aa r X i v : . [ phy s i c s . f l u - dyn ] J a n On cusped solitary waves in finite water depth
Shijun LiaoState Key Laboratory of Ocean EngineeringSchool of Naval Architecture,Ocean and Civil EngineeringShanghai Jiaotong University, Shanghai 200240, ChinaDepartment of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China( Email address: [email protected] )
Abstract
It is well-known that the Camassa-Holm (CH) equation admits both of thepeaked and cusped solitary waves in shallow water. However, it was an openquestion whether or not the exact wave equations can admit them in finite waterdepth. Besides, it was traditionally believed that cusped solitary waves, whose1st-derivative tends to infinity at crest, are essentially different from peakedsolitary ones with finite 1st-derivative. Currently, based on the symmetry andthe exact water wave equations, Liao [1] proposed a unified wave model (UWM)for progressive gravity waves in finite water depth. The UWM admits not onlyall traditional smooth progressive waves but also the peaked solitary waves infinite water depth: in other words, the peaked solitary progressive waves areconsistent with the traditional smooth ones. In this paper, in the frame ofthe linearized UWM, we further give, for the first time, the cusped solitarywaves in finite water depth, and besides reveal a close relationship between thecusped and peaked solitary waves: a cusped solitary wave is consist of an infinitenumber of peaked solitary ones with the same phase speed, so that it can beregarded as a special peaked solitary wave. This also well explains why and howa cuspon has an infinite 1st-derivative at crest. It is found that, like peakedsolitary waves, the vertical velocity of a cusped solitary wave in finite waterdepth is also discontinuous at crest ( x = 0), and especially its phase speed hasnothing to do with wave height, too. In addition, it is unnecessary to considerwhether the peaked/cusped solitary waves given by the UWM are weak solutionor not, since the governing equation is not necessary to be satisfied at crest. Allof these would deepen and enrich our understandings about the cusped solitarywaves. PACS:
Key words
Solitary waves, cusped crest, discontinuity
The smooth solitary surface wave was first reported by John Scott Russell [2] in1844. Since then, various types of solitary waves have been found. The mainstreammodels of shallow water waves, such as the Boussinesq equation [3], the KdV equa-tion [4], the BBM equation [5] and so on, admit dispersive smooth periodic/solitaryprogressive waves with permanent form: the wave elevation is infinitely differentiable everywhere . Especially, the phase speed of the smooth waves is highly dependent uponwave height: the larger the wave height of a smooth progressive wave, the faster itpropagates. Nowadays, the smooth amplitude-dispersive periodic/solitary waves arethe mainstream of researches in water waves.In 1993, Camassa and Holm [6] proposed the celebrated Camassa-Holm (CH)equation for shallow water waves, and first reported the so-called peaked solitarywave, called peakon, which has a peaked crest with a discontinuous (but finite) 1st-order derivative at crest. This is a breakthrough in water wave theories, since itopens a new field of research in the past 20 years. Physically, different from the KdVequation and Boussinesq equation, the CH equation can model phenomena of not onlysoliton interaction but also wave breaking [7]. Mathematically, the CH equation isintegrable and bi-Hamiltonian, therefore possesses an infinite number of conservationlaws in involution [6]. Besides, it is associated with the geodesic flow on the infinitedimensional Hilbert manifold of diffeomorphisms of line [7]. Thus, the CH equationhas lots of intriguing physical and mathematical properties. It is even believed thatthe CH equation “has the potential to become the new master equation for shallowwater wave theory” [8]. In addition, Kraenkel and Zenchuk [9] reported the cuspedsolitary waves of the CH equation, called cuspon. The so-called cuspon is a kind ofsolitary wave with the 1st derivative going to infinity at crest. Note that, unlike apeakon that has a finite infinite finite water depth. For example, the velocity distributionof peaked/cusped solitary waves in the vertical direction was unknown, since it cannot be determined by a wave model in shallow water (such as the CH equation).Currently, based on the symmetry and the exact wave equations, Liao proposed aunified wave model (UWM) for progressive gravity waves in finite water depth withpermanent form [1]. It was found that the UWM admits not only all traditionalsmooth periodic/solitary waves but also the peaked solitary waves in finite waterdepth, even including the famous peaked solitary waves of the CH equation as itsspecial case. Therefore, the UWM unifies both of the smooth and peaked solitarywaves in finite water depth, for the first time. In other words, the progressive peakedsolitary waves in finite water depth are consistent with the traditional smooth waves,and thus are as acceptable and reasonable as the smooth ones.In this article, using the linearized UWM, we give an closed-form expression ofcusped solitary waves in finite water depth, and illustrate that a cusped solitary waveis consist of an infinite number of peaked solitary ones. This reveals, for the first timeto the best of my knowledge, a simple but elegant relationship between the peakedand cusped solitary waves in finite water depth.
Let us first describe the UWM briefly. Consider a progressive gravity wave propagat-ing on a horizontal bottom in a finite water depth D , with a constant phase speed c anda permanent form. For simplicity, the problem is solved in the frame moving with thephase speed c . Let x, z denote the horizontal and vertical dimensionless co-ordinates(using the water depth D as the characteristic length), with x = 0 corresponding tothe wave crest, z = − z axis upward, respectively. Assumethat the wave elevation η ( x ) has a symmetry about the crest, the fluid in the interval x > x > x = 0 is not absolutely necessary to be irrotational. Let φ ( x, z ) denote the velocity potential. All of them are dimensionless using D and √ gD as the characteristic scales of length and velocity, where g is the acceleration due togravity. In the frame of the UWM, the velocity potential φ ( x, z ) and the wave eleva-tion η ( x ) are first determined by the exact wave equations (i.e. the Laplace equation ∇ φ = 0, the two nonlinear boundary conditions on the unknown free surface η , thebed condition and so on) only in the interval x ∈ (0 , + ∞ ), and then extended to thewhole interval ( −∞ , + ∞ ) by means of the symmetry η ( − x ) = η ( x ) , u ( − x, z ) = u ( x, z ) , v ( − x, z ) = − v ( x, z ) , which enforces the additional restriction condition v (0 , z ) = 0. Note that, in the frameof the UWM, the flow at x = 0 is not necessarily irrotational, so that the UWM ismore general: this is the reason why the UWM can admit both of the smooth andpeaked solitary waves.In the interval (0 , + ∞ ), the governing equation ∇ φ ( x, z ) = 0 with the bedcondition φ z ( x, −
1) = 0 has two kinds of general solutions [10], where the subscriptdenotes the differentiation with respect to z . One iscosh[ nk (1 + z )] sin( nkx ) , corresponding to the smooth periodic waves with the dispersive relation α = tanh( k ) k ≤ , (1)where α = c/ √ gD is the dimensionless phase speed, k is wave number and n is aninteger, respectively. The other iscos[ nk ( z + 1)] exp( − nkx ) , corresponding to the peaked solitary waves in finite water depth [1], with the relation α = tan( k ) k ≥ , (2)where k has nothing to do with wave number. Given α ≤ unique solution, as mentioned in thetextbook [10]. However, given α ≥ infinite number of solutions: α = tan k n k n , nπ ≤ k n ≤ nπ + π , n ≥ , (3)corresponding to an infinite number of peaked solitary waves [1] η n ( x ) = A n exp( − k n | x | ) (4)in the frame of the linear UWM, where A n denotes its wave height. For example,when α = 2 √ /π , the transcendental equation (2) has an infinite number of solutions k = π/ k = 4 . k = 7 . k = 10 . k = 14 . k = 17 . k = 20 . k = 23 . k = 26 . k = 29 . k = 32 . k n ≈ ( n + 0 . π, n > , (5)with less than 0.08% error. In general, k n ≈ ( n + 0 . π is a rather accurate approxi-mation of k n for large enough integer n .Obviously, the peaked solitary wave (4) is not smooth at crest, i.e. its first deriva-tive is discontinuous. Note that the well-known peaked solitary wave η = c exp( −| x | )of the CH equation is only a special case of (4) when A n = c and k n = 1. However,unlike η = c exp( −| x | ) that is a weak solution of the CH equation, it is unnecessary to consider whether or not the peaked solitary wave (4) is a kind of weak solution,because the CH equation is defined in the whole domain −∞ < x < + ∞ but thegoverning equation of the UWM is defined only in 0 < x < + ∞ . Physically, unlikethe CH equation and the fully nonlinear wave equations, waves in the frame of theUWM are not necessary to be irrotational at x = 0, therefore the governing equationholds only in the domain 0 < x < + ∞ , since the solution in the interval −∞ < x < x = 0 is a boundary of thegoverning equation, and it is well-known that solutions of differential equations canbe non-smooth at boundary, like a beam with discontinuous cross sections acted by aconstant bending moment. Therefore, in the frame of the UWM, it is unnecessary toconsider whether or not the peaked solitary waves (4) are weak solutions at all. Thisis the reason why, unlike the well-known peaked solitary wave η = c exp( −| x | ) of theCH equation whose phase speed is always equal to its wave height, the phase speed ofthe peaked solitary waves (4) given by the UWM has nothing to do with wave height!This is the most attractive novelty of the UWM.The above peaked solitary waves in finite water depth have some unusual char-acteristics, as revealed by Liao [1]. First, it has a peaked crest with a discontinuousvertical velocity v at crest. Besides, unlike the smooth waves whose horizontal velocity u decays exponentially from free surface to bottom, the horizontal velocity u of thepeaked solitary waves at bottom is always larger than that on free surface. Especially,different from the smooth waves whose phase speed depends upon wave height, thephase speed of the peaked solitary waves in finite water depth has nothing to do withwave height, i.e. it is non-dispersive . x η -10 -5 0 5 1000.020.040.060.080.1 H w = 0.1, k = π /6,Solid line: β = 1.9Dashed line: β = 1.5 Figure 1: Cusped solitary waves in finite water depth defined by (6) when H w = 0 . k = π/ α = 12 / / √ π ). Solid line: β = 1 .
9; Dashed line: β = 1 . x η -10 -5 0 5 1000.020.040.060.080.1 H w = 0.1, k = π /6Solid line: β = 10Dashed line: β = 2.5 Figure 2: Peaked solitary waves in finite water depth defined by (6) when H w = 0 . k = π/ α = 12 / / √ π ). Solid line: β = 10; Dashed line: β = 2 . α ≥
1, there exist an infinite number of peaked solitary waves A n exp( − k n | x | ) withthe same phase speed α but different wave amplitudes A n . Thus, we may have suchpeaked solitary waves η ( x ) = ∞ X n =0 A n exp( − k n | x | ) , where A n is a constant, which can be chosen with great freedom, as long as the aboveinfinite series is convergent in the whole interval ( −∞ , + ∞ ). As a special case of it,let us consider such a one-parameter family of wave elevations η ( x ) = H w ζ ( β ) + ∞ X n =1 n β exp( − k n − | x | ) , β > , (6)where H w denotes wave height, β > ζ ( β ) is the Riemann zeta function,and k n is determined by (2) for the given α ≥
1, respectively. Since β >
1, we have P + ∞ n =1 n − β = ζ ( β ) so that the above infinite series converges to the wave height H w at x = 0, and besides is convergent in the whole interval ( −∞ , + ∞ ). However, its 1stderivative at x = 0, i.e. η ′ (0) = ± H w ζ ( β ) + ∞ X n =1 k n − n β , (7)is convergent to a finite value when β >
2, but tends to infinity when 1 < β ≤ k n − ≈ ( n − . π for large enough integer n and the series P /n β − isconvergent when β > < β ≤
2. So, the infinite series(6) defines a cusped solitary wave in finite water depth when 1 < β ≤ peaked solitary wave when β >
2. Therefore, in essence, a cusped solitary wave in finite waterdepth is consist of an infinite number of peaked solitary waves (when 1 < β ≤
2) withthe same phase speed! To the best of the author’s knowledge, this reveals, for the firsttime, a simple but elegant relationship between the peaked and cusped solitary wavesin finite water depth! In addition, the infinite series (6) illustrates the consistency ofthe peaked and cusped solitary waves, and besides explains why and how a cusponhas an infinite 1st-derivative at crest. Since the phase speed of peaked solitary waves(4) in finite water depth has nothing to do with the wave height, it is straight forwardthat the phase speed of a cusped solitary wave in finite water depth also has nothingto do with the wave height, too.Note that, according to the definition of the wave elevation (6), given a dimen-sionless phase velocity α ≥ H w , there exist an infinite number of cusped solitary waves, dependent upon β ∈ (1 , α = 2 √ /π, H w = 1 /
10 when β = 1 . β = 1 . same expression (6) can define an infinite number of peakedsolitary waves in finite water depth, too, depending on β ∈ (2 , + ∞ ). For example, thetwo peaked solitary waves in finite water depth in the case of α = 2 √ /π, H w = 1 / β = 5 / β = 10 are as shown in Fig. 2. This well illustrates the consistencyof the peaked and cusped solitary waves in finite water depth.Theoretically speaking, given an arbitrary wave height H w and a dimensionlessphase speed α ≥
1, there are many different types of peaked/cusped solitary wavesin finite water depth. For example, a more generalized, two-parameter family ofpeaked/cusped solitary waves in finite water depth reads η ( x ) = H w ζ ( β, γ ) + ∞ X n =0 ,n = − γ n + γ ) β exp( − k n | x | ) , (8)where β > γ = 0 are constants to be chosen with great freedom, ζ ( β, γ ) is ageneralized Riemann zeta function, and k n is determined by (2) for the given α ≥ k n ≈ ( n + 0 . π for large enough integer n , the above infiniteseries defines a cusped solitary wave when 1 < β ≤ peaked ones when β > φ + (defined only inthe interval x >
0) corresponding to the peaked/cusped solitary wave elevation (6)reads φ + = − αH w ζ ( β ) + ∞ X n =1 cos[ k n − ( z + 1)] exp( − k n − x ) n β sin( k n − ) , (9)which gives, using the symmetry, the corresponding horizontal velocity u = α H w ζ ( β ) + ∞ X n =0 k n cos[ k n ( z + 1)] exp( − k n | x | )( n + 1) β sin( k n ) (10)in the whole interval x ∈ ( −∞ , + ∞ ), the vertical velocity v + = α H w ζ ( β ) + ∞ X n =0 k n sin[ k n ( z + 1)] exp( − k n x )( n + 1) β sin( k n ) (11)in the interval x ∈ (0 , + ∞ ), and the vertical velocity v − = − α H w ζ ( β ) + ∞ X n =0 k n sin[ k n ( z + 1)] exp( k n x )( n + 1) β sin( k n ) (12)in the interval ( −∞ , x → v + = − lim x → v − for the cusped (1 < β ≤
2) and peaked ( β >
2) solitary waves, although we alwayshave v = 0 at x = 0. Thus, like a peakon in finite water depth, a cuspon in finitewater depth has the velocity discontinuity at x = 0, too.Especially, at z = 0 and as x →
0, the corresponding vertical velocity readslim x → v + ( x,
0) = α H w ζ ( β ) + ∞ X n =0 k n ( n + 1) β , which is finite when β > < β ≤
2, since k n ≈ ( n + 0 . π for large enough integer n . Thus, unlike a peaked solitary wave in finite water depthwhose v is always finite, the vertical velocity of the cusped solitary waves in finitewater depth tends to infinity at z = 0 as x →
0. Mathematically, this is acceptable,since it is traditionally believed that a cuspon has a higher singularity than a peakon.Such kind of singularity leads to a more strong vortex sheet at x = 0 near z = 0.Physically, in reality such kind of singularity and discontinuity, “if it could ever beoriginated, would be immediately abolished by viscosity”, as mentioned by Lamb [11]. In summary, in the frame of the linearized UWM [1], we give, for the first time, thecusped solitary waves in finite water depth, and reveal that a cuspon is consist ofan infinite number of peaked solitary waves with the same phase speed. This kindof consistency also well explains why and how the 1st-derivative of a cusped solitarywave tends to infinity at crest. It is found that, like a peakon, the vertical velocity ofa cuspon is also discontinuous at x = 0, and besides, its phase speed also has nothingto do with wave height, too. All of these would deepen and enrich our understandingsabout the peaked and cusped solitary waves.It should be emphasized that, in the frame of the UWM, the governing equationis defined only in the domain 0 < x < + ∞ , since the solution at −∞ < x < whole domain −∞ < x < + ∞ . Physically, it means that the flow at crestis not absolutely necessary to be irrotational. Thus, mathematically, we need notconsider whether the peaked/cusped solitary waves are weak solutions or not. Thisis the reason why, unlike the well-known peaked solitary wave η = c exp( −| x | ) ofthe CH equation, whose phase speed is always equal to wave height, the phase speedof the peaked/cusped solitary waves (4) given by the UWM has nothing to do withwave height! This is the most attractive novelty of the UWM, which provides us asimple but elegant relationship between peaked and cusped solitary waves in finitewater depth.This work is partly supported by the State Key Laboratory of Ocean Engineering(Approval No. GKZD010061) and the National Natural Science Foundation of China(Approval No. 11272209). References [1] Liao, S.J. Do peaked solitary water waves indeed exist?
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