Surface tension plays a significant role as a restoration force in the setting of small-amplitude waves, leading to pure capillary and gravity-capillary waves. We show that within the framework of linear theory, the particle paths in a periodic gravity–capillary or pure capillary wave propagating at the surface of water over a flat bed are not closed.
The most common force for creating water waves is the wind: the friction between the air and the water deforms the originally flat water surface creating waves of small wavelength and amplitude, called ripples. These waves are capillary waves: surface tension is the dominant restoring force. In the presence of ripples, it is easier for the wind to grip the roughened surface, and thus increase the amplitude/wavelength of these waves, creating capillary–gravity waves where gravity acts as a restoring force in addition to the surface tension. As the waves achieve greater amplitude, the effects of surface tension become negligible: we encounter gravity water waves. The main three types of water waves (capillary, capillary–gravity and gravity waves) have generally different properties. In this paper, we are interested in the particle trajectories of periodic capillary and capillary–gravity waves within the framework of linear theory. Owing to the intractability of the nonlinear governing equations (some results on the nonlinear governing equations are available, see Amick et al. 1982; Jones & Toland 1986; Toland 1996; Groves & Toland 1997; Craig & Nicholls 2000; Constantin & Strauss 2002, 2004; Constantin & Escher 2004a,b; Ehrnström 2005; Constantin 2006; Constantin et al. 2006; Wahlen 2006a,b, but more detailed information is needed to study the particle paths in the fluid), the linear framework appears to be appropriate for a first study of the particle trajectories in capillary and capillary–gravity waves. Since most observed waves that are not near breaking appear to be two-dimensional periodic wave trains, we consider the case of periodic travelling waves. While formal considerations suggest that the particle trajectories in the fluid are closed (Stoker 1957; Lighthill 1978; Debnath 1994; Johnson 1997), and while there are special solutions to the nonlinear governing equations with all particle trajectories closed (Constantin 2001a,b), we will show that within linear capillary and capillary–gravity theory for steady waves, this is not the case: if the surface is not flat there are no closed orbits in the fluid, see theorem 3.1. Further studies might permit the extension of these results to the nonlinear governing equations—for gravity waves, the features observed within the linear theory in Constantin & Villari (in press) were recently proven (Constantin 2006) to hold true for the nonlinear governing equations.
In what follows we will deal with waves that are two dimensional, that is, the motion is identical in any direction parallel to the crest line. We consider a cross-section of the flow in the direction perpendicular to the crest line with Cartesian coordinates (x, y). The x-axis is the direction of wave propagation while the y-axis points vertically upwards. As such, let (u(t, x, y), v(t, x, y)) be the velocity field of the flow over the flat bed y=0 and y=h0+η(t, x) be the free surface of water, where h0>0 is the mean water level. It is physically reasonable to assume homogeneity (constant density) in the fluid for gravity–capillary and capillary waves (Lighthill 1978). This implies the equation of mass conservation(2.1)
Under the further assumption of inviscid flow, the governing equations for the motion of the waves are given by Euler's equation(2.2)where P(t, x, y) denotes the pressure and g is the gravitational constant of acceleration. In order to decouple the motion of the air from that of the free surface particles (Johnson 1997), we introduce the dynamic boundary condition(2.3)where P0 is the constant atmospheric pressure; the parameter Γ(>0) is the coefficient of surface tension; and 1/R is the curvature in the x-direction given in Cartesian coordinates by(2.4)Furthermore, since the free surface is always composed of the same particles, we have the kinematic boundary condition(2.5)On the flat bed, we have the kinematic boundary condition(2.6)which tells us that the rigid bed is impenetrable. The governing equations for the gravity–capillary water wave problem are encompassed by the nonlinear free boundary problems (2.1)–(2.6) (cf. Johnson 1997). We make the further assumption that(2.7)a condition which categorizes irrotational flows. It is reasonable to assume that water which is disturbed from a position of rest will remain irrotational at later times (Lighthill 1978; Johnson 1997).
(a) Linear gravity–capillary water waves
The problems (2.1)–(2.7) are non-dimensionalized using a typical wavelength λ and a typical amplitude a of the wave. We define the set of non-dimensional variableswhere, for example, we replace x by λx, with x now being a non-dimensionalized variable, thus avoiding new notation. We set the constant water density ρ=1 and the pressure in the new non-dimensional variables is given bywhere the non-dimensional pressure variable p measures the deviation from the hydrostatic pressure distribution. We end up with the following boundary value problem in non-dimensional variables:(2.8)where ϵ=a/h0 is the amplitude parameter and δ=h0/λ is the shallowness parameter. It is conventional to write Γ/(ρgλ2)=δ2We, with being the Weber number. This non-dimensional parameter measures the size of the surface tension contribution. From the fourth and the sixth equations in equation (2.8), it is obvious that both v and p, if evaluated on y=1+ϵη, are essentially proportional to ϵ. Indeed, physically as ϵ→0 we must have v→0 and p→0. This leads us to the scaling of the non-dimensional variablesavoiding again the introduction of new variables. Now problem (2.8) becomes(2.9)
The linearized problem is obtained by letting ϵ→0 in equation (2.9). The resulting equations are(2.10)
We will seek periodic travelling wave solutions of equation (2.10), that is, waves for which the (t, x) dependence of u, v, p, η is in the form of a periodic dependence in x−c0t, where c0>0 represents the non-dimensionalized speed of the wave. If we choose the Fourier modemanipulating the various equations in equation (2.10), we see thatand sofor some function f(x, y) which we choose to be equal to 0. For a fixed t, and under the change of variables , we get(2.11)
We solve this using the method of separable variables. Assuming(2.12)upon substituting this form of v into equation (2.11) we obtain, whenever X≠0≠Ywhere the prime denotes differentiation. As the left-hand side is independent of the y variable, and the right-hand side is similarly independent of the variable, it immediately follows thatfor some real constant K. Owing to the required periodicity in the -variable, we must have K=(2πα)2 for some . Thus,(2.13)the solutions of which are(2.14)where A, B, C, D are constants to be solved using the initial conditions. For y=0, we are told that v=0, and so it follows that C=−D as we cannot have A=B=0 unless v≡0 (and v≡0 in equation (2.10) is not admissible, since it forces p to be constant which is incompatible with our choice of η). Equation (2.12) now becomes(2.15)For y=1, v=ηt becomesThe functional part of the left-hand side must be equal to the functional part of the right-hand side, and this can only happen if A=−B and α=δ. Equating the constants on both sides, it follows that . Thus,It is a straightforward exercise in integration of the equations in equation (2.10) to show thatso long as
We return to the original physical variables, using the change of variables
The linear wave solution in the physical variables is(2.16)of amplitude ϵh0>0 and wavelength λ>0, propagating over the flat bed y=0 and with mean water level h0>0. Hereare the wavenumber and the frequency, respectively, and the dispersion relationdetermines the speed c of the linear wave. The period of this wave is
The two main classes of waves of interest in linear wave theory are shallow- and deep-water waves (Lighthill 1978). The case δ=h0/λ→0 corresponds to long waves, or shallow-water waves, whereas the deep-water limit is given by δ→∞. In the long waves case, kh0→0 and the dispersion relation for capillary–gravity waves yieldswhich is independent of both the wavelength λ and the coefficient of surface tension Γ. In the deep-water limit kh0→∞ and we obtain
This displays for capillary–gravity waves a strong relationship between the wavelength λ, the coefficient of surface tension Γ and the wave speed c. In addition, for short capillary–gravity waves, the wave speed is independent of the coefficient of gravity g. In deep water, the shorter the capillary–gravity wave the faster it travels, and the coefficient of surface tension is proportional to the square of the wave speed. It is interesting to note that the contrast in this aspect of the capillary–gravity waves with that of pure gravity waves: in deep water, the longer gravity waves propagate faster than shorter ones—see the discussion in Constantin & Villari (in press).
If (x(t), y(t)) is the path of the particle below the linear wave (2.16), thenso that the motion of the particle is described by the system(2.17)with initial data (x0, y0), where we denoted(2.18)
The right-hand side of the differential system (2.17) is smooth so that the existence of a unique local smooth solution is ensured (Hale 1969). In addition, since y is bounded, the right-hand side of equation (2.17) is bounded and therefore this unique solution is defined globally (Hale 1969).
If we now consider the case of pure capillary waves by letting g→0, the equations in (2.16) remain unchanged except for the last equation describing the pressure. This becomes(2.19)and we have
Therefore, the system (2.17) describes equally well the motion of particles in pure capillary waves, but with different c and M due to the different dependence of c and ω on Γ. If Γ tends to zero, the hydrostatic pressure becomes constant and the wave speed vanishes, which makes perfect physical sense. An examination of the pure capillary wave speed c in the case Γ>0 shows us that, in the deep-water limit, we obtain the same wave speed as the capillary–gravity waves. This fact is consistent with our previous observation that in waves with relatively small amplitude the surface tension effects dominate over gravity forces. Also in the long wave, or shallow water, limit the wave speed becomes zero: gravity plays an important role in the motion of waves with long wavelength.
3. Qualitative analysis of solutions
We now proceed to an analysis of the solutions of equation (2.17). Since the right-hand side of equation (2.17) is nonlinear, we will not try to solve this system of equations explicitly. Instead, we use phase plane analysis to examine the qualitative features of the solutions. Our aim is to show that there are no water particles travelling in closed orbits. In fact, we will see that every water particle experiences a forward drift as the wave progresses. We use the transformation(3.1)to give us the new system(3.2)
Since equation (3.2) is periodic in X, we need only consider the strip
Furthermore, as equation (3.2) is a description of our physical model, we can restrict our attention to the values Y>0.
The 0-isocline is defined to be the set where dY/dt=0, and the ∞-isocline is the set where dX/dt=0. Therefore, the 0-isocline is given byand the ∞-isocline is given by the curve (X, α(X)), for X∈(−(π/2), π/2), α(X)∈[Y*, ∞), where and α is defined as follows: on [0, π/2) we set α to be the inverse of the function defined on [Y*, ∞), and extend it by mirror symmetry to the interval (−(π/2), π/2). Now, by equation (2.18) we have(3.3)since s<sinh(s) for s>0 and we assume that ϵ<1 within the confines of linear theory. It follows for Y≥Y* that , and so α is well defined. Furthermore, the even function α is smooth, it takes on its infimum Y* at X=0, and satisfies
Now, for X∈(π/2, π) we have dX/dt<0, dY/dt>0. If X∈(0, π/2) then dX/dt<0 below the curve of α(X) and is positive above it, while dY/dt remains positive in this region. We obtain the corresponding signs for X∈(−π, 0) using the symmetric definition with respect to the Y-axis.
The only singular point of the system (3.2) in our region is P=(0, Y*). In order to show that P is a saddle point, we rewrite equation (3.2) as a Hamiltonian system(3.4)with the Hamiltonian function . Now P is a critical point of H, and as the Hessian of H at P isit follows that P is a non-degenerate singular point. By Morse's lemma (Milnor 1963) in a neighbourhood of P, there exists a diffeomorphic change of coordinates which sends the level lines of H to hyperbolas. Thus, P is a saddle point for H. If (X, Y) is a solution of equation (3.2), thenand so H is constant along the phase curves. Away from the critical point P, the seperatrix is a smooth curve, since we can apply the implicit function theorem (Hale 1969). It intersects the vertical line X=π at the point (π, β) if
Suppose, we have another point Q=(π, Y) on this line. If Y>β, then the positive trajectory γ+(Q) of the phase curve is unbounded, whereas γ+(Q) will intersect the line X=−π at (−π, Y) if Y∈(0, β).
At this point, we should take note of the restrictions necessary to ensure that solutions are compatible with our physical model. Namely, from the above discussion it is clear that we require(3.6)
This condition is ensured if ϵ cosh(h0(1+ϵ))<1 (Constantin & Villari in press), a relation which gives a quantitative meaning to the notion that ‘ϵ<1 is small’.
Let (X(t), Y(t)) be a solution of equation (3.2) with (X(0), Y(0))=(π, Y0), Y0∈[0, β). We denote by t−π(Y0) the time it takes for the phase curve (X(t), Y(t)) to intersect the line X=−π. Considerations similar to those made in the case of pure gravity water waves (Constantin & Villari in press) show that the only possible period τ for a periodic particle path (x(t), y(t)) is τ=2π/ω and, conversely, if t−π(Y0)=2π/ω then the corresponding particle path is periodic. We now prove our main result.
The system (2.17) has no periodic solutions.
As a consequence of the previous considerations, it suffices for us to show that t−π(Y0)>2π/ck for Y0∈[0,β) in order to prove the theorem.
We start with the case Y0=0. From equation (3.2), it follows that the phase curve of (X(t), Y(t)) with (X(0), Y(0))=(π, 0) remains on the line Y=0, and it can be obtained explicitly by solving the differential equation(3.7)
Keeping in mind that M<c, we integrate to get
Therefore,proving the theorem in the case Y0=0.
For the case Y0∈(0,β), we work as follows. Since dY/dt>0 in the region X∈(0,π), and dY/dt<0 when X∈(−π,0), then if Y1∈(Y0,Y*) is the value where the phase curve (X(t), Y(t)) intersects the line X=π/2, we have the phase curve lying below the line Y=Y1 if and lying above it if . Thus,(3.8)
Let us introduce the differential equation
It follows immediately from equation (3.8) and the fact that that for t≥0, and thus t−π(Y0)>t* where t* is the time when . However, we can now compute t* explicitly as being(3.9)in a manner similar to that of the solution of equation (3.7). Thus, t−π(Y0)>2π/ck, completing the proof.▪
The qualitative analysis performed above for the system (3.2) lets us describe the particle trajectories in linear capillary and capillary–gravity waves. We have, in view of equations (2.17) and (3.5),Hence, if we assume that at t=0 and X(0)=π a particle is at its greatest possible depth with y(0)=y0, then the particle moves back and up, then forwards up and down, and finally backwards and downwards, reaching the level y=y0 in the time t−π(Y0)>(2π/ω) with
The author gratefully acknowledges the hospitality of the Mittag-Leffler Institute, Sweden, for creating a pleasant research environment during the programme ‘Wave Motion’ in the autumn of 2005. This research was supported by the Science Foundation Ireland under the project 04/BRG/M0042.
One contribution of 13 to a Theme Issue ‘Water waves’.
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