3.1 Derivation of the asymptotic equations

We are interested in the behaviour of solutions of (3.2)–(3.5) in the limit $\unicode[STIX]{x1D716}\rightarrow 0$ . To construct an asymptotic expansion of a solution, we employ the method of multiple scales (e.g. Nayfeh Reference Nayfeh1973) and assume that the expansion has a form

where

is the slow time. On substituting these in (3.2)–(3.5) and collecting terms of the same order in $\unicode[STIX]{x1D716}$ , we obtain

at leading order and

at first order. Here $D_{0}=\{(x,y,z)\in \mathbb{R}^{3}|-\infty <x,y<\infty ,-1<z<\unicode[STIX]{x1D702}_{0}(x,y,\unicode[STIX]{x1D70F},t)\}$ .

Throughout the paper, we shall use the following fact: any bounded $2\unicode[STIX]{x03C0}$ -periodic function $g(\unicode[STIX]{x1D70F})$ can be presented in the form

where

is the averaged part of $g(\unicode[STIX]{x1D70F})$ and $\tilde{g}(\unicode[STIX]{x1D70F})=g(\unicode[STIX]{x1D70F})-\overline{g}$ is its oscillatory part (having zero mean).

Consider now the leading-order equations (3.8)–(3.11). It follows from (3.11) that $\unicode[STIX]{x1D702}_{0}$ does not depend on the fast time, i.e.

Note that this equation implies that $D_{0}$ does not depend on the fast time $\unicode[STIX]{x1D70F}$ , i.e. $D_{0}$ is the domain corresponding to the averaged (over the period in $\unicode[STIX]{x1D70F}$ ) position of the free surface.

Substituting (3.18) into (3.9) and separating the oscillatory part, we find that

The oscillatory parts of (3.8) and (3.10) yield

and

If function $\overline{\unicode[STIX]{x1D702}}_{0}(x,y,t)$ were known, we would be able to find $\tilde{\unicode[STIX]{x1D719}}_{0}$ by solving the Laplace equation (3.20) subject to the boundary conditions (3.19) and (3.21) and the periodicity or decay condition in $x$ and $y$ .

Consider now the first-order equations (3.12)–(3.15). On averaging (3.13) and (3.15), we obtain

Note that

and

So, (3.22) can be written as

where

Equation (3.26) implicitly depends on both $\tilde{\unicode[STIX]{x1D719}}_{0}$ and $\tilde{\unicode[STIX]{x1D702}}_{1}$ , which appear in (3.27). Function $\tilde{\unicode[STIX]{x1D702}}_{1}(x,y,\unicode[STIX]{x1D70F},t)$ can be found from the oscillatory part of (3.15) that can be written as

and this equation can be solved once $\tilde{\unicode[STIX]{x1D719}}_{0}$ is known.

It follows from (3.19)–(3.21) that $\tilde{\unicode[STIX]{x1D719}}_{0}(x,y,z,\unicode[STIX]{x1D70F},t)$ can be written in the form

where $\unicode[STIX]{x1D6F7}$ is a solution of the following boundary-value problem:

subject to (yet unspecified) additional conditions in variables $x$ and $y$ (periodicity or decay at infinity). Suppose that we can solve problem (3.30)–(3.32) and let $\unicode[STIX]{x1D6F7}(x,y,z,t)$ be its solution. Evidently, it will depend on $\overline{\unicode[STIX]{x1D702}}_{0}(x,y,t)$ . In what follows, we will use the following notation

It follows from (3.28)–(3.33) that

On substituting these into (3.27), we obtain

where

Parameter $\unicode[STIX]{x1D718}$ will be referred to as the vibrational parameter. In the absence of vibration, $\unicode[STIX]{x1D718}=0$ . For the harmonic vibration in the form $f(\unicode[STIX]{x1D70F})=\cos \unicode[STIX]{x1D70F}$ , $\unicode[STIX]{x1D718}=1/2$ .

Finally, averaging (3.8) and (3.10), we get

Equations (3.39), (3.40), (3.23) and (3.26) with $G$ given by (3.37) represent a closed system of equations governing slow evolution of waves on the free surface of a vibrating fluid.

3.2 Some properties of the asymptotic equations

Let

Then the averaged equations (3.23) and (3.26) can be written as

where $\unicode[STIX]{x1D713}$ satisfies

and where

and $\unicode[STIX]{x1D6F7}(x,y,z,t)$ is the solution of the problem

If there is no vibration, (3.42)–(3.44) reduce to the standard system of equations for irrotational water waves in a non-vibrating fluid. The vibration leads to the appearance of the extra term on the right-hand side of (3.42).

Equations (3.42)–(3.46) conserve the energy, given by

Here the domain of integration ${\mathcal{D}}$ is either the rectangle of periods, if periodic (in $x$ and $y$ ) solutions are considered, or the whole plane $\mathbb{R}^{2}$ . In the latter case, it is assumed that $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D701}$ decay at infinity so that the integrals in (3.47) exist.

Equations (3.42), (3.43) can be written in Hamiltonian form. If, following Zakharov (Reference Zakharov1968) (see also Miles Reference Miles1981), we introduce function

then $\unicode[STIX]{x1D713}(x,y,z,t)$ is uniquely determined by $\unicode[STIX]{x1D712}(x,y,t)$ as a solution of the problem

supplemented with appropriate boundary conditions in variables $x$ and $y$ .

It is also convenient to introduce the following notation

where $\unicode[STIX]{x1D713}(x,y,z,t)$ is a solution of problem (3.49). Note that $\hat{N}(x,y,t)$ is uniquely determined by $\unicode[STIX]{x1D712}(x,y,t)$ .

To rewrite (3.42), (3.43) in terms of $\unicode[STIX]{x1D712}$ and $\unicode[STIX]{x1D701}$ , we first observe that

Then, we use these to eliminate $\unicode[STIX]{x1D713}_{t}|_{z=\unicode[STIX]{x1D701}}$ and $\unicode[STIX]{x1D735}_{\shortparallel }\unicode[STIX]{x1D713}|_{z=\unicode[STIX]{x1D701}}$ from (3.42), (3.43). As a result, we obtain

It can be verified by direct calculation that (3.52)–(3.53) are equivalent to the canonical Hamiltonian equations

It is interesting that, in the short-wave limit ( $|\boldsymbol{k}|\gg 1$ ), the dispersion relation simplifies to

This would be the same as the dispersion relation for gravity–capillary waves on deep water, if $|\boldsymbol{k}|^{2}$ in the second term were replaced by $|\boldsymbol{k}|^{3}$ . Note also that for sufficiently large $|\boldsymbol{k}|$ , the first term in the above formula is small in comparison with the second one. If it is discarded, we obtain non-dispersive waves.

4.1 Small-amplitude vibrations

The long-wave model discussed in this subsection can be obtained directly from (2.4)–(2.7). However, it is much easier to derive asymptotic equations from the averaged equations of § 3, and this is what we shall do now.

One of the standard recipes to derive the shallow-water equations is to re-scale the independent variables $x$ , $y$ , $t$ and parameter $\unicode[STIX]{x1D6FE}_{0}$ as follows. Let

where $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D707}^{2}$ (and $\unicode[STIX]{x1D707}$ is the long-wave parameter defined in § 2), $\unicode[STIX]{x1D6FF}$ is assumed to be small ( $\unicode[STIX]{x1D6FF}\ll 1$ ) and $\unicode[STIX]{x1D6E4}=O(1)$ as $\unicode[STIX]{x1D6FF}\rightarrow 0$ .

In terms of the new variables, equations (3.42)–(3.46) become

In (4.2)–(4.4), $\unicode[STIX]{x1D735}_{\shortparallel }=(\unicode[STIX]{x2202}_{X},\unicode[STIX]{x2202}_{Y})$ and

and $\unicode[STIX]{x1D6F7}(X,Y,z,T)$ is the solution of the problem

Now we assume that

On substituting these in (4.4) and collecting terms of the same order in $\unicode[STIX]{x1D6FF}$ , we obtain

From (4.8), we deduce that $\unicode[STIX]{x1D713}_{0}$ does not depend on $z$ :

In what follows, we only need $\unicode[STIX]{x1D713}_{1z}|_{z=\unicode[STIX]{x1D701}_{0}}$ . So, we integrate (4.9) in $z$ and apply the boundary condition at $z=-1$ . As a result, we get $\unicode[STIX]{x1D713}_{1z}=-(1+z)\unicode[STIX]{x1D6FB}_{\shortparallel }^{2}\unicode[STIX]{x1D713}_{0}$ , so that

Similarly, equation (4.6) yields

Hence,

Finally, we substitute (4.7) in (4.2) and (4.3) and employ (4.10)–(4.13). At leading order, we obtain

These are the shallow-water equations we were looking for. Note that if $\unicode[STIX]{x1D718}=0$ (i.e. there is no vibration), then these equations reduce to the standard shallow-water model.

4.2 Vibrations of finite amplitude

Now we drop our earlier assumption that the amplitude of vibrations is small (compared with the fluid depth). Namely, we assume that (cf. (4.2))

where $\unicode[STIX]{x1D6FE}_{0}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$ . In other words, our assumptions are: (i) the typical wavelength is much larger that the fluid depth, (ii) the amplitude of vibrations is of the same order of magnitude as the fluid depth, (iii) the amplitude of the waves is of the same order of magnitude as the fluid depth and (iv) the frequency of vibrations is high enough for the acceleration due to vibrations to be much higher than the gravitational acceleration (but not as high as in § 4.1, cf. (4.16)). Note that the model discussed below cannot be derived from the general model of § 3, because the latter works only for small-amplitude vibrations. So, we need to start with (2.4)–(2.7).

Substituting (4.19) into (2.4)–(2.7), we obtain

The asymptotic expansion for $\unicode[STIX]{x1D716}\ll 1$ is constructed in appendix A using the same procedure as in § 3. Here we only describe the results. The expansion has the form

where $\tilde{\unicode[STIX]{x1D719}}_{0}=-f^{\prime }(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D701}$ and functions $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D701}$ satisfy the following closed system of evolution equations:

In (4.24)–(4.27), $\unicode[STIX]{x1D718}$ is the vibrational parameter, defined by (3.38), and

is the slow time. Note that this slow time is different from the slow time employed in § 4.1, but the same as the slow time of § 3.

Evidently, if we replace $\unicode[STIX]{x1D6FE}_{0}$ in (4.26) with $\unicode[STIX]{x1D6E4}$ (defined by (4.16)), then (4.26) and (4.27) become exactly the same as (4.14) and (4.15). It is quite remarkable that the same asymptotic equations describe two physically different situations (small-amplitude vibrations and vibrations of finite amplitude).

5.1 Solitary waves

Solitary travelling waves correspond to finite, but non-periodic motions of the particle. In view of figure 2, these requirements are satisfied if the particle’s energy, $\hat{E}={R^{\prime }}^{2}/2+\hat{V}(R)$ , is equal to the value of $\hat{V}$ at the local maximum, i.e. $\hat{E}=\hat{V}(R_{2})$ . Thus, for each value of parameter $q\in (0,2)$ , we obtain a solitary wave solution of (5.2) and (5.3). In what follows, we restrict our attention to solitary waves satisfying the conditions

which, in terms of $H(s)$ and $U(s)$ , have the form

When $\unicode[STIX]{x1D70E}\rightarrow \pm \infty$ , the particle approaches the point $R=R_{2}$ . This fact and (5.20a,b ) imply that

It also follows from conditions (5.20a,b ) and (5.9) and (5.10) that

These and (5.17) give us the following relationship between parameter $q$ and the wave speed ${\hat{c}}$ :

Note that (5.21) and (5.23) are not independent: if (5.23) holds, so does (5.21), or vice versa. The requirement that $0<q<2$ and (5.23) imply that $0<{\hat{c}}<1$ , which means solitary wave can propagate only with speed less than the phase speed of small-amplitude gravity waves, $\sqrt{\unicode[STIX]{x1D6FE}_{0}}$ .

Equation (5.15) was solved numerically for various values of $q$ using the MATLAB built-in ordinary differential equation (ODE) solvers. Then, equation (5.14) was employed to compute $H(s)$ and the wave amplitude, $A$ , defined as

Typical shapes of the solitary waves are presented in figure 3. Figure 3 shows that the only type of possible solitary waves are depression waves. It is known that the depression capillary–gravity solitary waves are possible (see Korteweg & de Vries Reference Korteweg and de Vries1895; Benjamin Reference Benjamin1982; Vanden-Broeck & Shen Reference Vanden-Broeck and Shen1983) and have been observed experimentally in fluids with sufficiently strong surface tension (Falcon, Laroche & Fauve Reference Falcon, Laroche and Fauve2002). This gives us one more argument in favour of our earlier conclusion that the effect of vibrations is similar to that of surface tension.

The wave speed, ${\hat{c}}=c/\sqrt{\unicode[STIX]{x1D6FE}_{0}}$ , versus the amplitude, $A$ , is shown in figure 4 as a dashed curve. It approaches $1$ when the amplitude decreases to zero and $0$ when the amplitude approaches $1$ . The solid curves in figure 4 correspond to periodic travelling waves discussed below.

Figure 3.

Typical profiles of solitary waves.

Figure 4.

Wave speed ${\hat{c}}=c/\sqrt{\unicode[STIX]{x1D6FE}_{0}}$ versus amplitude $A$ . Solid curves show the wave speed of periodic waves for several values of parameter $q$ . The dashed curve corresponds to solitary waves. Circles represent the limit points of the wave speed of periodic waves when the wavelength goes to infinity.

5.2 Periodic waves

It follows from figure 2 that the motion of the particle is periodic if the particle’s energy, $\hat{E}={R^{\prime }}^{2}/2+\hat{V}(R)$ , satisfies the inequality

Periodic motions of the particle correspond to periodic travelling waves solutions of the original equations (5.2) and (5.3). For each $q\in (0,2)$ , there is a family of periodic solutions corresponding to values of the particle’s energy, $\hat{E}$ , satisfying inequality (5.25). In what follows, we restrict our attention to waves for which the mass flux (equivalently, the momentum density), $M=HU$ , averaged over the period of the wave, is zero, i.e.

Here $S$ is the period (wavelength) of the wave. Also, to be consistent with our non-dimensionalisation, we require that the averaged (over the period) fluid depth is equal to $1$ , i.e.

On averaging (5.10) over the period and taking into account (5.26) and (5.27), we find the relationship between $B$ and $c$ :

Since the particle’s energy is a constant of motion ( $\text{d}\hat{E}/\text{d}s=0$ ), we have

Hence, the period of motion of the particle is given by

where $R_{max}$ and $R_{min}$ are the maximum and minimum values of $R$ corresponding to periodic motion of the particle with energy $\hat{E}$ (in other words, $R_{max}$ and $R_{min}$ are solutions of the equation $\hat{V}(R)=\hat{E}$ ). In view of (5.14), the period of function $H(s)$ (the wavelength) is then given by

Further, on rewriting the integral in (5.27) in terms of $R$ and $\unicode[STIX]{x1D70E}$ , we obtain

Now all properties of the periodic solutions can be found as follows. First, we fix $q\in (0,2)$ . Then for each $\hat{E}$ satisfying (5.25), we compute ${\hat{C}}$ , $S$ (using (5.32) and (5.31), respectively) and ${\hat{c}}$ (using (5.17) and (5.28)), as well as the wave amplitude, $A$ , defined by

As a result, we get the period, $S$ , and the wave speed, ${\hat{c}}$ , as functions of the amplitude,  $A$ .

Figure 5.

Period $S$ versus amplitude $A$ for several values of parameter $q$ . Dashed vertical lines represent the asymptotes of the curves $S(A)$ . These asymptotes correspond to solitary waves.

The period versus the amplitude for several values of parameter $q$ are shown in figure 5. The figure shows that for each value of $q$ , $S(A)$ increases with $A$ and goes to infinity as $A$ approaches a certain critical value, which corresponds to the amplitude of the solitary wave for the same value of $q$ . The dashed vertical asymptotes of the curves $S(A)$ , shown in figure 5, represent the amplitudes of the corresponding solitary waves. The wave speed as a function of the amplitude for several values of $q$ is shown in figure 4: solid curves. For each $q$ , ${\hat{c}}$ decreases as $A$ varies from $0$ up to the critical amplitude (corresponding to a solitary wave). The end points of the curves, ${\hat{c}}(A)$ are indicated by circles in figure 4. Examples of the wave shapes for various values of $q$ and $A$ are shown in figures 6 and 7.

Figure 6.

Periodic waves for $q=0.25$ and several values of the amplitude $A=\max |H(s)-1|$ ; ${\hat{c}}=c/\sqrt{\unicode[STIX]{x1D6FE}_{0}}$ where $c$ is the wave speed.

Figure 7.

Periodic waves of the same amplitude $A=0.5$ for several values of parameter $q$ .

5.3 Very strong vibration ( $\unicode[STIX]{x1D6FE}_{0}=0$ )

In the case of $\unicode[STIX]{x1D6FE}_{0}=0$ , equation (5.12) can be rewritten as

where (cf. (5.13))

In terms of new variables $R$ and $\unicode[STIX]{x1D70E}$ , defined as (cf. (5.14))

Equation (5.34) reduces to the equation of motion of a particle in a potential:

Note that the potential $\hat{V}(R)$ does not contain free parameters. It has a global minimum at $R=1$ , so that only periodic motion of the particle is possible. As before, we restrict our attention to waves for which the mass flux, $M=HU$ , averaged over the period of the wave, is zero ( $\langle M\rangle =0$ ) and require that the mean depth is equal to $1$ ( $\langle H\rangle =1$ ). Let

where $\hat{E}={R^{\prime }}^{2}/2+\hat{V}(R)$ is the particle’s energy. Periodic solutions are possible for $9/2=\hat{V}(1)<\hat{E}<\infty$ . This implies that parameter $\unicode[STIX]{x1D706}$ , defined by (5.38), can vary from $0$ to $\infty$ . The equation of motion (5.37) can be integrated analytically, albeit in implicit form. (An integral, which is similar to (5.30), is computed using the change of the variable of integration: $R^{2/3}=\unicode[STIX]{x1D706}\sin \unicode[STIX]{x1D703}+\sqrt{1+\unicode[STIX]{x1D706}^{2}}$ .) Here we present only final results. It can be shown that the period, $S$ , and the amplitude, $A$ , can be expressed in terms of $\unicode[STIX]{x1D706}$ as

The wave amplitude does not depend on the wave speed. It goes to $0$ as $\unicode[STIX]{x1D706}\rightarrow 0$ and to $1$ when $\unicode[STIX]{x1D706}\rightarrow \infty$ . The period (wavelength) is inversely proportional to the wave speed and tends to a non-zero limit, depending on $c^{\ast }$ , as $\unicode[STIX]{x1D706}\rightarrow 0$ and goes to $0$ when $\unicode[STIX]{x1D706}\rightarrow \infty$ . For each $c^{\ast }$ , equation (5.39) defines a curve on the $(A,S)$ -plane. These curves for several values of the wave speed are shown in figure 8.

Figure 8.

Periodic travelling waves for $\unicode[STIX]{x1D6FE}_{0}=0$ : the wavelength, $S$ , versus the amplitude, $A$ , for several values of the wave speed.