2. The Mathieu equation for surface waves

We consider a container partly filled with a Newtonian fluid, moving up and down in a purely sinusoidal motion of angular frequency ${\it\Omega}$ and amplitude $\mathscr{A}$ , so that the forcing acceleration is ${\it\Omega}^{2}\mathscr{A}\cos ({\it\Omega}t)$ . In the (non-Galilean) reference frame moving with the vessel, the fluid experiences a vertical acceleration due to the apparent gravity $G(t)\equiv g-{\it\Omega}^{2}\mathscr{A}\cos ({\it\Omega}t)$ , $g$ being the acceleration due to gravity in the laboratory (Galilean) frame of reference and $t$ being the time.

Let $\boldsymbol{x}=(x_{1},x_{2})$ and $y$ be respectively the horizontal and upward vertical Cartesian coordinates moving with the vessel. The ordinates $y=-d$ , $y=0$ and $y={\it\eta}(\boldsymbol{x},t)$ correspond respectively to the horizontal impermeable bottom, the liquid level at rest and the impermeable free surface. The Fourier transform of the latter is ${\it\zeta}(\boldsymbol{k},t)\equiv \iint _{-\infty }^{\infty }{\it\eta}(\boldsymbol{x},t)\exp (-\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x})\,\text{d}^{2}\boldsymbol{x}$ , where $\text{i}^{2}=-1$ and $\boldsymbol{k}$ is the wavevector, with $k=|\boldsymbol{k}|$ .

For parametrically driven infinitesimal surface waves, ${\it\zeta}$ is described by a damped Mathieu equation (Benjamin & Ursell Reference Benjamin and Ursell1954; Ciliberto & Gollub Reference Ciliberto and Gollub1985),

(2.1) $$\begin{eqnarray}{\it\zeta}_{tt}+2{\it\sigma}{\it\zeta}_{t}+{\it\omega}_{0}^{2}[1-F\cos ({\it\Omega}t)]{\it\zeta}=0,\end{eqnarray}$$

where ${\it\sigma}={\it\sigma}(k)$ is the viscous attenuation, ${\it\omega}_{0}={\it\omega}_{0}(k)$ is the angular frequency of linear waves without damping and without forcing, and $F=F(k)$ is a dimensionless forcing. For pure gravity waves in finite depth, we have

(2.2a,b ) $$\begin{eqnarray}{\it\omega}_{0}^{2}=gk\tanh (kd),\quad F=g^{-1}{\it\Omega}^{2}\mathscr{A},\end{eqnarray}$$

while for capillary–gravity waves of surface tension $T$

(2.3a,b ) $$\begin{eqnarray}{\it\omega}_{0}^{2}=(gk+{\it\rho}^{-1}Tk^{3})\tanh (kd),\quad F={\it\rho}{\it\Omega}^{2}\mathscr{A}({\it\rho}g+Tk^{2})^{-1}.\end{eqnarray}$$

In (2.1), the damping coefficient ${\it\sigma}$ originates from the bulk viscous dissipation and the viscous friction with the bottom in the case of shallow water. For free gravity waves in the limit of small viscosity, we have (Hough Reference Hough1896; Hunt Reference Hunt1964)

(2.4) $$\begin{eqnarray}{\it\sigma}={\it\nu}k^{2}\left[2+\frac{\coth (2kd)}{\sinh (2kd)}\right]+\sqrt{\frac{k{\it\nu}\sqrt{gd}}{8d^{2}}}\frac{2kd}{\sinh (2kd)},\end{eqnarray}$$

where ${\it\nu}$ is the fluid kinematic viscosity. The first term on the right-hand side of (2.4) represents the bulk dissipation, while the second one models the friction with the bottom. For infinite depth, the dissipation ${\it\sigma}$ reduces to the bulk term $2{\it\nu}k^{2}$ , while the bottom wall term becomes prominent in the case of shallow water. More sophisticated dissipation terms could of course be considered, but the simple model (2.4) is sufficient here.

It should be noted that the damped Mathieu equation (2.1) holds only for infinitesimal waves. For steep waves, nonlinear effects play a crucial role, but the exact treatment is far beyond existing mathematical methods. Nonetheless, as shown below, the simple model (2.1) allows some fundamental physical properties of Faraday waves to be recognised.

3. Periodic solutions of the damped Mathieu equation

It is well known that systems obeying a Mathieu equation with excitation angular frequency ${\it\Omega}$ exhibit a series of resonance conditions for response angular frequencies ${\it\omega}$ equal to $n{\it\Omega}/2$ , $n$ being an integer (Abramowitz & Stegun Reference Abramowitz and Stegun1965).

Indeed, on introducing the change of independent variable $t\mapsto {\it\tau}\equiv {\it\Omega}t/2$ and of dependent variable ${\it\zeta}\mapsto {\it\xi}\equiv {\it\zeta}(\boldsymbol{k},t)\exp ({\it\sigma}t)$ , (2.1) becomes the undamped Mathieu equation

(3.1) $$\begin{eqnarray}{\it\xi}_{{\it\tau}{\it\tau}}+[p-2q\cos (2{\it\tau})]{\it\xi}=0,\quad p\equiv 4({\it\omega}_{0}^{2}-{\it\sigma}^{2}){\it\Omega}^{-2},q\equiv 2F{\it\omega}_{0}^{2}{\it\Omega}^{-2}.\end{eqnarray}$$

According to the Floquet theorem, (3.1) admits solutions of the form

(3.2) $$\begin{eqnarray}{\it\xi}({\it\tau})=\exp (-\text{i}{\it\mu}{\it\tau})P({\it\tau};p,q),\end{eqnarray}$$

where $P$ is a ${\rm\pi}$ -periodic function and the (generally complex) parameter ${\it\mu}={\it\mu}(p,q)$ is the so-called Floquet exponent which depends on the parameters $p$ and $q$ , themselves depending on the wavevector $\boldsymbol{k}$ via ${\it\omega}_{0}$ , $F$ and  ${\it\sigma}$ .

Since we are interested in periodic solutions of (2.1), they correspond to aperiodic solutions of (3.1). From (3.2) and the relation ${\it\zeta}={\it\xi}\exp (-2{\it\sigma}{\it\tau}/{\it\Omega})$ , these solutions are obviously such that $\text{Re}({\it\mu})=2{\it\omega}/{\it\Omega}=n$ ( $n$ an integer) and $\text{Im}({\it\mu})=2{\it\sigma}/{\it\Omega}$ , i.e.

(3.3) $$\begin{eqnarray}{\it\mu}(p,q)=n+2\text{i}{\it\sigma}/{\it\Omega}.\end{eqnarray}$$

Equation (3.3) is transcendent and cannot be expressed in a simpler form, in general. This is the (implicit) dispersion relation relating the wavenumber $k=|\boldsymbol{k}|$ and the angular frequency ${\it\omega}=n{\it\Omega}/2$ via ${\it\omega}_{0}(k)$ , $F(k)$ and ${\it\sigma}(k)$ . A key point here is to recognise that the wave angular frequency is ${\it\omega}=n{\it\Omega}/2$ and not ${\it\omega}_{0}$ , as is so often assumed in the literature. Actually, ${\it\omega}_{0}$ is the angular frequency only for unforced undamped waves, i.e. only if $F={\it\sigma}=0$ . Therefore, for Faraday waves, taking the equation ${\it\omega}_{0}(k)=n{\it\Omega}/2$ for the dispersion relation has led, in the past, to miscalculations of the wavenumber and to incorrect physical interpretations.

The parameters ${\it\Omega}$ , $n$ , $F$ and ${\it\sigma}$ being given, the dispersion relation (3.3) generally admits up to two solutions for ${\it\omega}_{0}$ . This means that several (two or more depending on how ${\it\omega}_{0}$ , $F$ and ${\it\sigma}$ depend on  $k$ ) wavenumbers are solutions of the dispersion relation for each response frequency $n{\it\Omega}/2$ . As the dispersion relation (3.3) involves higher transcendent functions, this multivaluation can be seen only via intensive numerical computations. However, as shown below, the exact dispersion relation (3.3) can be approximated by tractable closed-form expressions in the limit of weak forcing and dissipation.

4. Approximate dispersion relation

For weak forcing and damping, the exact dispersion relation (3.3) can be approximated by simple closed-form expressions using a standard perturbation scheme (see appendix A for details). Assuming $F\ll 1$ and ${\it\sigma}\sim O(F)$ , an approximate dispersion for the subharmonic response ( $n=1$ ) is

(4.1) $$\begin{eqnarray}{\it\omega}_{0}/{\it\omega}\approx 1\pm \sqrt{(F/4)^{2}-({\it\sigma}/{\it\omega})^{2}},\quad {\it\omega}={\it\Omega}/2,\end{eqnarray}$$

where ${\it\omega}_{0}$ is related to $k$ via (2.2) or (2.3). One condition to obtain stationary waves is that ${\it\omega}$ is real, thus defining a threshold – i.e.  $F>F_{\downarrow }$ with $F_{\downarrow }\equiv 4{\it\sigma}/{\it\omega}$ – for the forcing in order to obtain time-periodic waves. Interestingly, we note that there are two wavenumbers $k$ corresponding to the same wave angular frequency ${\it\omega}$ (for ${\it\Omega}$ , $F$ and ${\it\sigma}$ given), whatever the relation ${\it\omega}_{0}={\it\omega}_{0}(k)$ .

Assuming now $F\ll 1$ and ${\it\sigma}\sim O(F^{2})$ , an approximate dispersion for the harmonic response ( $n=2$ ) is

(4.2) $$\begin{eqnarray}{\it\omega}_{0}/{\it\omega}\approx 1+F^{2}/12\pm \sqrt{F^{4}/64-({\it\sigma}/{\it\omega})^{2}},\quad {\it\omega}={\it\Omega}.\end{eqnarray}$$

The condition of reality for ${\it\omega}$ defines the threshold $F^{2}\geqslant 8{\it\sigma}/{\it\omega}$ to obtain harmonic surface waves. Similarly, analogue approximations for all $n$ can be easily derived. All of these approximations show that there is more than one ${\it\omega}_{0}$ (and therefore more than one  $k$ ) for each response frequency ${\it\omega}=n{\it\Omega}/2$ .

Despite a limited range of validity, these approximate relations clearly demonstrate that two wavenumbers (i.e. two ${\it\omega}_{0}\equiv {\it\omega}_{0}^{\pm }$ ) correspond to the angular frequency ${\it\omega}=n{\it\Omega}/2$ . Equations (4.1) and (4.2) result from a linear model, hence their validity is restricted to waves of infinitesimal amplitude. However, nonlinearities play a significant role for waves of finite amplitude, so we look now at the nonlinear effects in an amplitude equation.

5. Weakly nonlinear model

Seeking an approximation in the form ${\it\eta}(\boldsymbol{x},t)=\text{Re}\{A(t)\}\cos (\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x})+O(A^{2})$ , assuming $|kA|\ll 1$ together with a weak forcing and dissipation – i.e.  $F\sim O(|A|^{2})$ and ${\it\sigma}\sim O(|A|^{2})$ – an equation for the slowly modulated amplitude $A$ can be derived in the form (Meron Reference Meron1987; Milner Reference Milner1991; Zhang & Viñals Reference Zhang and Viñals1997)

(5.1) $$\begin{eqnarray}\frac{\text{d}A}{\text{d}t}+({\it\sigma}-\text{i}{\it\omega}_{0})A+\frac{\text{i}F{\it\Omega}}{8}\text{e}^{\text{i}{\it\Omega}t}A^{\ast }+\frac{\text{i}K{\it\Omega}k^{2}}{2}|A|^{2}A=0,\end{eqnarray}$$

a star denoting the complex conjugate. It is obvious that the sign of the nonlinear term in (5.1), via the sign of  $K$ , plays a key role in the stability of the solutions. For pure gravity waves in finite depth, we have (Tadjbakhsh & Keller Reference Tadjbakhsh and Keller1960)

(5.2) $$\begin{eqnarray}K=\frac{2-6s-9s^{2}-5s^{3}}{16(1+s)(1-s)^{2}},\quad s\equiv \text{sech}(2kd).\end{eqnarray}$$

It is noteworthy that $K$ changes sign with the depth: $K>0$ for short waves ( $K\approx 1/8$ if $kd\gg 1$ ), $K<0$ for long waves ( $K\approx -9/64(kd)^{4}$ if $kd\ll 1$ ) and $K=0$ for $kd\approx 1.058$ (with $\tanh (1.058)\approx 0.785$ ).

Introducing $B\equiv A\exp ((\text{i}/4){\rm\pi}-(\text{i}/2){\it\Omega}t)$ , (5.1) is recast into the autonomous equation

(5.3) $$\begin{eqnarray}\frac{\text{d}B}{\text{d}t}=\left(\text{i}{\it\omega}_{0}+\frac{{\it\Omega}}{2\text{i}}-{\it\sigma}\right)B+\frac{F{\it\Omega}}{8}B^{\ast }+\frac{K{\it\Omega}k^{2}}{2\text{i}}|B|^{2}B,\end{eqnarray}$$

which is a more convenient form for the subsequent analysis.

6. Stationary weakly nonlinear solutions

We focus now on two solutions of (5.3) that are of special interest here: the rest solution $B=0$ and the standing wave of constant amplitude. The first one is trivial and we investigate its stability below. The second one is obtained by seeking solutions of the form $B=a\exp ((\text{i}/4){\rm\pi}-\text{i}{\it\delta})$ , $a$ and ${\it\delta}$ being constant. Equation (5.3) yields thus

(6.1a,b ) $$\begin{eqnarray}\frac{{\it\omega}_{0}}{{\it\omega}}=1+K(ka)^{2}\pm \sqrt{\frac{F^{2}}{16}-\frac{{\it\sigma}^{2}}{{\it\omega}^{2}}},\quad \sin (2{\it\delta})=\frac{4{\it\sigma}}{{\it\omega}F},\end{eqnarray}$$

with ${\it\omega}={\it\Omega}/2$ . As $a\rightarrow 0$ , the approximate dispersion relation (4.1) is recovered. As before, we denote ${\it\omega}_{0}^{\pm }$ the two solutions of (6.1a ). If $F={\it\sigma}=0$ , the dispersion relation of weakly nonlinear unforced standing waves in finite depth is recognised too. Therefore, compared with free nonlinear waves, the dispersion relation of parametrically forced waves is characterised by the shift in angular frequency ${\rm\Delta}{\it\omega}=\pm \sqrt{(F{\it\omega}/4)^{2}-{\it\sigma}^{2}}$ . It should be noted that this shift is independent of the wave amplitude  $a$ .

In the subsequent discussion, we consider that the parameters $F$ , ${\it\sigma}$ , ${\it\omega}={\it\Omega}/2$ and $K$ are fixed. We also limit our study to the case $K>0$ (for $K<0$ the analysis is similar, replacing ${\it\omega}_{0}-{\it\omega}$ by ${\it\omega}-{\it\omega}_{0}$ ). According to (6.1a ), we have

(6.2) $$\begin{eqnarray}K(ka)^{2}={\it\omega}_{0}/{\it\omega}-1\mp \sqrt{(F/4)^{2}-({\it\sigma}/{\it\omega})^{2}},\end{eqnarray}$$

with the constraint that $K(ka)^{2}$ must be real and positive. As the last term on the right-hand side of (6.2) is real, the forcing $F$ must exceed a minimum value $F_{\downarrow }=4{\it\sigma}/{\it\omega}$ to generate at least one stationary non-zero-amplitude wave, as already mentioned in the previous section. The condition $F>F_{\downarrow }$ being fulfilled, we have moreover

1. (i) if ${\it\omega}_{0}-{\it\omega}>\sqrt{({\it\omega}F/4)^{2}-{\it\sigma}^{2}}$ , or equivalently

   (6.3) $$\begin{eqnarray}F<F_{\uparrow }\quad \text{with}~F_{\uparrow }\equiv 4{\it\omega}^{-1}\sqrt{({\it\omega}_{0}-{\it\omega})^{2}+{\it\sigma}^{2}},\end{eqnarray}$$
   there are two stationary solutions of non-zero amplitude of the dispersion relation (6.1a ) (in addition to the solutions with the opposite phase and to the rest solution $B=0$ );

2. (ii) if ${\it\omega}_{0}-{\it\omega}<-\sqrt{({\it\omega}F/4)^{2}-{\it\sigma}^{2}}$ there are no solutions of the dispersion relation;

3. (iii) if $-\sqrt{({\it\omega}F/4)^{2}-{\it\sigma}^{2}}<{\it\omega}_{0}-{\it\omega}<\sqrt{({\it\omega}F/4)^{2}-{\it\sigma}^{2}}$ (i.e.  $F>F_{\uparrow }$ ), there is only one solution of (6.1a ) (the one with the minus sign).

An important question to address now is whether or not these stationary solutions are stable.

7. Stability analysis

In order to perform a stability analysis of the stationary solutions of the amplitude equation (5.3), we introduce a small perturbation to the solutions, and we look for the eigenvalues of the linearised system of the dynamical equations obeyed by the perturbation. The stability analysis that we conduct below resembles that carried out by Fauve (Reference Fauve, Godr\`eche and Manneville1998) for the parametric pendulum. Nonetheless, the relevance of this comparison is limited. Indeed, a major difference is that the eigenfrequency of a freely oscillating pendulum is unique, whereas free unforced water waves exhibit a continuous spectrum of mode frequencies. Moreover, and contrary to the case of the pendulum equation where the nonlinear terms merely proceed from the series expansion of the sine function, the sign of the nonlinear terms in the water wave equation depends on the depth.

7.1. Bifurcation from rest

First, we study the bifurcation from rest (i.e., the stability of the trivial solution $B=0$ ). The linearised equation (5.3) has two eigenvalues ${\it\lambda}_{1}$ and ${\it\lambda}_{2}$ such that

(7.1) $$\begin{eqnarray}{\it\lambda}_{j}=-{\it\sigma}+(-1)^{j}\sqrt{(F{\it\omega}/4)^{2}-({\it\omega}-{\it\omega}_{0})^{2}}.\end{eqnarray}$$

If $(F{\it\omega}/4)^{2}<({\it\omega}-{\it\omega}_{0})^{2}+{\it\sigma}^{2}$ , the real parts of both eigenvalues are negative. Therefore, the rest state is stable. If $(F{\it\omega}/4)^{2}>({\it\omega}-{\it\omega}_{0})^{2}+{\it\sigma}^{2}$ , the eigenvalue ${\it\lambda}_{2}$ is real and positive. Therefore, the rest state is unstable and

(7.2) $$\begin{eqnarray}F_{\uparrow }=4\sqrt{(1-{\it\omega}_{0}/{\it\omega})^{2}+({\it\sigma}/{\it\omega})^{2}}\end{eqnarray}$$

corresponds to the minimal forcing necessary to destabilise the rest state and to generate surface waves.

7.2. Stability of stationary solutions

Second, we analyse the stability of the permanent solutions of finite amplitude $a>0$ of the amplitude equation (5.3). We consider, for simplicity, small perturbations in the form $B=[a+b(t)]\exp \text{i}({\rm\pi}/4-{\it\delta})$ , $a$ , ${\it\delta}$ and ${\it\omega}_{0}$ being given in (6.1), and $b$ being a complex amplitude to be determined such that $|b|\ll a$ . To the linear approximation, the eigenvalues of the resulting equation are (with $j=1,2$ )

(7.3) $$\begin{eqnarray}{\it\lambda}_{j}=-{\it\sigma}+(-1)^{j}\sqrt{{\it\sigma}^{2}-K(2{\it\omega}ka)^{2}[1-{\it\omega}_{0}^{\pm }/{\it\omega}+K(ka)^{2}]}.\end{eqnarray}$$

The criterion for having both eigenvalues real and negative is, obviously, $1-{\it\omega}_{0}^{\pm }/{\it\omega}+K(ka)^{2}>0$ . This inequality is to be coupled with (6.2). Thus, it appears clearly that, for the case $K>0$ , the two eigenvalues are both negative if ${\it\omega}_{0}={\it\omega}_{0}^{-}$ . The corresponding stationary solution is therefore stable. The other stationary solution ${\it\omega}_{0}={\it\omega}_{0}^{+}$ , existing in the range $F_{\downarrow }<F<F_{\uparrow }$ , namely

(7.4) $$\begin{eqnarray}\frac{{\it\omega}_{0}^{+}}{{\it\omega}}=1+K(ka)^{2}+\sqrt{\frac{F^{2}}{16}-\frac{{\it\sigma}^{2}}{{\it\omega}^{2}}},\end{eqnarray}$$

corresponds to ${\it\lambda}_{1}<0$ and ${\it\lambda}_{2}>0$ and is therefore unstable.

It should be noted that the neutrally stable limiting case ${\it\lambda}_{2}=0$ is obtained for $F=F_{\downarrow }$ , $ka=0$ or $K=0$ . The two first cases correspond to the rest state (i.e. no waves), while the third one requires a higher-order equation to conclude on the stability. It should be noted also that the opposite conclusions hold for $K<0$ : the stable solution corresponds then to ${\it\omega}_{0}={\it\omega}_{0}^{+}$ (i.e.  ${\it\omega}_{0}>{\it\omega}$ ).

7.3. Wavenumber selection and nature of the transition from rest

We are now in a position to determine the wavenumbers selected at the instability onset. The minimal forcing required to destabilise the free surface from rest is given by (7.2), where ${\it\omega}_{0}$ is related to the wavenumber $k$ by (2.2), the dissipation factor ${\it\sigma}$ being given by (2.4). The first wave to emerge from rest is the one requiring the smaller value of $F_{\uparrow }$ , i.e. this wave corresponds to the wavenumber such that $\partial F_{\uparrow }/\partial k=0$ , i.e.

(7.5) $$\begin{eqnarray}\frac{\partial F_{\uparrow }}{\partial k}=\frac{16}{{\it\omega}^{2}F_{\uparrow }}\left[({\it\omega}_{0}-{\it\omega})\frac{\partial {\it\omega}_{0}}{\partial k}+{\it\sigma}\frac{\partial {\it\sigma}}{\partial k}\right]=0,\end{eqnarray}$$

together with ${\it\omega}={\it\Omega}/2$ .

In the limiting case of deep water (i.e.  $d=\infty$ , ${\it\omega}_{0}=\sqrt{gk}$ , ${\it\sigma}=2{\it\nu}k^{2}$ ), the most unstable wavenumber $k$ given by (7.5) is, after some elementary algebra, defined by the equation

(7.6) $$\begin{eqnarray}2{\it\omega}_{0}={\it\omega}+\sqrt{{\it\omega}^{2}-16{\it\sigma}^{2}}.\end{eqnarray}$$

In the opposite limit of shallow water (i.e.  $kd\ll 1$ , ${\it\omega}_{0}=\sqrt{gd}k$ , ${\it\sigma}=(gd)^{1/4}\sqrt{k{\it\nu}/8d^{2}}$ ), the most unstable wavenumber corresponds to

(7.7) $$\begin{eqnarray}{\it\omega}_{0}={\it\omega}-16{\it\nu}/d^{2}.\end{eqnarray}$$

In both cases, the first mode emerging from rest is such that ${\it\omega}_{0}<{\it\omega}$ (since ${\it\nu}>0$ ). The same conclusion arises for arbitrary depth and with surface tension under quite general assumptions. Indeed, if ${\it\omega}_{0}$ and ${\it\sigma}$ are both increasing functions of $k$ (as is the case with gravity–capillary surface waves), their derivatives with respect to $k$ are both positive. As the definition (7.2) of $F_{\uparrow }$ and the condition $\partial F_{\uparrow }/\partial k=0$ yield

(7.8) $$\begin{eqnarray}{\it\omega}_{0}={\it\omega}-{\it\sigma}\frac{\partial {\it\sigma}}{\partial k}\left(\frac{\partial {\it\omega}_{0}}{\partial k}\right)^{-1},\end{eqnarray}$$

we conclude that the critical mode $k$ selected at the destabilisation threshold $F_{\uparrow }$ of the rest state fulfils the inequality ${\it\omega}_{0}(k)<{\it\omega}={\it\Omega}/2$ , in cases of both short and long waves. Thence, the selected mode corresponds to the solution ${\it\omega}_{0}^{-}$ , whatever the sign of  $K$ .

As mentioned above, the sign of the nonlinear term in (5.1) depends on the depth, $K$  being positive for short waves (so the solution ${\it\omega}_{0}^{-}$ is stable) and negative for long waves (so the solution ${\it\omega}_{0}^{-}$ is unstable). Therefore, we conclude that the transition from rest to the wavy state is supercritical (i.e. smooth) for short waves, while it is subcritical (i.e. has hysteresis) for long waves (figure 1).

It should be noted that, according to this theoretical model, the hysteresis (observed experimentally by Rajchenbach et al. (Reference Rajchenbach, Leroux and Clamond2011, Reference Rajchenbach, Clamond and Leroux2013)) exists only because ${\it\omega}_{0}\neq {\it\omega}$ , i.e. because the frequency of a Faraday wave is not that of a free wave. Indeed, if ${\it\omega}_{0}={\it\omega}$ then, from (6.3), $F_{\uparrow }=4{\it\sigma}/{\it\omega}=F_{\downarrow }$ , so the hysteresis region vanishes. Therefore, consideration of the correct dispersion relation not only leads to quantitative corrections but, more importantly, also yields qualitatively different behaviours.

Figure 1.

The bifurcation diagram from rest. (Only the stable branches are displayed by thick blue lines.)

It should be noted also that, at the instability threshold $F_{\uparrow }$ , the relative deviation between ${\it\omega}_{0}$ and ${\it\omega}$ can be written from (6.3) as

(7.9) $$\begin{eqnarray}|{\it\omega}_{0}-{\it\omega}|{\it\omega}^{-1}={\textstyle \frac{1}{4}}(F_{\uparrow }^{2}-F_{\downarrow }^{2})^{1/2},\end{eqnarray}$$

which is practically convenient because $F_{\downarrow }$ and $F_{\uparrow }$ are easily measured experimentally in the subcritical regime. In the experiments by Rajchenbach et al. (Reference Rajchenbach, Clamond and Leroux2013) this deviation is approximately 10 %. In the supercritical regime, the deviation seems to be up to the order of 5 % according to the experiments by Edwards & Fauve (Reference Edwards and Fauve1994, figure 7).