2 Geometric interpretation of the Luke variational principle for steady periodic gravity waves

Consider a steady two-dimensional irrotational flow of an ideal incompressible fluid in an infinite $\unicode[STIX]{x1D706}$ -periodic domain bounded by a $\unicode[STIX]{x1D706}$ -periodic line $y=\unicode[STIX]{x1D702}(x)$ from above and by a horizontal bottom $y=0$ from below. One period $\unicode[STIX]{x1D6FA}_{z}$ of such a flow is shown in figure 1(a). The function $\unicode[STIX]{x1D702}(x)$ satisfies the conditions

Let $\unicode[STIX]{x1D719}(x,y)$ be the potential of the flow. For the harmonic function $\unicode[STIX]{x1D719}(x,y)$ , we have the following boundary conditions

where $\boldsymbol{n}$ is the unit outer normal to the boundaries of the flow domain. The velocity vector $\boldsymbol{q}$ and its components $u$ , $v$ are

where $\unicode[STIX]{x1D735}$ is the nabla operator. The volume flux $Q$ is

We assume that $Q>0$ , i.e. the fluid moves from left to right.

In the flow domain, the components of the velocity vector $u(x,y)$ , $v(x,y)$ are $\unicode[STIX]{x1D706}$ -periodic functions with respect to the variable $x$ , but the potential $\unicode[STIX]{x1D719}(x,y)$ increases by an increment $C$ (by a circulation) on every period:

For a fixed upper boundary the flow is defined uniquely by specifying the circulation  $C$ .

Consider now a system of two-dimensional irrotational periodic waves of wavelength $\unicode[STIX]{x1D706}$ propagating with phase velocity $c$ under the influence of gravity $g$ without change of form from right to left on the surface of a fluid of finite depth. In the wave-fixed frame of reference the flow appears to be steady. Hence, we have a $\unicode[STIX]{x1D706}$ -periodic flow which moves from left to right, and its potential $\unicode[STIX]{x1D719}(x,y)$ satisfies the boundary conditions (2.2).

In addition to (2.2), on the unknown free surface $y=\unicode[STIX]{x1D702}(x)$ the Bernoulli equation should be fulfilled:

where $q=|\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}|$ is the modulus of the velocity vector $\boldsymbol{q}$ and $R$ is the Bernoulli constant. Thus, the steady gravity wave problem is fully determined by the boundary conditions (2.2), (2.6) at specified values of $\unicode[STIX]{x1D706}$ , $C$ and $R$ .

For the $\unicode[STIX]{x1D706}$ -periodic flows, consider the Luke Lagrangian, which we take as the integral of pressure over one period (see Luke Reference Luke1967):

where $\unicode[STIX]{x1D70C}$ is the density of the fluid, $p$ is the pressure. At fixed $R$ and $C$ by virtue of the Bernoulli equation,

so the functional $L$ depends only on $\unicode[STIX]{x1D702}(x)$ because $q(x,y)=|\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}|$ is fully determined by the boundary condition (2.2). Hence, we can write

Varying the functional $L[\unicode[STIX]{x1D702}]$ at a fixed $R$ , we obtain

where $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D719}(x,y)$ is the dependent variation of the potential $\unicode[STIX]{x1D719}(x,y)$ corresponding to the independent variation $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}(x)$ .

Applying Green’s first identity to the second integral in (2.10) and taking into account that the function $\unicode[STIX]{x1D719}(x,y)$ is harmonic, we get

where $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{z}$ is the boundary of $\unicode[STIX]{x1D6FA}_{z}$ , $\boldsymbol{n}$ is the unit outer normal to $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{z}$ , $\text{d}s$ is the line element along $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}_{z}$ . By virtue of (2.2), the contour integrals over $y=\unicode[STIX]{x1D702}(x)$ and $y=0$ vanish. Further, the integrals along the vertical lines $x=0$ and $x=\unicode[STIX]{x1D706}$ cancel each other. Indeed, because the circulation $C$ is fixed,

Besides, the function $u(x,y)$ is $\unicode[STIX]{x1D706}$ -periodic with respect to $x$ , and on the vertical lines, we have

Thus, the variation

under the constrain that $C$ is fixed. It follows from (2.14) that for $\unicode[STIX]{x1D706}$ -periodic flows with a fixed circulation $C$ , the dynamic boundary condition (2.6) will be satisfied if and only if $\unicode[STIX]{x1D6FF}L[\unicode[STIX]{x1D702}]$ =0.

It is to be noted that for any $\unicode[STIX]{x1D706}$ -periodic flow

(see, e.g. paper 1, p. 297).

Let $z=x+\text{i}y$ be the complex variable. In the plane $w=\unicode[STIX]{x1D719}+\text{i}\unicode[STIX]{x1D713}$ , consider the rectangle  $\unicode[STIX]{x1D6FA}_{w}$ of length $C$ and width $Q$ , which corresponds to the flow period $\unicode[STIX]{x1D6FA}_{z}$ in the physical $z$ -plane (see figure 1 a,b). By means of the function

we map conformally $\unicode[STIX]{x1D6FA}_{w}$ onto an annulus with an outer radius of unity and inner radius of $r_{0}$ (see figure 1 b,c). It follows from (2.16) that

Taking into account (2.9), (2.15) and (2.17), we come to the formula

where

is the averaged horizontal component of velocity at any level $y=\text{const.}$ completely within the fluid.

As in Longuet-Higgins (Reference Longuet-Higgins1975) and many other works, we non-dimensionalize all flow parameters by choosing $\unicode[STIX]{x1D706}/(2\unicode[STIX]{x03C0})$ , $\sqrt{g\unicode[STIX]{x1D706}/(2\unicode[STIX]{x03C0})}$ and $\unicode[STIX]{x1D70C}$ as the scales for length, velocity and density, respectively. In what follows, all physical quantities will be dimensionless with accordance to the chosen scales. Now, the dimensionless wavelength is $2\unicode[STIX]{x03C0}$ , $\unicode[STIX]{x1D70C}=g=1$ , the potential $\unicode[STIX]{x1D719}$ , circulation $C$ and flux $Q$ are scaled by $\sqrt{g\unicode[STIX]{x1D706}^{3}/(2\unicode[STIX]{x03C0})^{3}}$ . The relation between $Q$ and $c_{a}$ is

The Luke functional $L$ , scaled by $g\unicode[STIX]{x1D706}^{2}/(2\unicode[STIX]{x03C0})^{2}$ , takes the form (for convenience, we double the functional and divide it by $\unicode[STIX]{x1D706}$ )

where $h_{a}$ and $H_{a}$ are the mean and root-mean-square depths:

Any $2\unicode[STIX]{x03C0}$ -periodic domain can be transformed into a doubly connected one by means of the conformal mapping $t=\exp (\text{i}z)$ . The conformal invariant $M$ , that is defined as the ratio $r_{1}/r_{2}$ ( $r_{1}>r_{2}$ ) of the radii of an annulus onto which the doubly connected domain can be mapped, is called the modulus of the domain (see Bergman Reference Bergman1950, p. 102). Taking into account that in our case $M=1/r_{0}$ , one can see that the Luke Lagrangian  $L$ in the form (2.21) contains only geometric characteristics of $2\unicode[STIX]{x03C0}$ -periodic domains: the mean depths and the modulus. In terms of these geometric characteristics, the variational principal can be formulated in the form of the following proposition.

The parameter $r_{0}$ is responsible for the wave depth-to-length ratio and ranges between 0 and 1. For the waves of infinite depth $r_{0}$ =0, for the solitary waves $r_{0}=1$ . Application of the functional (2.21) is especially convenient when the parameter $r_{0}$ is given, as for example, in computations by Schwartz (Reference Schwartz1974) and Cokelet (Reference Cokelet1977). Indeed, in this case, at fixed $c_{a}$ the variation $\unicode[STIX]{x1D6FF}(c_{a}\log r_{0})=0$ , and the functional $L[\unicode[STIX]{x1D702}]$ is equivalent to

So we can formulate the following proposition.

We consider Propositions 1 and 2 as a geometric interpretation of the Luke variational principle.

3 System of quadratic equations for the Stokes coefficients

Let us assume that the waves are symmetric and the axis of ordinate is that of symmetry which goes through one of the wave crests. One period of the flow is shown in figure 2. We shall seek the conformal mapping of the annulus (figure 1 c) in the parametric $\unicode[STIX]{x1D701}$ -plane onto the flow domain of one wave period $\unicode[STIX]{x1D6FA}_{z}$ in the form

The representation (3.1) is a variant of the second Stokes method, suggested by Schwartz (Reference Schwartz1974). By virtue of the wave symmetry, the Stokes coefficients $y_{n}$ ( $n=1,2,3\ldots$ ) are real. Moreover, for the representation (3.1) the condition on the bottom $\text{Im}z=0$ is automatically fulfilled. The parametric equations of the free surface are

where $\unicode[STIX]{x1D6FC}_{n}=1+r_{0}^{2n}$ , $\unicode[STIX]{x1D6FD}_{n}=1-r_{0}^{2n}$ , $\unicode[STIX]{x1D6FE}$ is the polar angle in the $\unicode[STIX]{x1D701}$ -plane.

Figure 2.

One wave period in the physical $z$ -plane.

It is to be noted that the mapping (3.1) specifies a family of $2\unicode[STIX]{x03C0}$ -periodic domains with the same modulus $M=1/r_{0}$ . Thus, we are under the conditions of Proposition 2. In appendix A to the paper, it will be proved that

where

Inserting (3.3) into (2.24) yields

It follows from (3.4) and (3.5) that $L_{M}$ is a function of the Stokes coefficients $y_{n}$ ( $n=1,2,3,\ldots$ ) and the parameter $r_{0}$ . In accordance with (2.25) of Proposition 2, we variate $L_{M}$ with respect to $y_{n}$ at fixed $r_{0}$ to get

where

In appendix A to the paper, we shall prove that

With allowance for (3.9) we deduce

and at $n=1$ the first sum in (3.10) vanishing. The system (3.10) is an infinite system of quadratic equations with respect to the Stokes coefficients $y_{n}$ .

Equation (2.23) of Proposition 1 allows us to express $c_{a}^{2}$ in terms of the Stokes coefficients. Indeed, fixing $y_{n}$ and varying only $r_{0}$ , we obtain

where

The conformal mapping of the annulus $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D701}}$ onto the rectangle $\unicode[STIX]{x1D6FA}_{w}$ in the $w$ -plane (see figure 1 b,c) is determined by the equation

If the depth of the fluid is infinite, then $r_{0}\rightarrow 0$ , $\unicode[STIX]{x1D6FD}_{n}=1$ , $\unicode[STIX]{x1D6E4}_{m,n}=m+n$ , and the system (3.10) becomes simpler:

Equation (3.11) transforms to the relation

If $r_{0}\rightarrow 0$ , then $\log r_{0}\rightarrow -\infty$ , and the formula of the conformal mapping (3.1) loses its sense. To avoid this, it is enough to transfer the $x$ -axes of the $z$ -plane onto the mean level of waves by setting $z_{1}=z-\text{i}h_{a}$ . It follows from (3.3a ) that $\log r_{0}=\unicode[STIX]{x1D6EC}/2-h_{a}$ , and then for the infinite depth,

Thus, for the fixed parameters $K=2(R+\log r_{0})$ and $r_{0}$ , any non-trivial solution to system (3.10) and formulae (3.1), (3.14) define a $2\unicode[STIX]{x03C0}$ -periodic wave domain and the parametric equations for the complex potential $w$ .

We should remark that Schwartz (Reference Schwartz1974), and later Cokelet (Reference Cokelet1977), located the $x$ -axes in the physical plane at the level $d$ above the bottom, defining $d$ as the depth of uniform stream flowing with speed $c_{a}$ whose mass flux equals that of the wave. Hence, $d=Q/c_{a}$ . Since $Q=-c_{a}\log r_{0}$ , it is clear that $d=-\log r_{0}$ . This elucidates the physical sense of the parameter $K$ in system (3.10). Indeed, by virtue of (2.6) and (3.8), on the free surface we have

This means that the parameter $K$ is simply the Bernoulli constant as it was defined and designated in Schwartz (Reference Schwartz1974) and Cokelet (Reference Cokelet1977). If in solving system (3.10) we specify another parameter instead of $K$ , for example, $y_{1}$ as in Stokes (Reference Stokes1880), then (3.10) is a pure quadratic system of equations with respect to the unknowns $K$ and  $y_{n}$ .

4 Algorithm for solving system (3.10)

Let $a$ be the amplitude of the wave (one half the vertical distance from the crest to the trough). The goal of this section is to develop an algorithm for finding expansions for the Stokes coefficient $y_{n}$ in powers of the wave amplitude $a$ with coefficients depending on the mean depth $h_{a}$ .

Because in system (3.10) the coefficients $\unicode[STIX]{x1D6FD}_{n}$ and $\unicode[STIX]{x1D6E4}_{m,n}$ depend only on $r_{0}^{2}$ , it is convenient to introduce

Now in (3.10), the coefficients $\unicode[STIX]{x1D6FD}_{n}$ , $\unicode[STIX]{x1D6E4}_{m,n}$ depend on $r$ , formula (3.11) for $c_{a}^{2}$ transforms to

and in (3.12), (3.13) we need to replace $r_{0}$ by  $r$ .

At the first step, we follow Stokes (Reference Stokes1880), namely, we assume that $y_{1}=b$ , where $b$ is a specified small parameter. Since $y_{1}$ is given, the parameter $K$ is now unknown. We shall seek the coefficients $y_{n}$ ( $n=2,3,\ldots$ ) and the parameter $K$ in the form of expansions in powers of $b$ . Let an odd number $N$ be the greatest power in these expansions, then

where $[\;\;]$ denotes the integer part of a real number, and the coefficients $K_{2n}$ and $b_{i,2n}$ are to be found. The general number of unknown coefficients is $(N+1)^{2}/4$ . The procedure of their consecutive search appears as follows.

(i) We substitute the expressions for $K$ and $y_{i}$ from (4.3) in the first $N$ equations of system (3.10) and expand each of the equations into the series in powers of $b$ up to the power $N$ inclusively. We obtain $N$ equations of the form $P_{i}(b)=0$ , $i=1,2,\ldots ,N$ , where $P_{i}(b)$ are $N$ -degree polynomials in $b$ . The coefficients of these polynomials depend on the unknowns $K_{2n}$ and $b_{i,2n}$ . Equating these coefficients to zero, we obtain a system of $(N+1)^{2}/4$ equations with respect to $(N+1)^{2}/4$ unknowns $K_{2n}$ and $b_{i,2n}$ . It is to be noted that if $N$ is chosen to be even, then the number of unknowns will be greater by one than the number of equations, and not all coefficients will be found.

(ii) We organize the external loop for $j=1,2,\ldots ,N$ , where $j$ is the power of the parameter $b$ labelling the coefficients at $b^{j}$ in the polynomials $P_{i}(b)$ . Let the function $\lceil x\rceil$ designate a so called ‘ceiling’, i.e. the least integer greater than or equal to $x$ . The unknown coefficients $K_{2n}$ and $b_{i,2n}$ are determined by portions with $\lceil j/2\rceil$ coefficients in each portion. In the external loop over $j$ , we organize the internal loop for $k=1,2,\ldots ,\lceil j/2\rceil$ . For each $k$ we find

We equate the coefficient at $b^{j}$ in the polynomial $P_{i_{k}}(b)$ to zero and from the obtained equation we find either

and in so doing all equations for finding $K_{j-1}$ or $b_{i_{k},j-i_{k}}$ being linear.

Formulae (4.4), (4.5) define exactly the consecutive procedure of determining $K_{2n}$ , $b_{i,2n}$ , and for each pair $(j,k)$ , allow us to express a new unknown coefficient in terms of the coefficients which have been already found. Note, by the way, that as follows from (4.5a ), if $N$ is even, then $K_{2N}$ remains indefinite.

The above algorithm results in the expansions (4.3) with coefficients which are rational functions of $r=r_{0}^{2}$ . Inserting these expansions into (4.2) gives

where the coefficients $c_{2n}$ are also rational functions of $r$ .

The second step is to express the parameter $b$ in terms of the amplitude $a$ and to reconstruct the expansions (4.3) and (4.6) in powers of $a$ . It follows from (3.2b ) that

Taking into account that the coefficients $y_{2n-1}$ contain only odd powers of $b$ , we shall seek the parameter  $b$ as

Substituting (4.8) into (4.7) and expanding the left-hand side of (4.7) in powers of $a$ up to the power $N$ inclusively, we obtain an $N$ -degree polynomial in odd powers of  $a$ with $(N+1)/2$ coefficients depending on the parameter $r$ and the unknowns $b_{2n-1}$ . Equating these coefficients to zero, we sequently find $b_{1},b_{3},\ldots ,b_{N}$ , which again turn out to be rational functions of $r$ . Now we reconstruct the expansions (4.3) and (4.6) in powers of $a$ by inserting in these expansions the expression (4.8) for $b$ . As a result we have

where the coefficients $K_{2n}^{\prime }$ , $c_{2n}^{\prime }$ and $b_{i,2n}^{\prime }$ are rational functions of $r=r_{0}^{2}$ .

The third and last step is to express the parameter $r=r_{0}^{2}$ in terms of the mean depth $h_{a}$ and to reconstruct the expansions (4.9) and (4.10) so that they would be in powers of $a$ with coefficients depending $h_{a}$ . Let $\unicode[STIX]{x1D700}=\exp (-2h_{a})$ . It follows from (3.3a ) and (3.4a ) that

We shall seek the dependence between $r$ and $\unicode[STIX]{x1D700}$ in the form

where $\unicode[STIX]{x1D708}_{2n}$ are unknown coefficients depending on $\unicode[STIX]{x1D700}=\exp (-2h_{a})$ .

Substituting (4.12) into (4.11) and expanding the left-hand side of (4.11) in powers of $a$ up to the power $N-1$ inclusively, we obtain an $(N-1)$ -degree polynomial in even powers of $a$ with $(N-1)/2$ non-zero coefficients depending on the parameter $\unicode[STIX]{x1D700}$ and the unknowns $\unicode[STIX]{x1D708}_{2n}$ . The coefficient at $a^{0}$ in this polynomial is zero. Equating the remainder coefficients to zero, we sequently find $\unicode[STIX]{x1D708}_{2},\unicode[STIX]{x1D708}_{4},\ldots ,\unicode[STIX]{x1D708}_{N-1}$ , which are now rational functions of $\unicode[STIX]{x1D700}$ . Reconstruction of the expansions (4.9) and (4.10) by means of (4.12) yields

where the coefficients $\unicode[STIX]{x1D705}_{2n}$ , $\unicode[STIX]{x1D70E}_{2n}$ and $a_{i,2n}$ are known rational functions of $\unicode[STIX]{x1D700}=\exp (-2h_{a})$ .

This algorithm has been realized by means of Mathematica package which allows symbolic computations and exact arithmetics to be applied (see Wolfram Reference Wolfram2003). In so doing, the routines Series, Coefficient and Solve have been used systematically. It is worth noticing that steps 1 and 2 of the algorithm correspond to the formulations by Schwartz (Reference Schwartz1974) and Cokelet (Reference Cokelet1977), when the depth $d=-\log r_{0}$ is fixed. The third step correspond to the formulations by De (Reference De1955) and Fenton (Reference Fenton1985), when instead of the parameter $d=-\log r_{0}$ , whose physical meaning is not quite clear, the mean wave depth $h_{a}$ is specified.

We should remark that the first two steps can be programmed in such computer languages as Fortran or C++, but the most time-consuming third step requires necessarily the symbolic computations because in (4.11) we need to know the coefficients $y_{n}$ as functions of $r$ and $a$ . Realizing all three steps, we have carried out the computations at $N=21$ .

5 Expansions for the wave resistance and other wave properties in terms of the geometric wave parameters: $h_{a}$ and $a$

Applying the algorithm of § 4, we obtain the expansions (4.13) and (4.14) in powers of $a$ with coefficients which are rational functions of the parameter $\unicode[STIX]{x1D700}=\exp (-2h_{a})$ . As we shall see in what follows, finding the expansion for the wave resistance requires the knowledge of the expansions for the mean potential energy $V$ and the root-mean-square velocity $c_{b}$ at the bottom in steady motion, which are defined as

It can be easily seen from the definitions of $V$ and $H_{a}^{2}$ that $V=(H_{a}^{2}-h_{a}^{2})/2$ . This relation and equations (3.3) yield

Using (3.4) and (4.14), we get the expansion for $V$ . In so doing, the coefficients $\unicode[STIX]{x1D6FD}_{n}$ and  $\unicode[STIX]{x1D6E4}_{mn}$ , which depend on $r$ , should be also expanded in powers of $a$ by means of formula (4.12).

For the parameter $c_{b}^{2}$ , it is possible to prove the identity

(see equation (3.22) in paper 1). From this identity and equations (3.3a ), (3.8) we conclude that

which allows us to find the expansion for $c_{b}^{2}$ .

Implementation of the algorithm of § 4 and the above manipulations leads to expansions for the wave parameters $c_{a}^{2}$ , $c_{b}^{2}$ , $V$ , $\unicode[STIX]{x1D6EC}$ , which are needed to get the expansion for the wave resistance. All these expansions are in powers of $a$ with coefficients which are rational functions of $\unicode[STIX]{x1D700}=\exp (-2h_{a})$ . Besides, the expansions for $c_{a}^{2}$ , $c_{b}^{2}$ , $V$ and $\unicode[STIX]{x1D6EC}$ contain only even powers of $a$ , moreover, they possess a very useful feature: the numerators of their coefficients (but not denominators) are so-called palindromic polynomials of $\unicode[STIX]{x1D700}$ . By definition, a polynomial $P_{n}(\unicode[STIX]{x1D700})$ of degree $n$ is palindromic if

This feature allows us to shorten representations of polynomials by one-half, for example,

and so on.

In appendix B to the paper, formulae (B 1)–(B 4) present the expansions for $c_{a}^{2}$ , $c_{b}^{2}$ , $\unicode[STIX]{x1D6EC}$ and $V$ up to $a^{6}\,(N=7)$ . In spite of the fact that we have carried out the computations at $N=21$ , expansions for $N\geqslant 9$ become too long to be printed. For example, the formula for $c_{a}^{2}$ at $N=21$ after conversion to LaTeX takes approximately 75 kilobytes, i.e. approximately 25 journal pages. For completeness, an expansion for the mean kinetic energy $T_{c_{a}}$ , computed under the assumption that the waves propagate with the speed $c_{a}$ in the bottom-fixed reference frame, is also presented by (B 5). The expansion for $T_{c_{a}}$ can be easily obtained from (3.3a ) and formula (5.21) from the paper by Cokelet (Reference Cokelet1977):

In paper 1, we have derived formula (3.13) for the wave resistance $R_{w}$ of a two-dimensional body that moves horizontally at a constant speed $c$ in a channel of finite depth $h$ . In dimensionless form, this formula can be written as

where $\unicode[STIX]{x1D6FF}_{1}=h_{a}-h$ is the defect of levels (the difference between the mean level far downstream and the undisturbed level far upstream). The scale for $R_{w}$ in (5.8) is $\unicode[STIX]{x1D70C}g\unicode[STIX]{x1D706}^{2}/(2\unicode[STIX]{x03C0})^{2}$ . The formula has been derived under assumptions that in the body-fixed reference frame the flow is steady and irrotational.

Assume that for the waves far downstream we know the mean depth $h_{a}$ and amplitude  $a$ . The knowledge of these parameters is enough to compute all wave properties, but to find the wave resistance by means of (5.8), we should determine the parameters $c$ and $h$ . In a sense, this is an inverse problem: in the body-fixed reference frame find the speed $c$ and depth $h$ of a uniform stream which without dissipation is able to create a given system of waves far downstream due to some disturbance located in the stream.

By virtue of the fact that the Bernoulli constant $R$ and the flow flux $Q$ far upstream and far downstream are the same, the parameters $c$ and $h$ satisfy the following system of equations

where $R$ and $Q$ are known wave properties. With the help of some results obtained in the works by Keady & Norbury (Reference Keady and Norbury1975) and Benjamin (Reference Benjamin1995), it has been demonstrated in paper 1 that system (5.9) always has two solutions, $(c_{1},h_{1})$ and $(c_{2},h_{2})$ . For the first solution, the Froude number $Fr=c_{1}/\sqrt{h_{1}}<1$ , for the second one, the Froude number $Fr=c_{2}/\sqrt{h_{2}}>1$ . So, for the first solution, the upstream flow is subcritical, for the second one, it is supercritical.

In this paper, for convenience, we rename the designations for these solutions. The first, subcritical solution, we designate simply as $(c,h)$ , and the second, supercritical one, we designate as $(c_{conj},h_{conj})$ , taking into consideration that two uniform streams whose speed $c$ and depth $h$ satisfy (5.9) with the same $R$ and $Q$ are called conjugate streams (see, e.g. Keady & Norbury Reference Keady and Norbury1975). Now, the subcritical Froude number is simply $Fr$ , and supercritical one is $Fr_{conj}$ . The sketches of the subcritical and supercritical flows are shown in figure 3(a,b). It is worthwhile to stress that for both cases the waves far downstream are the same, but the bodies which generate these waves can differ in shape, size and their location with respect to the bottom.

Figure 3.

Sketches of subcritical flows (a), supercritical flows (b) and hydraulic falls (c).

In paper 1, we have derived explicit analytical formulae (4.11), (4.12), (3.28b,c) for the solutions to system (5.9). In dimensionless form, the formulae can be rewritten as

where

6 Concluding remarks

In this paper, for steady periodic gravity waves in water on finite depth, we have derived a quadratic system of equations with respect to the coefficients of the second Stokes method and developed an effective computer algorithm for solving the system. Making use of exact arithmetics and symbolic calculations, we have obtained explicit analytical expansions in powers of the wave amplitude $a$ for the wave resistance and other wave properties to $a^{21}$ . The coefficients of these expansions depend only on the mean wave depth $h_{a}$ .

To our regret, the expansions of order higher than seven are too long to be printed, and in appendix B to the paper we have presented the results only up to $a^{7}$ . Having taken a look at these formulae, the reader can easily imagine the length of expansions of order $a^{9}$ and higher. In spite of this evident deficiency, the formulae have also an important advantage: because even their LaTeX forms for printout have been generated by computer, they cannot contain any typos or calculational errors. On the other hand, these long analytical formulae, which sometimes take several hundreds of journal pages, are a logical consequence of accurate application of the famous analytical Stokes approach.

A usual way to enhance the convergence of the high-order Stokes expansions is the use of Padé approximants. However, this approach is effective when the Stokes coefficients are numbers, decimal as in Schwartz (Reference Schwartz1974) and Cokelet (Reference Cokelet1977) or rational as in Dallaston & McCue (Reference Dallaston and McCue2010). In our case, the coefficients are rational functions of $\unicode[STIX]{x1D700}=\exp (-2h_{a})$ (at best) or a mixture of $h_{a}$ and $\unicode[STIX]{x1D700}$ (at worst). By specifying $a$ and $h_{a}$ , we are able to compute these coefficients and represent them as numbers with any quantity of decimal places, but then all analyticity of our approach will be lost and we return to the previous investigations.

We have compared our asymptotic expansions with those obtained by the fifth-order theory of Fenton (Reference Fenton1985) and found that the results are in full coincidence. Comparison also has been made with the results obtained in paper 1, where the waves resistance $R_{w}$ was computed by the accurate numerical method of the work by Maklakov (Reference Maklakov2002). The conclusion is as follows: analytical expansions to $a^{7}$ can be applied if the depth-to-wavelength ratio is not less than 0.15 and the wave amplitude is approximately 60 % of its limiting value.

All expansions used in the paper are stored in the supplementary materials and are available at https://doi.org/10.1017/jfm.2017.62 in the form of Mathematica expressions. The order of the stored expressions is 21. The materials contain the file ‘Instruction.nb’, which explains how to use the stored information.