Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth: Derivation and Numerical Solution

Abstract

The nonlinear two-dimensional problem describing periodic steady gravity waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko’s equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the $2 \pi$-periodic Hilbert transform ${\mathcal {C}}$ used in the equation for deep water, the equation obtained here contains a certain operator ${\mathcal {B}}_r$, which is the sum of ${\mathcal {C}}$ and a compact operator depending on a parameter related to the depth of water. Numerical computations are based on an equivalent form of Babenko’s equation derived by virtue of the spectral decomposition of the operator ${\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t$. Bifurcation curves and wave profiles of the extreme form are obtained numerically.

Appendices

Appendix A

Here, we outline how Eq. (28) differs from the Babenko equation for finite depth obtained by Constantin, Strauss and Vărvărucă [18] (see Remark 4 in their paper), and what these equations have in common.

First, we describe the Babenko equation derived by Constantin, Strauss and Vărvărucă, which has the following form (see (2.50) in [18]):

$$\begin{aligned} {\tilde{\mu }} {{\mathcal {C}}}_d ({\tilde{v}}') = {\tilde{v}} + {\tilde{v}} {{\mathcal {C}}}_d ({\tilde{v}}') + {{\mathcal {C}}}_d ({\tilde{v}}' {\tilde{v}}) , \end{aligned}$$

(47)

where $d > 0$ is the so-called conformal mean depth, which is defined uniquely by the fluid domain. However, this depth, generally speaking, is not equal to the nondimensional mean depth of the water domain D introduced in Sect. 2.2; see, in particular, formulae (10). By analogy with the conformal mean depth, it would be natural to characterise the parameter $r \in (0, 1)$, on which the operator ${{\mathcal {B}}}_r$ depends in (28), as the conformal mean radius of the water domain D. Furthermore, the conjugation operator ${{\mathcal {C}}}_d$ is defined for $2 \pi$-periodic functions on $\mathbb {R}$ as follows:

$$\begin{aligned} ({{\mathcal {C}}}_d f) (x) = \sum _{n=1}^\infty \coth n d \, (a_n \sin n x - b_n \cos n x) , \quad x \in \mathbb {R}, \end{aligned}$$

provided f has zero mean value over a $2 \pi$ interval; that is, its Fourier series has the form

$$\begin{aligned} f (x) = \sum _{n=1}^\infty (a_n \cos n x + b_n \sin n x) , \quad x \in \mathbb {R}. \end{aligned}$$

This definition is similar to that of ${{\mathcal {B}}}_r$ in (4), but with the multiplier $\coth n d$ instead of $(1 + r^{2 n}) / (1 - r^{2 n})$ appearing in our operator. Moreover, ${{\mathcal {C}}}_d$ has a representation analogous to $\mathcal {B}_r = \mathcal {C} + \mathcal {K}_r$ with $\mathcal {K}_r$ given by (23); see formulae (A.9) and (A.12) in [18]. Thus, despite a similarity between ${{\mathcal {C}}}_d$ and ${{\mathcal {B}}}_r$, the essential point that distinguishes these operators is that the latter operator is defined for all $2 \pi$-periodic functions, whereas the domain of ${{\mathcal {C}}}_d$ is orthogonal to constants.

Having described how our equation (28) differs from (47), let us turn to demonstrating what these equations have in common. It happens that the nondimensional parameter ${\tilde{\mu }}$ in equation (2.50), [18], coincides with $\mu$ in (28); that is, the bifurcation parameter is the same in both equations (47) and (28).

To demonstrate that ${\tilde{\mu }} = \mu$, we begin with the definition

$$\begin{aligned} \tilde{\mu } = \frac{2R}{g} - 2 d - 2 \beta . \end{aligned}$$

(48)

Here, R is the Bernoulli constant that appears in (8), and d is the parameter in ${{\mathcal {C}}}_d$, whereas $\beta$ is defined by Constantin, Strauss and Vărvărucă in terms of the solution to their equation, but, fortunately, its exact value is of no importance for our considerations. From the formulae used in Sect. 2.2 for the derivation of the nondimensional problem, it follows that $l = \pi$ and $H = h$, and so

$$\begin{aligned} \mu = \frac{2R}{g} - 2 h . \end{aligned}$$

Comparing this formula and (48), we see that to prove $\tilde{\mu }= \mu$ it is sufficient to show that $h = d + \beta$. To prove this equality, we notice that the definition of ${{\mathcal {C}}}_d$ implies that the functions ${{\mathcal {C}}}_d ({\tilde{v}}')$ and ${{\mathcal {C}}}_d ({\tilde{v}}' {\tilde{v}})$ are orthogonal to constants. Then, equation (47) implies

$$\begin{aligned} \int _{-\pi }^{\pi } {\tilde{v}} (x) [ 1 + {{\mathcal {C}}}_d ({\tilde{v}}') (x) ] \, \mathrm {d}x = 0 \quad \Longleftrightarrow \quad \int _0^{\pi } {\tilde{v}} (x) [ 1 + {{\mathcal {C}}}_d ({\tilde{v}}') (x) ] \, \mathrm {d}x = 0 , \end{aligned}$$

where the second relation follows from the fact that ${\tilde{v}}$ is an even solution of (47). To transform this relation, we consider the parametric representation of the free surface profile used in [18]:

$$\begin{aligned} X (x) = U (x, 0) = x + \mathcal {C}_d (\tilde{v} + \beta ) , \quad Y (x) = V (x, 0) = \tilde{v} + d + \beta , \quad x \in \mathbb {R};\qquad \end{aligned}$$

(49)

see formulae (2.7), (2.8), (2.10) on p. 202 and the definition of $\beta$ in Remark 4. Hence, we have

$$\begin{aligned} \int _0^{\pi } {\tilde{v}} (x) \frac{\mathrm {d}X}{\mathrm {d}x} (x) \, \mathrm {d}x = \int _0^{\pi } {\tilde{v}} (x (X)) \, \mathrm {d}X = 0 , \end{aligned}$$

where it is taken into account that X(x) is invertible on $(0, \pi )$. Averaging the second formula (49) over $(0, \pi )$, we obtain

$$\begin{aligned} h = \frac{1}{\pi } \int _0^\pi Y(x (X)) \, \mathrm {d}X = \frac{1}{\pi } \int _0^\pi \tilde{v} (x (X)) \, \mathrm {d}X + d + \beta = d + \beta . \end{aligned}$$

This completes the proof of the equality $\tilde{\mu } = \mu$.

Appendix B

Here, we compare our numerical method (described in Sect. 4) with that used by Clamond and Dutykh [17] for computing waves with the help of a Babenko-like equation which involves an operator that up to notation coincides with ${{\mathcal {C}}}_d$ in equation (47) derived in [18].

The advantage of the equation considered in [17] is that the very fast Petviashvili iteration algorithm [37] is applicable for its numerical solution. This algorithm has another merit, namely, it is insensitive to the initial approximation of a solution. Indeed, it has been discovered that Petviashvili iterations always converge to the correct solution if the linear approximation is taken as the initial guess. Moreover, it is found that the algorithm ‘works efficiently [...] for quite large steepnesses, up to approximately 99% of the maximum steepness for all wavelengths’. Thus, the following conclusion is made in [17]: ‘To the best of our knowledge, it is the first algorithm that is uniformly valid for all wavelength-over-depth ratios’ (see p. 512).

However, the authors point out that their algorithm has a drawback that makes it inappropriate for computing bifurcation diagrams (the topic of the present paper). It happens that the Petviashvili algorithm does not work close to the limiting solutions, thus being unsuitable for computing waves with different crests like that plotted in Fig. 12. Moreover, the advantage mentioned above—that the algorithm does not need a good initial guess—gives rise to difficulties when following the bifurcation curve.

In this aspect, the software package SpecTraVVave used for our computations has an advantage because its part referred to as Navigation (see [23], Fig. 12) serves for the purpose of choosing good initial approximations which allow us not only to follow the bifurcation curve, but also to find secondary bifurcations. Of course, our numerical method based on the SpecTraVVave package is not as multipurpose as that in [17], but it is better suited for computation of bifurcation diagrams.