Journal of Mathematical Sciences, Vol. 150, No. 1, 2008
ON UNIQUENESS OF A SOLUTION TO THE PLANE PROBLEM ON INTERACTION OF SURFACE WAVES WITH OBSTACLE N. G. Kuznetsov Institute of Problems of Mechanical Engineering RAS St. Petersburg, Russia
[email protected]
UDC 517.95
We consider the plane linear boundary value problem describing the behavior of time-harmonic water waves with an obstacle formed by partially and totally immersed bodies of infinite length and also by the part of bottom the topography of which is different from that in the plane case. We study a special two-dimensional case where the crests of waves incoming on the obstacle are parallel to the obstacle generators. Bibliography: 10 titles.
1. Introduction We consider the plane linear boundary value problem describing the behavior of time-harmonic water waves in the presence of an obstacle formed by partially and totally immersed bodies of infinite length and by the part of bottom the topography of which is different from that in the plane case. The water-wave problems admit two-dimensional and three-dimensional statements, but we deal only with a special two-dimensional problem. Namely, we consider the plane case, where the crests of waves incoming on the obstacle are parallel to the obstacle generators. A rather complete bibliography concerning this problem can be found in the recent paper by the author [1]. The fact that the question about the uniqueness of a solution to the above problem is nontrivial became clear in 1996 after the publication of the paper [2], where was shown that this problem has the so-called trapped modes i.e., nontrivial solutions to the homogeneous problem. The first
Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 69–75.
c 2008 Springer Science+Business Media, Inc. 1072-3374/08/1501-1860
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uniqueness criterion was obtained by John [3] in 1950. His proof is based on geometric conditions which will be formulated below under the assumption that the domain occupied by water has finite depth (such domains will be studied below). (i) A finite collection of immersed cylindrical bodies (with generators parallel to the horizontal free water surface) which together with analogous rough parts of bottom form an obstacle, includes only one body intersecting the free surface. (ii) The obstacle is entirely contained between two vertical planes passing through the lines along which the partially immersed body intersects the free surface.
A modification of the John method in [4] allows one to replace condition (ii) with a weaker condition that the obstacle is contained in the corner formed by the lines passing through the intersection points of the body contour and the free surface, but inclined at the angle of π/4. The case of an infinitely deep domain was considered in [5], where a uniqueness criterion was announced. This results concerns the case of only one partially immersed body for which it is admissible that at least one of the angles formed by the body with the free surface is less than π/4. The goal of this paper is to generalize the last result to the case where the depth of domain occupied by water is finite and there are several immersed cylinders, but only one of them is partially immersed in water. We apply a method that was used earlier by the author (cf., for example, [1]) and can be described as follows. First, we apply a special conformal mapping to the original plane domain (the cross–section of the layer occupied by water) and then we use the integral identity due to Vainberg and Maz’ya [6] combined with a relation expressing that the potential energy is equal to the kinetic energy.
2. Two Equivalent Statements of the Problem 2.1. Hydrodynamic statement. The cross–section of a domain occupied by an ideal incompressible heavy fluid (water) is denoted by W . We assume that W = R2− \ D, where R2− = {−∞ < x < +∞, y < 0} and D = D0 ∪ D1 ∪ . . . ∪ Dn ∪ D∞ . Here, D0 , . . . , Dn and D∞ are simply connected pairwise disjoint domains in R2− possessing the following properties. 1) The unbounded domain D∞ (the cross–section of the solid base of the domain occupied by water) is such that R2− \ D∞ is a curvilinear strip L with the upper boundary ∂R2− , and the lower boundary B (the curve which is the cross–section of the bottom of the domain occupied by water) coincides with the line {x ∈ R, y = −d}, d > 0, at infinity (i.e., for |x| > b0 , where b0 is a positive number). 2) The closures of bounded domains D1 , . . . , Dn (the cross–sections of the totally immersed cylinders) are contained in the strip L. 3) The bounded domain D0 (the cross–section of a single partially immersed cylinder) is adjoined to the interval {x ∈ (−a, a), y = 0}, a > 0, of the abscissa axis in such a way that the points (+a, 0) and (−a, 0) are corner points of the domain W . 4) The boundary ∂W consists of arcs of class C 2 which are joined in such a way that there are no cusps on ∂W . 1861
Denote by S0 , S1 , . . . , Sn the cross–sections of wetted surfaces of the immersed cylinders, i.e., ∂D0 ∩ L, ∂D1 . . . , ∂Dn respectively, and F = {|x| > a, y = 0} is the cross–section of the free water surface. Assume that the water motion is irrotational and presents small oscillations with angular frequency ω. If we neglect the surface tension, then for describing the wave process we can use the complex-valued velocity potential ϕ(x, y) satisfying the boundary value problem ∇2 ϕ = 0 in W, ϕy = νϕ on F, ∂ϕ/∂n = f
(2.1)
on S ∪ B
which must be completed by the radiation condition ϕ|x| − ikϕ = o(1)
as |x| → ∞ uniformly with respect to y ∈ (−d, 0)
(2.2)
and the condition that the Dirichlet integral of ϕ is locally finite. The function f in the Neumann condition is compactly supported on S ∪B; moreover, the form of this function depends on the type of problem (radiation or scattering of waves). The parameter ν = ω 2 /g, where g is the acceleration of gravity, plays the role of a spectral parameter in the homogeneous problem (with f = 0). The parameter k in the radiation condition (2.2) is a unique positive root of the equation ν = k th kd. This problem will be abbreviated as WWP (water-wave problem). Since we are interested in the uniqueness of a solution and the problem is linear, it suffices to assume that f ≡ 0 on S ∪ B. Moreover, the solution satisfies the following asymptotic formulas (cf. the proof in [7, Section 2.2]): ϕ(x, y), |∇ϕ(x, y)| = O(|x|−1 ) as |x| → ∞, which means that the solution satisfies some conditions stronger than the radiation condition (2.2), namely:
|∇ϕ|2 d x d y < ∞,
W
ϕ2 d x < ∞.
(2.3)
F
These conditions mean that the kinetic energy and potential energy of waves per the unit of length of the cylinder generators are finite. Thus, for the homogeneous problem we can require the conditions (2.3) instead of the condition (2.2). Thereby the potential ϕ can be assumed to be real-valued, but not a complex-valued function. Indeed, by (2.3), the real and imaginary parts of ϕ are solutions to the homogeneous problem. Finally, for solutions to the homogeneous problem the kinetic energy is equal to the potential energy:
2
|∇ϕ| d x d y = ν W
ϕ2 d x,
(2.4)
F
which can be established by using the Green formula, the Laplace equation, and the homogeneous boundary conditions in (2.1). Of course, we should begin with the Green formula applied to a finite 1862
domain obtained by eliminating two half-strips W ∩ {±x > b}, b > b0 and then pass to the limit as b → +∞ (in a similar situation, this procedure is presented in more detail in Section 3). 2.2. Conformal mapping and equivalent formulation of the problem. We transform the water-wave problem by passing from the Cartesian coordinates (x, y) to the bipolar coordinates with poles at the points (+a, 0) and (−a, 0). These bipolar coordinates are connected with the Cartesian coordinates by the following formulas (cf., for example, [8, Section 2.01] or [9, Section 10.1]): a sin v a sh u , y= . (2.5) x= ch u − cos v ch u − cos v The metric coefficients of the bipolar coordinates are equal to a2 /(ch u − cos v)2 . As is known, the passage to the bipolar coordinates is equivalent to a conformal mapping under which the image of the lower half-plane R2− is the strip {−∞ < u < +∞, −π < v < 0} and formula (2.5) determines the inverse conformal mapping. We list some sets on the (x, y)– and (u, v)–planes which correspond each other. The points (+a, 0) and (−a, 0) on the (x, y)-plane turn out to be at infinity in the (u, v)-plane, whereas the origin of the (u, v)-plane corresponds to infinity in the (x, y)–plane. Each coordinate line {±u > 0, v = σ},
σ = const,
−π < σ < 0,
(2.6)
±x > 0,
(2.7)
on the (u, v)-plane goes to a circle arc on the (x, y)-plane x2 + (y − a ctg σ)2 = a2 (ctg2 σ + 1),
y < 0,
with (±a, 0) for one of the endpoints. Although the coordinate lines of the second family v = const will not be used in this context, we note that they go to semicircles that are centered at the abscissa axis and are orthogonal to the semicircles (2.7). Further, the sets {−∞ < u < +∞, v = −π},
{±u > 0, v = 0}
are the images of the sets {|x| < a, y = 0},
{±x > a, y = 0}
respectively, and conversely. Thus, under a conformal mapping, the free water surface F is transformed to the entire u-axis (one of the reasons, why it is more convenient to use the bipolar coordinates). This axis bounds from above the image W of the domain W . The image S0 of the curve S0 is the boundary of domain W. Note that S0 is located between the u-axis and the parallel line v = −π. Furthermore, the lines v = −α± are the asymptotes of S0 as u → ±∞, where α± ∈ (0, π) is the angle between S0 and F at the point (±a, 0). Finally, the bottom B goes to a closed curve B that is contained between the curve S0 and the u-axis and touch the axis at the origin. Furthermore, the images S1 , . . . , Sn of the curves S1 , . . . , Sn are located between the curve S0 and the u-axis (but not touch them). We set Φ(u, v) = ϕ(x(u, v), y(u, v)), where x(u, v) and y(u, v) are given by formula (2.5). Then the boundary value problem (2.1) with the homogeneous Neumann condition takes the form ∇2 Φ = 0
in W,
(ch u − 1)Φv = νa Φ for v = 0, ∇Φ · n = 0
on S ∪ B,
(2.8) (2.9) (2.10) 1863
where n is the unit outward normal to S ∪ B relative to W. Further, the condition (2.3) implies
+∞
2
|∇Φ| d u d v < ∞, −∞
W
Φ2 d u < ∞. 1 + ch u
(2.11)
Thus, the problem (2.8)–(2.11) is equivalent to the homogeneous water-wave problem formulated in Subsection 2.1. Finally, the equality +∞ Φ2 2 du (2.12) |∇Φ| d u d v = νa 1 + ch u −∞
W
is similar to formula (2.4) expressing the fact that the kinetic energy is equal to the potential energy.
3. Integral Identities We use the Vainberg–Maz’ya identity [6] (cf. also [7, Section 2.2.2]) (2uΦu + Φ)∇2 Φ = ∇ · (2uΦu + Φ)∇Φ − 2Φ2u − u|∇Φ|2 u
(3.1)
which can be easily verified by differentiation. We integrate this identity over the domain W = W ∩ {|u| < b}, where b is sufficiently large (so that the set SL = S1 ∪ . . . ∪ Sn ∪ B lies in the strip {|u| < b}). By the Laplace equation, the integral on the left-hand side of the obtained relation vanishes. Using the divergence theorem and the boundary condition (2.10), we obtain the equality 2
Φ2u d u d v
W
2
u · n |∇Φ| d S +
+ S0
u · n |∇Φ|2 d S
SL
+b ± (2uΦu + Φ) Φu d v, = [2uΦu (u, 0) + Φ(u, 0)] Φv (u, 0) d u + ±
−b
where S0 = S0 ∩ {|u| < b}, u = (u, 0),
±
(3.2)
C±
is the sum of two terms corresponding to the upper and
lower sign, and C± = W ∩ {u = ±b}. We express Φv (u, 0) in the first integral on the right-hand side by using the boundary condition (2.9) and integrate by parts. Then u=+b +b +b Φ(u, 0) u Φ2 (u, 0) u sh u 2 du = [2uΦu (u, 0) + Φ(u, 0)] Φ (u, 0) d u + . ch u − 1 (ch u − 1)2 ch u − 1 u=−b
−b
−b
Since (+a, 0) and (−a, 0) are corner points of the domain W , in the view of the first condition in (2.3) we can use the known results of the theory of elliptic boundary value problems in domains 1864
with piecewise smooth boundary (cf., for example, [10, Chapter 2]). They describe the asymptotic behavior of solutions to the boundary value problems for the Laplace equation in a neighborhood of corner points of domain and guarantee that the potential ϕ(x, y) tends to some constant as (x, y) → (±a, 0). Consequently, the function Φ(u, v) possesses the same property as u → ±∞. Therefore, in the last equality, the term outside the integral tends to zero as b → ∞. By the second condition in (2.11), the integral on the right-hand side of the last equality converges as b → ∞. Passing to the limit, we find +∞ +∞ Φ(u, 0) u sh u du = [2uΦu (u, 0) + Φ(u, 0)] Φ2 (u, 0) d u. ch u − 1 (ch u − 1)2
−∞
(3.3)
−∞
Finally, by the first condition in (2.11), the integrals over C± in formula (3.2) tend to zero as b runs over some sequence {bn }∞ 1 tending to infinity as n → ∞. Passing to the limit in the equality (3.2) for b = bn as n → ∞ and taking into account the relation (3.3), we obtain the integral identity 2
Φ2u
u · n |∇Φ| d S = νa
dudv +
W
+∞
2
−∞
S0 ∪SL
u sh u Φ2 (u, 0) d u. (ch u − 1)2
(3.4)
Subtracting it from the equality (2.12) multiplied by 2, we find 2 W
Φ2v
dudv −
2
+∞
u · n |∇Φ| d S + νa −∞
S0 ∪SL
u sh u − 2(ch u − 1) 2 Φ (u, 0) d u = 0. (ch u − 1)2
(3.5)
This identity is a basis of the proof of the uniqueness theorem for the water-wave problem. Remark 3.1. The above method for obtaining the integral identity (3.4) based on the differential equality (3.1) is suitable only for the plane water-wave problem. There is a more general two-dimensional problem, known as the problem about oblique waves incoming on an obstacle (cf. [7, Section 5.4]). If we try to use the above method for this problem, some difficulties arise because the modified Helmholtz equation ∇2 ϕ = m2 ϕ used in this problem takes the following form under the conformal mapping (cf. formula (2.5)): ∇2 Φ =
(ma)2 Φ . (ch u − cos v)2
Then an extra term appears on the left-hand side of the equality (3.2), and one should apply the divergence theorem to this term. Then we have a term of sign opposite to the sign of the first term on the left-hand side, which creates some difficulties for proving Theorem 4.1 below. However, regarding the original modified Helmholtz equation, there are no such difficulties if we use (3.1) to prove the uniqueness criterion for the problem about oblique waves incoming on an obstacle. However, the presence of bodies immersed in water is not admissible in this case (cf. [6, Proposition, Section 1]). 1865
4. Uniqueness Theorems It is easy to check the inequality u sh u − 2(ch u − 1) 0 ∀u ∈ R;
(4.1)
moreover, equality takes place only for u = 0. If u · n 0 on S0 ∪ SL , then taking into account (4.1), we can conclude that the expression on the left-hand side of (3.5) is strictly positive if Φ does not vanish identically. Consequently, the identity (3.5) leads to a contradiction if Φ does not vanish identically. Thus, we have proved the following assertion. Theorem 4.1. Let u · n 0 on S0 ∪ SL . Then the boundary value problem (2.8)–(2.11) in the domain W has only the trivial solution for all positive values of the parameter ν. Remark 4.1. The requirement that the inequality u · n 0 holds on S0 ∪ SL has the following geometric sense. Each half-line (2.6) starting on the v-axis and going to ±∞ intersects transversally the curves S0 ∪ SL at most once. The proof of the following result concerning the uniqueness of a solution to the water-wave problem is based on the fact that the curves (2.7) are the images of half-lines (2.6) under the conformal mapping (2.5). Corollary 4.1. Suppose that the set of all transversal intersections of the curves (2.7) starting on the axis of ordinates and terminating at the points (+a, 0) and (−a, 0) with the lines S0 , S1 , . . . , Sn and B is the set of input points of the curves (2.7) into the domain W occupied by water. Then the homogeneous water-wave problem has only the trivial solution for all positive values of the parameter ν.
5. Examples Illustrating Uniqueness 5.1. We begin to consider examples of domains satisfying the assumptions of Corollary 4.1 with the simplest case: assume that there are no totally immersed bodies, i.e., D1 ∪ . . . ∪ Dn = ∅, and the bottom {x ∈ R, y = −d} is horizontal. Hence the intersecting arcs (2.7) come in the domain occupied by water. Moreover, it is natural to begin the consideration with a body such that the contour S0 of the cross–section is an arc of circle of radius r centered at the axis of ordinates with the endpoints (+a, 0) and (−a, 0) (i.e., described by formula (2.7)). It is easy to check that such a circle determines the domain W , i.e., it is located above the horizontal bottom if r
0, can be replaced with any 1866
convex curve located below the arc in the example from Subsection 5.1 for S0 and is contained between the vertical lines x = ±b0 . We indicate some other variants of curvilinear part of the bottom joining the horizontal parts in such a way that the assumption of Corollary 4.1 is satisfied. In particular, if such a part of bottom is located above its horizontal part, it is not necessarily convex and can go beyond the strip {|x| < ±b0 }. On the other hand, this part of bottom can be replaced with the corresponding arc of circle (2.7) which, obviously, will be located below the horizontal part of bottom. Clearly, between the above curves determining the bottom B and the wetted contour S0 of the partially immersed body it is possible to place curves S1 , . . . , Sn bounding totally immersed bodies, but each of these closed contours must satisfy the assumption of Corollary 4.1. a2 + d2 in the example in Subsection 5.1, then the set W is not Finally, we note that if r > 2d connected. Hence for a domain occupied by water we can consider only one subset (for example, the right one), where S0 divides W . Moreover, Corollary 4.1 expresses a uniqueness criterion for the water-wave problem in a neighborhood of the curvilinear slope of the bottom. 5.3. The last example illustrates Theorem 4.1. It is easy to see that, on the plane (u, v), the image of horizontal bottom {x ∈ R, y = −d} is a convex closed curve B which is defined as follows: u = ± arch
d cos v − a sin v , d
where v ∈ [−2 arc tan(a/d), 0]. Moreover, for the curve S0 which, together with B, bounds the domain W satisfying Theorem 4.1 we can take the graph of the function v=
1 − α, β + u2
where α ∈ (2 arc tan(a/d), π) and β is an arbitrary number such that a −1 . β ∈ 0, α − 2 arc tan d It is easy to find intervals of parameters for which the angle α between the curve S0 and the free water surface is less than π/4. As in above examples, between the curves S0 and B, one can place closed contours S1 , . . . , Sn such that any line v = const intersects them at most two times, which guarantees the validity of the assumptions of Theorem 4.1. The above examples demonstrate methods for constructing other examples.
References 1. N. G. Kuznetsov, “Nodal lines and uniqueness of solutions to linear water-wave problems” [in Russian], Tr. S. Peterb. Mat. O-va 13 (2007), 73–91; English transl.: Am. Math. Soc., Providence, 2008. 2. M. McIver, “An example of non-uniqueness in the two-dimensional linear water-wave problem,” J. Fluid Mech. 315 (1996), 257–266. 3. F. John, “On the motion of floating bodies. II ,” Comm. Pure Appl. Math. 3 (1950), 45–101. 1867
4. M. J. Simon and F. Ursell, “Uniqueness in linearized two-dimensional water-wave problems,” J. Fluid Mech. 148 (1984), 137–154. 5. N. Kuznetsov, “Uniqueness in the water-wave problem for bodies intersecting the free surface at arbitrary angles,” C. R. Mecanique 332 (2004), 73–78. 6. B. R. Vainberg and V. G. Maz’ya, On the problem of the steady-state oscillations of a fluid layer of variable depth [in Russian], Tr. Mosk. Mat. O-va 28 (1973), 57–74. 7. N. Kuznetsov, V. Maz’ya, and B. Vainberg, Linear Water Waves: A Mathematical Approach, Cambridge Univ. Press, Cambridge, 2002. 8. P. H. Moon and D. E. Spencer, Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1988. 9. Ph. M. Morse and H. Feshbach, Methods of Theoretical Physics, II, McGraw-Hill Book Co., New York, 1953. 10. S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundary [in Russian], Nauka, Moscow, 1991.
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