aa r X i v : . [ m a t h - ph ] N ov On the absence of water wavestrapped near a cliffed cape
Nikolay Kuznetsov
Laboratory for Mathematical Modelling of Wave Phenomena,Institute for Problems in Mechanical Engineering, Russian Academy of Sciences,V.O., Bol’shoy pr. 61, St. Petersburg 199178, Russian FederationE-mail: [email protected]
Abstract
The water wave problem is considered for a class of semi-infinite domains eachhaving its shore shaped as a cliffed cape. In particular, the free surface of a waterdomain is supposed to be an infinite sector whose vertex angle is greater than π ,whereas the water layer lying under the free surface is of constant depth. Underthese assumptions, it is shown that there are no trapped mode solutions of theproblem for all values of a non-dimensional spectral parameter; in other words, nopoint eigenvalues are embedded in the problem’s continuous spectrum. Let a non-dissipative medium occupy an unbounded domain, where waves can propagate.By trapped modes (they are also referred to as resonance modes or bound states) oneunderstands localised free oscillations of the medium that cannot radiate to infinity.More precisely, a time-harmonic solution of a boundary value problem describing wavesis called a trapped mode provided this solution decays at infinity so that the total energyof the wave motion is finite. We just mention a couple of recent articles that show asubstantial progress in this area of research during the past two decades. The absenceof trapped modes in locally perturbed open acoustical waveguides was considered in[1], whereas [2] provides a survey of localised modes in acoustics and elasticity. Somerelevant results concerning water waves are described in § π + 2 α with some α > π , and so the coastline is cape-shaped. Itis also assumed that the water layer under the free surface is of constant depth, say h . Apart from this layer, the water domain can include some subdomains covered fromabove by the rigid underwater surface of the cape. In this case, its boundary has the formof an overhanging cliff on one or both sides of the cape. Moreover, underwater tunnelsthrough the rigid body of the cape that connect its opposite sides are admissible. Thiskind of geometry essentially distinguishes from those for which the absence and existenceof trapped modes were studied earlier (see § In the framework of the linear theory of water waves, the question of absence/existenceof trapped modes has a long history. Its initial point is the classical paper [3] publishedby John in 1950. In this article, he proved the first results guaranteeing the absence oftrapped modes near bounded immersed obstacles. In the case of a fixed one, the resultis valid for all frequencies provided some geometric restriction (now referred to as John’scondition) is imposed on the obstacle. Also, this condition yields the result for a freelyfloating obstacle, but only for sufficiently large frequencies. In the three-dimensionalcase, John’s restriction on the geometry is as follows: the whole obstacle is confinedwithin the finite number of vertical cylinders through the lines, where surface-piercingbodies (at least one such body must be a part of the obstacle) intersect the free surfaceof water; moreover, the bottom (when the depth is finite) is horizontal outside of thesecylinders.During the second half of the 20th century the main effort was devoted to findingvarious conditions on geometry of fixed bodies guaranteeing the absence of trappedmodes (see the summarising monograph [4] by Kuznetsov, Maz’ya and Vainberg). Inparticular, it was Maz’ya who found the first such condition for totally submerged bodies(see his papers [5, 6] and § § Let a Cartesian coordinate system ( x , y ) ( x = ( x , x ) for the sake of brevity) be suchthat the y -axis is directed upwards and the free surface at rest, say F , lies in the plane2 FC W ( α )1 W c The part ofcape withoverhangingcliff Rigid bottomFigure 1: A sketch of the cross-section of the water domain by a plane orthogonal tothe coastline. The point, where the plane intersects the coastline, is marked with C above it. Various other cross-sections are marked as follows: F denotes the free surface,whereas W c and W ( α )1 stand for the water subdomains located under the overhangingcliff and the free surface respectively; the latter has the constant depth equal to unity. { y = 0 } . Moreover, we assume that F = { ( x , y ) : x ∈ F ( α ) , y = 0 } , where F ( α ) = { x ∈ R : x > −| x | tan α } and α ∈ (0 , π/ . Thus, the free surface is a horizontal infinite sector whose vertex angle is π + 2 α , whereasthe mainland forms an infinite cape within the coast-line C = { ( x , y ) : x ∈ ∂F ( α ) , y = 0 } . It is convenient to formulate the problem of time-harmonic oscillations of water ina dimensionless form. For this purpose we scale the coordinates ( x , x , y ) so that thedepth of the water layer under F is equal to unity. Thus, a water domain, say W , is theunion W ( α )1 ∪ W c (see Figure 1) of the layer W ( α )1 = { ( x , y ) : x ∈ F ( α ) , y ∈ ( − , } (1)and a region W c , which consists of one or more Lipschitz domains. This region is suchthat | x | and | y | are bounded for all its points. Moreover, W c is adjoint to the rigidsurface of the cape’s overhanging cliff, that is, above the level y = −
1, each of thedomains forming W c is confined between the vertical half-planes, that go through thecoast-line C and form the dihedral angle π − α , but below the level y = − W c is arbitrary. It is admissible that W c is not simply connected; this takes place inthe case of underwater tunnel through the cape’s body.Following the procedure described in [14], §
4, we consider the real-valued, non-dimensional velocity potential φ as the problem’s unknown. It is obtained by using p h g as the scaling parameter; here g > y -axis and h is the dimensional depth of the water layerunder the free surface. Then the potential φ describes a time-harmonic mode trappedin the domain W provided it is a finite energy solution of the homogeneous problem: ∇ φ = 0 in W, (2) φ y − νφ = 0 on F, (3) ∂φ/∂n = 0 on ∂W \ ¯ F . (4)Here ∇ = ( ∂ x , ∂ x , ∂ y ) is the gradient operator; ν = ω h/g is the non-dimensionalspectral parameter related to the radian frequency ω of water oscillations; n denotes thenormal on ∂W \ ¯ F pointing outside of W . 3t is convenient to understand the finiteness of energy in the sense that φ belongs tothe usual Sobolev space H ( W ), in which the norm is as follows: (cid:20)Z W |∇ φ | d x d y + Z W | φ | d x d y (cid:21) / . It is explained in [16] (see p. 3619) that this norm is equivalent to the following one: (cid:20)Z W |∇ φ | d x d y + Z F | φ | d x (cid:21) / , where the first (second) term in the square brackets is proportional to the kinetic (poten-tial) energy of the wave motion. Hence the fact that φ ∈ H ( W ) expresses the finitenessof energy. Under this assumption problem (2)–(4) is usually formulated in the weaksense: Z W ∇ φ ∇ ψ d x d y = ν Z F φ ψ d x . (5)It is sufficient to require that this integral identity holds for all smooth functions ψ compactly supported in W .It is well known (see, for example, the book [17], ch. 8), that the Laplace equationholds in the classical sense provided φ ∈ H ( W ) satisfies (5). The same concerns thefollowing boundary conditions: φ y − νφ = 0 on F and φ y = 0 on ∂W ∩ { y = − } . (6)Now we are in a position to formulate the main result. Theorem 1.
Let W = W ( α )1 ∪ W c be the water domain, where W ( α )1 is defined byformula (1) with α ∈ (0 , π/ and W c is described after that formula, and let ν be apositive number. If φ ∈ H ( W ) satisfies problem (2) – (4) in the weak sense presented byidentity (5) , then φ vanishes identically in W . This theorem means that there are no trapped water waves near a cliffed cape.
The basic idea of the proof is to combine the classical method of John [3] (see also [4], § Theorem 2.
Let k be a positive number. If w ∈ L ( F ( α ) ) , where α ∈ (0 , π/ , satisfiesthe Helmholtz equation ∇ x w + k w = 0 in F ( α ) in the distributional sense, then w vanishes identically; by ∇ x the gradient operator in R is denoted. To prove that φ ∈ H ( W ) vanishes identically when it satisfies problem (2)–(4) inthe domain W , on which some geometric assumptions are imposed, it is sufficient toderive an inequality that contradicts the equipartition of energy equality, namely: Z W |∇ φ | d x d y = ν Z F | φ | d x . (7)Indeed, this relation is a direct consequence of the integral identity (5) with ψ = φ .4o implement this approach John based his method on properties of the followingauxiliary function: w ( x ) = Z − φ ( x , y ) cosh k ( y + 1) d y, (8)where k is the unique positive zero of the function ν − k tanh k . Geometric restrictionsimposed on W in Theorem 1 yield that this function is defined for all x ∈ F ( α ) . In viewof equation (2), we have that ∇ x w = − Z − φ yy ( x , y ) cosh k ( y + 1) d y in F ( α ) , and so after integration by parts twice in the right-hand side we obtain that ∇ x w + k w = 0 in F ( α ) . (9)Of course, the boundary conditions (6) must be taken into account for cancelling theintegrated terms.For applying Theorem 2 to w , it remains to show that this function belongs to L ( F ( α ) ), which is true because φ ∈ L ( W ). Indeed, the Schwarz inequality gives | w ( x ) | ≤ (cid:18)Z − | φ ( x , y ) | d y (cid:19) (cid:18)Z − cosh k ( y + 1) d y (cid:19) = (2 k ) − [ k + sinh k cosh k ] Z − | φ ( x , y ) | d y . Therefore, R F ( α ) | w ( x ) | d x is finite being estimated by R W ( α )1 | φ ( x , y ) | d x d y with somepositive constant, and we conclude that Z − φ ( x , y ) cosh k ( y + 1) d y = 0 for all x ∈ F ( α ) . Integrating by parts again, we obtain φ ( x ,
0) sinh k = Z − φ y ( x , y ) sinh k ( y + 1) d y for all x ∈ F ( α ) , and the Schwarz inequality gives | φ ( x ,
0) sinh k | ≤ (cid:18)Z − | φ y ( x , y ) | d y (cid:19) (cid:18)Z − sinh k ( y + 1) d y (cid:19) = 12 (cid:20) sinh k ν − (cid:21) Z − | φ y ( x , y ) | d y , where the definition of k is applied. An immediate consequence of this is the inequality ν Z F | φ ( x , | d x ≤ Z W ( α )1 |∇ φ | d x d y , (10)which is obviously incompatible with (7). The proof of Theorem 1 is complete.5 Concluding remarks
The absence of trapped modes has been established for all values of a non-dimensionalspectral parameter under some geometrical restrictions on the water domain surroundinga cliffed cape. The first key point of the proof is the equipartition of energy equality(7), whereas the auxiliary function (8) used for deriving an inequality incompatiblewith (7) is the second key point. Finally, it is essential that the function (8) vanishesidentically being a solution of equation (9) and this follows from Theorem 2 concerningthe Helmholtz equation.Since the proof of Theorem 1 does not involve any property of W c other than thefact that it lies partly between W ( α )1 and the rigid cliff and partly below W ( α )1 , it isadmissible to place a finite number of totally submerged bounded rigid bodies into theregion W c , in which case Theorem 1 is still true. Indeed, the equipartition of energyequality (7) remains valid in this case as well as the considerations leading to inequality(10).Let us discuss how Theorem 1 can be extended to a full-fledged uniqueness theoremin the case when W = W ( α )1 ; that is, the water domain is the sectorial layer, whosedepth is constant and the vertex angle is π + 2 α . By k , k , . . . we denote the sequenceof positive zeroes of the function ν + k tan k ; this sequence up to the factor i coincideswith that of zeroes of ν − k tanh k lying on the positive imaginary axis. It is known (see,for example, [19], § { ψ n ( y ) } ∞ n =0 with ψ ( y ) = cosh k ( y + 1) , ψ n ( y ) = cos k n ( y + 1) , n = 1 , , . . . , is an orthogonal and complete system in L ( − , φ ( x , y ) = ∞ X n =0 v n ( x ) ψ n ( y ) in W ( α )1 . It is clear that we have in F ( α ) : ∇ x v + k v = 0 and ∇ x v n − k n v n = 0 for n = 1 , , . . . . (11)Assuming that the norm of φ in L (cid:0) W ( α )1 (cid:1) is finite, we see that Z W ( α )1 | φ | d x d y = ∞ X n =0 N n Z F ( α ) | v n | d x , where each integral on the right-hand side is finite and N = Z − cosh k ( y + 1) d y = 12 (cid:18) k k (cid:19) , whereas N n = Z − cos k n ( y + 1) d y = 12 (cid:18) k n k n (cid:19) , n = 1 , , . . . . Then Theorem 2 yields that v vanishes identically in F ( α ) . In the case of the modifiedHelmholtz equation valid for v n , n = 1 , , . . . , the uniqueness theorem analogous to6heorem 2 can be obtained in the same way as in [18] (the proof is even simpler forthis equation). Thus, every v n with n ≥ F ( α ) . Therefore, if φ belongsto L (cid:0) W ( α )1 (cid:1) , then φ vanishes identically regardless which boundary condition holds onthe vertical part of the domain’s boundary, that is, on ∂W ( α )1 \ ( F ∪ { y = − } ).Of course, in various uniqueness theorems proved for water wave problems involvingbounded obstacles in a layer of constant depth, a solution is supposed to belong to awider class than the corresponding L space. Hence the requirement that the L -normof a solution is finite must be changed to a certain radiation condition complementingproblem (2)–(4) and guaranteeing that it has a trivial solution only. In particular, sucha condition can be formulated similarly to that used for the two-dimensional Helmholtzequation in F ( α ) ; see, for example, [20], § References [1] C. Hazard, On the absence of trapped modes in locally perturbed open waveguides.
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