimated from the set of loaded rods. Results are very close to those obtained from the direct analysis (see figure 10). It confirms the robustness of numerical and mechanical models. Parameters deduced from inverse method seem to represent the main trends: percentage of unloaded rods and global shape of the distribution. In terms of reliability analysis, these results are more questionable. In fact loads can reach value of 800 kN which has never been met in measures. It is due to the fact that a percentage of unloaded rods must be balanced by extreme values. Mean and standard deviation of measures are shown not to be sufficient to describe the overall problem with accuracy. In this case, the problem is badly conditioned. In fact, the number of parameters ( 3) is upper than the number of inputs (2). Table 4 gives results of measured statistics (target values) compared to those obtained from direct analysis. It is shown that the competition between the terms of cost-function (Eq. 4) leads to a bias in the evaluation of mean and standard deviation. For further works it could be interesting to modify the cost function with a different weight for the three terms depending of the confidence on each. Finally, the technological play obtained is consistent with those published for the deep water harbour of Calais [START_REF] Delattre | Etude expérimentale du comportement du quai en eaux profondes du port de Calais[END_REF] where the play is about 6 mm.

CONCLUSION

Reliability of complex structures such as wharves require analyses on basic variables selection and uncertainties modeling. After a detailed analysis on structural behavior and sensitivity, the paper focuses on two basic variables which allow to understand output distributions: play in clamping device and equivalent soil-rod stiffness. Direct simulations through a simplified finite element model are performed as sensitivity studies. They allows to consider these variables as basic dominant ones. Then an inverse method based on the simplex algorithm allows the estimate of input distribution parameters. A bias in the evaluation of these parameters appears if the percentage of unloaded rod is introduced in the inverseproblem. It illustrates that a more complete information is required as the displacement field for example.

Figure 2 .

 2 Figure 2. Sketch of a wharf section.

Figure 3 .

 3 Figure 3. 3D finite element model.

Figure 4 .

 4 Figure 4. Sketch of the clamping device and constitutive law of the rods. This 3-D FE model allows to discuss the complexity level to be considered in modelling and the role of each component : -complexity level for platform FE model: model presented in figure 3 gives the following percentage of relative stiffness for → z

Figure 5 .

 5 Figure 5. Rod instrumented in three sections: couples of vibrating wire (VW) and cell.

Figure 6 .

 6 Figure 6. Loading variation with time according to sensors presented in figure 5.

  spring with stiffness 58.2 MN/m. It is obtained from soil modulus Es adapted for each row of pile wit a special care to the 3 first rows near the river which support 78 % of lateral loading (see section 4.1).

Figure 7 .

 7 Figure 7. Equivalent beam model.

Figure 8 .

 8 Figure 8. Load average for embankment loading along wharf length.

Figure 9 .

 9 Figure 9. Probability density of loading in rod for embankment.

Figure 10 .

 10 Figure10. Probability density of loading in rod for embankment loading for several hypothesis of statistics for play.

  : uniform, log-normal with mean value at 60 MN/m and standard deviation at 34.6 MN/m. This distribution allow to keep positive values for stiffness. The aim is to analyze how distributions are transferred. Figure 11 shows the theoretical p.d.f. and the p.d.f. obtained from samples of size 38. Values smaller and higher than distribution theoretical bounds are obtained due to the technique of equalization kernel. Load distribution obtained for this hypothesis are shown in figure 12.

Figure 11 .

 11 Figure 11. Experimental and theoretical distribution selected for soil-stiffness distribution.

Figure 12 .

 12 Figure 12. Load distribution in rods for two distributions types of soil-rod stiffness.A pure translation of platform should lead to the same distribution of rod type. Due to the stiffness of the platform, only a rotation is observed and the initial distribution types are quite preserved.

  m F and σ F are the computed mean and standard deviation of load in rods and m Fmes and σ Fmes the corresponding measured values.

  F and σ * F are mean and standard computed on the set of values for loaded rods. Due to the small size of sample, the solution is not unique. In the following, standard deviation of δ 0 statistics and associated 95 % confidence interval are given: they are obtained from 10 successive numerical resolutions. For the first problem (Eq. 3), target values are m Fmes = 134 kN et σ Fmes = 178 kN. Results are presented in table 1.

For

  the second problem (Eq. 4), two bounds of target values are selected for taking into account the reduced percentage of instrumented rods: -p mes = 0.5, m* Fmes = 296 kN et σ * Fmes = 94 kN. -p mes = 0.27, m* Fmes = 296 kN et σ * Fmes = 94 kN.

Figure 13 .

 13 Figure 13. P.d.f obtained from direct simulations.

Table 1 :

 1 solution of optimization for target values m Fmes = 134 kN, σ Fmes = 178 kN

	Parameter Mean	Standard devia-	95% confidence interval
		[mm]	tion [mm]	[mm]
	mδ 0	12.6 0.4	[12.2 ; 13.0]
	σδ 0	5.1	0.35	[4.8 ; 5.4]

  Table 2 and 3 presents respectively results obtained for first and second conditions.

Table 2 :

 2 solution of optimization with p=0.5

	Parameter Mean	Standard devia-	95% confidence interval
		[mm]	tion [mm]	[mm]
	mδ 0	13	3.6	[9.9 ; 16.1]
	σδ 0	5.4	1.6	[3.9 ; 6.8]

Table 3

 3

		: solution of optimization with p=0.27
	Parameter Mean	Standard devia-	95% confidence interval
		[mm]	tion [mm]	[mm]
	mδ 0	11.5 0.3	[11.2 ; 11.8]
	σδ 0	3.5	1.5	[2.2 ; 4.8]
	6.2 Direct simulation from the results of the
		inverse problem
	We now analyze the effect of results obtained previ-
	ously on the output distribution. Direct simulation of
	Monte-Carlo is used. Statistics of rods are deduced
	from the mean of 50 samples. Results obtained for
	the three hypotheses upon in tables 1, 2 and 3 and
	called respectively first, second and third case are
	presented in figure 13.

Table 4 :

 4 statistics of direct problem compared to target values.

		First case	Second case		Third case
		m	σ	m	σ	p	m	σ	p
	Target value 134	178	296 94 0.5 296 94 0.27
	Simultation	134	176	246 178 0.47 198 133 0.31

ACKNOWLEDGMENT

The authors would like to thank Harbour Authorities of Nantes St Nazaire for their technical support and expert judgement.