3.1.1. Governing equations

In the absolute frame of reference, the centre of the wavemaker at the mean depth $\tilde {h}$ below the unperturbed water level moves vertically with the velocity

(3.2)\begin{equation} \tilde{U}(\tilde{t})=\tilde{A}\tilde{\omega}\cos \tilde{\omega} \tilde{t}. \end{equation}

The wavemaker frequency $\tilde {\omega }$ is in the vicinity of the $n$th natural frequency of the cavity $\tilde {\omega }_n=\sqrt {g\tilde {k}_n}$. Horizontally, the wavemaker is at the first antinode of the natural eigenmode, at the distance of $\tilde {L}/n$ from the left wall. In the non-inertial frame of reference connected to the wavemaker, the effective gravity is $g+\tilde {A}\tilde {\omega }^2\sin \tilde {\omega }\tilde {t}$. Cartesian coordinates are used; the origin of the system is located above the wavemaker at the unperturbed water level and the $\tilde {x}$, $\tilde {y}$ and $\tilde {z}$ axes are directed along the long and short cavity walls and upward, respectively. The flow is in $(\tilde {x},\tilde {z})$ plane.

The dimensionless parameters characterizing the forcing frequency $\alpha$, the wavemaker amplitude $\varepsilon$ and the effective gravitational acceleration $w$ are defined using the dimensional $\tilde {\omega }_n^{-1}$, $\tilde {k}_n^{-1}$ and the wavemaker amplitude $\tilde {A}$ as the scales of time, spatial coordinates and the surface elevation

(3.3ac)\begin{equation} \alpha=\frac{\widetilde{\omega^2}}{g\tilde{k}_n}=\frac{\tilde{\omega}^2}{\tilde{\omega}_n^2},\quad \varepsilon=\tilde{k}_n\tilde{A}, \quad w=\alpha^{{-}1}+\varepsilon\sin t. \end{equation}

The dimensionless coordinates are denoted by the same symbols without tildes, and the horizontal coordinates of the walls are

(3.4a,b)\begin{equation} x_L={-}{\rm \pi},\quad x_R=(n-1){\rm \pi}, \end{equation}

while the dimensionless wavemaker immersion depth and radius are

(3.5a,b)\begin{equation} h=\tilde{k}_n\tilde{h},\quad R=\tilde{k}_n\tilde{R}. \end{equation}

For $n=2,\ 3$ in the present experiments (see table 1), $h\sim 10^{-1}$, $R\sim 10^{-1}$, $\varepsilon \sim 10^{-2}$, $|\alpha -1|\sim 10^{-2}$. The dimensionless potential $\varPhi$ related to the dimensional one $\tilde {\varPhi }$ by

(3.6)\begin{equation} \tilde{\varPhi}=\tilde{A}\tilde{\omega}\tilde{k}_n^{{-}1}\varPhi, \end{equation}

satisfies the Laplace equation

(3.7)\begin{equation} \nabla^2\varPhi=0. \end{equation}

The following boundary conditions are imposed. At infinite depth, the disturbances vanish, and only the frame movement is retained, thus

(3.8)\begin{equation} \boldsymbol{\nabla}\varPhi={-}\cos t \boldsymbol{e}_z\quad \text{at } z\to-\infty. \end{equation}

The non-penetration conditions at the cavity walls and the wavemaker lead to vanishing derivatives of the potential in the normal to the solid surface direction

(3.9)\begin{gather} \frac{\partial \varPhi}{\partial x}=0\quad x=x_L,\ x_R, \end{gather}

(3.10)\begin{gather}\frac{\partial \varPhi}{\partial n}=0\quad \text{at } x^2+(z+h)^2=R^2. \end{gather}

The free surface $z=\varepsilon \eta$ is defined in the moving frame of reference as

(3.11)\begin{equation} \eta=\frac{\tilde{\zeta}}{\tilde{A}}-\sin t, \end{equation}

the kinematic and dynamic boundary conditions are

(3.12a)\begin{gather} \frac{\partial \varPhi}{\partial t}+\frac{\varepsilon}{2}(\boldsymbol{\nabla} \varPhi)^2 + (\alpha^{{-}1}+\varepsilon \sin t)\eta = 0, \end{gather}

(3.12b)\begin{gather}\frac{\partial \eta}{\partial t} + \varepsilon\frac{\partial \eta}{\partial x}\frac{\partial \varPhi}{\partial x} = \frac{\partial \varPhi}{\partial z}, \end{gather}

or, equivalently, in the Newman (Reference Newman2018) form

(3.13a)\begin{align} &\frac{\partial^{2} \varPhi}{\partial t^{2}}+(\alpha^{{-}1}+\varepsilon \sin t)\frac{\partial \varPhi}{\partial z}+2\varepsilon\boldsymbol{\nabla}\varPhi\boldsymbol{\cdot}\boldsymbol{\nabla}\frac{\partial \varPhi}{\partial t} +\frac{\varepsilon^2}{2}\boldsymbol{\nabla}\varPhi\boldsymbol{\cdot}\boldsymbol{\nabla}(\boldsymbol{\nabla}\varPhi\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi)\nonumber\\ &\quad -\frac{\varepsilon \cos t}{\alpha^{{-}1}+\varepsilon \sin t} \left[ \frac{\partial \varPhi}{\partial t}+\frac{\varepsilon}{2}\boldsymbol{\nabla}\varPhi\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi\right] =0, \end{align}

(3.13b)\begin{equation} \eta={-}\frac{1}{\alpha^{{-}1}+\varepsilon \sin t}\left[\frac{\partial \varPhi}{\partial t} +\frac{\varepsilon}{2}\boldsymbol{\nabla}\varPhi\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi\right]. \end{equation}

The two evolution equations (3.12) are more convenient for simulations of transient waves, while (3.13) is preferable for semi-analytical studies of periodic regimes and used in the sequel.

Since the time derivatives appear in the free-surface boundary condition only, the solution is represented as a linear combination of functions of spatial variables with time-dependent coefficients. This linear combination is decomposed into two parts: forcing potential $\varPhi _0$ that corresponds to fluid motion in a cavity of infinite vertical extent due to the prescribed forcing motion of the cylinder, and the eigenmodes $\varPhi _m$ that satisfy linear homogeneous boundary conditions at $z=0$ and correspond to the eigenfrequencies determined together with the eigenmodes.

The solutions for $\varPhi$ and $\eta$ are assumed to have the following form:

(3.14)\begin{gather} \varPhi={-}\varPhi_0 \cos t+\sum_{m=1}^\infty c_m(t)\varPhi_m, \end{gather}

(3.15)\begin{gather}\eta={-}\sin t+\sum_{m=1}^\infty b_m(t)\cos\left(\frac{m}{n}(x+{\rm \pi})\right). \end{gather}

The first term in (3.15) represents the contribution of the frame's motion in absolute coordinates; the unperturbed surface corresponds to $b_m=0$ for all $m$. For the periodic regime, the instant amplitudes of the spatial modes $c_m$ and $b_m$ are decomposed into Fourier series

(3.16)\begin{gather} \varPhi=\frac{1}{2}\left[-\varPhi_0 \,{\rm e}^{{\rm i}t} +\sum_{l=1}^{\infty} \sum_{m=1}^\infty c_m^{(l)}\varPhi_m \,{\rm e}^{{\rm i}lt}+\text{c.c.}\right], \end{gather}

(3.17)\begin{gather}\eta=\frac{1}{2}\left[{\rm i} {\rm e}^{{\rm i}t}+\sum_{l=0}^{\infty}\sum_{m=1}^\infty b_m^{(l)}\cos\left(\frac{m}{n}(x+{\rm \pi})\right){\rm e}^{{\rm i}lt}+\text{c.c.}\right], \end{gather}

where $\text {c.c.}$ stands for complex conjugate. For numerical calculations, the infinite series in (3.16), (3.17) are truncated. The term ‘harmonic’ is used in the sequel for the temporal Fourier decomposition, while ‘modes’ and ‘eigenmodes’ denote the corresponding spatial decomposition into the series of $\cos (m(x+{\rm \pi} )/n)$ and $\varPhi _m$, respectively, at any given instant.

3.1.2. Forcing potential

The forcing potential $\varPhi _0$ is a solution of the problem

(3.18a)\begin{gather} \nabla^2\varPhi_0=0\quad x_L< x< x_R, \end{gather}

(3.18b)\begin{gather}\frac{\partial \varPhi_0}{\partial x}=0\quad x=x_L,\ x_R, \end{gather}

(3.18c)\begin{gather}\frac{\partial \varPhi_0}{\partial z}\to 1\quad z\to-\infty, \end{gather}

(3.18d)\begin{gather}\frac{\partial \varPhi_0}{\partial x}\to 0\quad z\to-\infty, \end{gather}

(3.18e)\begin{gather}\frac{\partial \varPhi_0}{\partial n}=0\quad \text{at } x^2+(z+h)^2=R^2, \end{gather}

that does not account for boundary conditions at the free surface and thus is not unique. For definiteness, assume the symmetry relative to $z=-h$ plane and consider a superposition of the uniform flow and an infinite series of multipoles placed at the centre of the wavemaker and symmetrical points relative to the walls $(x_q,-h)$, as shown in figure 3

(3.19a)\begin{gather} \displaystyle\varPhi_0(x,z)=z+a_0+\sum_{p=1}^\infty\sum_{q={-}\infty}^\infty \left(a_{pq}\frac{\cos p\vartheta_q}{r_q^p}+b_{pq}\frac{\sin p\vartheta_q}{r_q^p}\right), \end{gather}

(3.19b)\begin{gather}x-x_q=r_q\cos\vartheta_q,\quad z+h=r_q\sin\vartheta_q, \end{gather}

(3.19c)\begin{gather}x_0=0, \end{gather}

(3.19d)\begin{gather}x_q=2x_L-x_{|q|-1}\quad q<0, \end{gather}

(3.19e)\begin{gather}x_q=2x_R-x_{-|q|+1}\quad q>0, \end{gather}

where $r_q$ and $\vartheta _q$ are polar coordinates with the centre at points ($x_q,-h)$. For $R\ll 1$, the contribution of high-order multipoles is negligible, and (3.18) is equivalent to the classical problem of a uniform flow past a cylinder, so that the solution $\varPhi _0^0$ corresponds to the superposition of a uniform flow with velocity ${\partial \varPhi _0}/{\partial z} = 1$ and a dipole in the centre of the cylinder

(3.20)\begin{equation} \varPhi_0^0=z+\frac{R^2}{x^2+(z+h)^2} (z+h). \end{equation}

The function $\varPhi _0^0$ serves as an initial approximation that is improved by an iterative numerical procedure (see Appendix A) that treats the boundary conditions at the walls in the presence of a finite-sized cylinder more accurately. Once all coefficients with $a_{pq}$ and $b_{pq}$ in (3.19) are determined, a constant $a_0$ is found from the condition

(3.21)\begin{equation} \int_{x_L}^{x_R}\varPhi_0\,{\rm d}\kern 0.06em x=0. \end{equation}

Figure 3.

Locations of the multipoles and auxiliary polar coordinates. The real flow domain is shaded.

The forcing potential produces the terms of the order of $\varPhi _0(x,0)$ in the free surface boundary conditions (3.13); they decrease with the increase of $h$. Denote the reference value of $\varPhi (x,0)$ as the transmitting factor $\kappa$; it depends on the geometry only

(3.22)\begin{equation} \varPhi_0(x,0)\sim\kappa=\frac{R^2}{h}=\tilde{k}_n\frac{\tilde{R}^2}{\tilde{h}}. \end{equation}

For the parameters of the experiment, $\kappa \sim 10^{-1}$.

3.1.3. Eigenmodes

The eigenmodes $\varPhi _m$ are non-trivial solutions of the eigenvalue problem with zero boundary conditions at the walls (3.9), at the wavemaker (3.10) and at $z\to -\infty$; the linear boundary conditions at $z=0$ correspond to a periodic flow with an unknown frequency $\alpha _m^{1/2}$

(3.23a)\begin{gather} \nabla^2\varPhi_m=0\quad x_L< x< x_R,\ z<0, \end{gather}

(3.23b)\begin{gather}\frac{\partial \varPhi_m}{\partial n}=0\quad \text{at } x^2+(z+h)^2=R^2, \end{gather}

(3.23c)\begin{gather}\frac{\partial \varPhi_m}{\partial x}=0\quad \text{at } x=x_L,\ x_R, \end{gather}

(3.23d)\begin{gather}-\alpha_m\varPhi_m+\frac{\partial \varPhi_m}{\partial z}=0\quad \text{at } z=0, \end{gather}

(3.23e)\begin{gather}\boldsymbol{\nabla}\varPhi_m\to 0 \quad \text{at } z\to-\infty. \end{gather}

General properties of the solutions of the Laplace equation ensure that the eigenvalues $\alpha _m$ are real and correspond to real frequencies. Functions $\varPhi _m$ form a complete orthogonal system; applying the Green theorem, one obtains

(3.24)\begin{equation} \int_{-{\rm \pi}}^{(n-1){\rm \pi}}\varPhi_{m1}(x,0)\varPhi_{m2}(x,0)\,{{\rm d}\kern 0.06em x}=0\quad m_1\neq m_2, \end{equation}

and

(3.25)\begin{equation} \int_{-{\rm \pi}}^{(n-1){\rm \pi}}\varPhi_{m}(x,0)\,{{\rm d}\kern0.05em x}=0\quad \forall m. \end{equation}

The eigenmodes $\varPhi _m$ and the eigenvalues $\alpha _m$ differs from those for the pure rectangular cavity

(3.26a,b)\begin{equation} \varPhi_m^0=\cos\left(\frac{m}{n}(x+{\rm \pi})\right)\exp\left(\frac{m}{n}z\right), \quad \alpha_m^0=\frac{m}{n}, \end{equation}

by the values of the order of $\kappa$. Appendix B provides the proof and the details of the numerical method. The normalized by the parameter $\kappa$ forcing potential, the eigenfunctions $\varPhi _m$ at the free surface and their deviation from $\varPhi _m^0$ are shown in figure 4.

Figure 4.

Normalized forcing potential (a,d), eigenmodes (b,e), and their normalized deviations from cosines (c,f) for non-symmetrical case I (a–c) and symmetrical case II (d–f) described in table 1.

The eigenvalues are calculated analytically in the limit $R\ll 1$ (Appendix C). The deviation of $\alpha _n$ from unity in this limit is given as

(3.27)\begin{align} 1- \alpha_n&=\kappa\frac{2\exp({-}h)}{{\rm \pi} n }\left[ ({-}2h+2h^2-h^3)\tan^{{-}1}\frac{\sqrt{2}}{h}-2\sqrt{2}h+\sqrt{2}h^2 \right]\nonumber\\ &\quad +O(\kappa^2). \end{align}

Figure 5 shows this deviation for the natural frequencies $n=2,\ 3$. The approximate expression (3.27) is quite accurate for $h$ exceeding unity, where the effect of the finite wavemaker size decays with its depth of immersion $h$. However, in the whole range of the parameters used: $R< h<5R$, $R\sim 10^{-1}$, the relative error in applying the asymptotic (3.27) remains below 15 %.

Figure 5.

Absolute (a) and normalized (b) values of the detuning for $\tilde {R}=21$ mm and $n=2,\ 3$ calculated numerically and analytically by (3.27).

3.2.1. Potential

Linearization of (3.13) and shift of the free-surface boundary conditions to the unperturbed level lead to the simplified boundary conditions

(3.28a)\begin{gather} \displaystyle\frac{\partial^{2} \varPhi}{\partial t^{2}}+\alpha^{{-}1}\frac{\partial \varPhi}{\partial z}=0\quad \text{at } z=0, \end{gather}

(3.28b)\begin{gather}\displaystyle\eta={-}\alpha\frac{\partial \varPhi}{\partial t}\quad \text{at } z=0. \end{gather}

The modulation of gravity and the displacement of the free surface due to the motion of the frame of reference are negligible at this order.

The potential and the surface elevation for the periodic flow regime are expressed through their complex-valued amplitudes $\hat {\varPhi }$ and $\hat {\eta }$, respectively,

(3.29a,b)\begin{equation} \varPhi=\tfrac{1}{2}\hat{\varPhi}\,{\rm e}^{{\rm i}t} +\text{c.c.},\quad \eta =\frac{1}{2}\hat{\eta}\,{\rm e}^{{\rm i}t} + \text{c.c.} \end{equation}

The boundary conditions for those amplitudes $\hat {\varPhi }$, $\hat {\eta }$ are given by

(3.30a)\begin{gather} \displaystyle-\alpha \hat{\varPhi}+\frac{\partial \hat{\varPhi}}{\partial z}=0\quad \text{at } z=0, \end{gather}

(3.30b)\begin{gather}\hat{\eta}={-}{\rm i}\alpha \hat{\varPhi}\quad \text{at } z=0, \end{gather}

(3.30c)\begin{gather}\boldsymbol{\nabla}\hat{\varPhi}={-}\boldsymbol{e}_z\quad \text{at } z\to-\infty. \end{gather}

Applying the decomposition (3.16)

(3.31)\begin{equation} \hat{\varPhi}={-}\varPhi_0 \,{\rm e}^{{\rm i}t} +\sum_{m=1}^\infty c_m^{(1)}\varPhi_m, \end{equation}

and using the boundary conditions at the free surface yields the equations for the coefficients $c_m^{(1)}$ in the expansion of the potential

(3.32)\begin{equation} -\left(-\alpha\varPhi_0+\frac{\partial \varPhi_0}{\partial z}\right) +\sum_{m=1}^{\infty}c_m^{(1)}\varPhi_m(-\alpha+\alpha_m)=0. \end{equation}

For brevity, define the scalar product $\langle {\cdot },{\cdot }\rangle$ and norm $\|{\cdot }\|$ for the functions of two spatial variables via their values at $z=0$

(3.33a,b)\begin{equation} \langle\, f(x,z),g(x,z)\rangle =\int_{-{\rm \pi}}^{(n-1){\rm \pi}}f(x,0)g(x,0) \,{{\rm d}\kern 0.06em x},\quad \|\,f\|^2=\int_{-{\rm \pi}}^{(n-1){\rm \pi}}f^2(x,0)\, {{\rm d}\kern 0.06em x}. \end{equation}

The coefficients $c_m^{(1)}$ are obtained from (3.32) invoking the orthogonality of the system $\varPhi _m$

(3.34)\begin{equation} c_m^{(1)}=\frac{1}{-\alpha+\alpha_m}\frac{1}{\|\varPhi_m\|^2} \left\langle-\alpha\varPhi_0+\frac{\partial \varPhi_0}{\partial z},\varPhi_m\right\rangle. \end{equation}

For $\alpha$ close to unity, all coefficients $c_m^{(1)}$ except for $m=n$ change insignificantly with $\alpha$, while $c_n^{(1)}$ tends to infinity at $\alpha =\alpha _n$. Once the potential $\hat {\varPhi }$ is known, the surface elevation can be found from (3.30).

3.2.2. Effect of viscosity

In the presence of weak kinematic viscosity $\nu$, dissipation may appear due to internal friction both in the bulk and at rigid boundaries. The bulk dissipation manifests itself in the dynamic free-surface boundary condition by a dimensionless term of the order $\tilde {k}_n^2\nu /\tilde {\omega }\sim 10^{-5}$ and can be neglected. The wall friction results in thin Stokes layers at all rigid surfaces. The reference thickness of the Stokes layer is $\sqrt {\nu /\tilde {\omega }}$; for water waves in the present experiments with frequencies $\tilde {\omega }\sim 10$ s$^{-1}$, it is approximately 0.3 mm. Considering the dimensionless Stokes layer thickness as a small parameter, the correction to the potential solution is introduced.

Following Mei (Reference Mei1989), Kit & Shemer (Reference Kit and Shemer1989) and Hill (Reference Hill2003), a method to account for the Stokes layer corrections for potential flow with arbitrary time-periodic potential $\varPhi$ is applied. To satisfy the no-slip boundary conditions in the Stokes layers at all rigid boundaries, a rotational correction to the potential solution needs to be introduced

(3.35a,b)\begin{equation} \frac{\partial V}{\partial t}=\frac{\partial^{2} V}{\partial \xi^{2}},\quad V|_{\xi=0}={-}\frac{\partial \varPhi}{\partial \tau}, \end{equation}

where $V$ is the tangential velocity, and $\xi$ and $\tau$ are the coordinates normal and tangential to the surface, respectively, normalized by

(3.36)\begin{equation} s=\sqrt{\frac{\tilde{k}_n^2\nu}{\tilde{\omega}}}\sim 10^{{-}2}. \end{equation}

For time-periodic flows with unit radian frequency, the amplitude $\hat {\varPhi }$ is defined by (3.30). Introducing complex amplitude $\hat {V}$

(3.37)\begin{equation} V=\tfrac{1}{2}\hat{V}\,{\rm e}^{{\rm i}t}+\text{c.c.}, \end{equation}

yields

(3.38)\begin{equation} \hat{V}={-}\left.\frac{\partial \hat{\varPhi}}{\partial \tau}\right|_{\xi=0}\exp\left(-\frac{1+i}{\sqrt{2}}\xi\right). \end{equation}

This correction needs to be calculated for two linearly independent tangential velocity components at any rigid surface. Two types of rigid surfaces need to be considered separately: (i) the sidewalls and wavemaker, which both have a normal in the $(x,z)$ plane; and (ii) the front and back walls with the normal that is orthogonal to the plane of the basic flow.

For type (i) surfaces, there is only one velocity component with a non-zero value of $\partial {\hat {\varPhi }}/\partial \tau$. In this case, there is a mass exchange between the Stokes layer and the bulk flow, resulting in a correction $s({(1- i)}/{\sqrt {2}})\hat {\varPhi }^v$ for the potential $\hat {\varPhi }$. The amplitude of the correction $\hat {\varPhi }^v$ is the solution of the problem

(3.39a)\begin{gather} \nabla^2\hat{\varPhi}^v=0, \end{gather}

(3.39b)\begin{gather}\displaystyle\frac{\partial \hat{\varPhi}^v}{\partial n}={-}\frac{\partial^{2} \hat{\varPhi}}{\partial \tau^{2}}\quad \text{at rigid surfaces}, \end{gather}

(3.39c)\begin{gather}\displaystyle\frac{\partial \hat{\varPhi}^v}{\partial n}=0\quad z=0, \end{gather}

(3.39d)\begin{gather}\hat{\varPhi}^v\to 0\quad z\to -\infty. \end{gather}

The non-trivial Neumann boundary conditions are imposed at the sidewalls and the wavemaker surface; the solution is found numerically.

For type (ii) boundaries (the front and back walls), the velocity in the Stokes layer has non-zero $x$ and $z$ components. For any fixed $y$, these components are proportional to $\boldsymbol {\nabla }\hat {\varPhi }$, ensuring that the $y$-component of the velocity is identically zero. The Stokes layer affects the bulk flow through the kinematic free surface boundary condition. The liquid velocity at the free surface deviates from the bulk value $\partial {\hat {\varPhi }}/\partial z$ in the vicinity of the walls by the value

(3.40)\begin{equation} \hat{W}={-}\frac{\partial \hat{\varPhi}}{\partial z}\exp\left(-\frac{1+i}{\sqrt{2}} \xi \right). \end{equation}

The corrected kinematic free-surface boundary condition reads

(3.41)\begin{equation} {\rm i}\hat{\eta}=\frac{\partial \hat{\varPhi}}{\partial z}+\hat{W}. \end{equation}

Integrating across the cavity yields

(3.42)\begin{equation} {\rm i}\hat{\eta}=\frac{\partial \hat{\varPhi}}{\partial z}\left[1- \frac{2s}{B} \frac{1-i}{\sqrt{2}}\right], \end{equation}

where $B=\tilde {k}_n\tilde {B}\sim 1$ is the dimensionless cavity size in the $y$-direction. Note that the correction $\hat {\varPhi }^v$ is not involved in the free-surface boundary condition since $\partial \hat {\varPhi }^v/\partial z=0$ at $z=0$.

Corrected due to the presence of the Stokes layer boundary condition for the potential, (3.30) is

(3.43)\begin{equation} -\!\alpha\left(\hat{\varPhi}+s\frac{(1-i)}{\sqrt{2}}\hat{\varPhi}^v\right) +\frac{\partial \hat{\varPhi}}{\partial z}\left[1- \frac{2s}{B} \frac{1-i}{\sqrt{2}}\right]=0\quad \text{at } z=0. \end{equation}

To solve the problem, the corrections forcing potential $\varPhi _0^v$ and eigenmodes $\varPhi _m^v$ are calculated from (3.39) substituting $\varPhi _0$ and $\varPhi _m$ as $\hat {\varPhi }$, respectively. Following (3.16), the solution is represented as

(3.44a,b)\begin{equation} \hat{\varPhi}={-}\varPhi_0+\sum_{m=1}^{\infty}c_m^{(1)}\varPhi_m,\quad \hat{\varPhi}^v={-}\varPhi_0^v+\sum_{m=1}^{\infty}c_m^{(1)}\varPhi_m^v, \end{equation}

with the identical coefficients $c_m^{(1)}$ in both decompositions. The values of $c_m^{(1)}$ are obtained similarly to the inviscid case.

The relative contribution of the different dissipation mechanisms is estimated: dissipation at the sidewalls and the wavemaker associated with $\hat {\varPhi }^v$ and at the front and back walls associated with the additional term in the kinematic boundary condition that is proportional to $B^{-1}$. The problem (3.43) is solved using three assumptions:

1. (i) full dissipation with all terms proportional to $s$ retained;

2. (ii) dissipation on sidewalls and the wavemaker corresponding to an infinitely wide cavity, keeping terms with $\hat {\varPhi }^v$ but omitting those proportional to $B^{-1}$;

3. (iii) dissipation at the front and back walls only corresponding to the narrow cavity, keeping the term with $B^{-1}$ but omitting those with $\hat {\varPhi }^v$.

In figure 6, the dependence of the inverse amplitudes $|c_n^{(1)}|^{-1}$ of the resonant eigenmode obtained under these three assumptions is compared with the results of the inviscid model. The dissipation limits the maximum amplitude and downshifts the effective resonant frequency. The minimum inverse amplitude is zero for the inviscid model and is finite in the presence of any dissipation mechanism. The dissipation at the front and back walls dominates over the other mechanisms. Keeping the terms corresponding to the assumption (iii) only, the equation for potential becomes

(3.45)\begin{equation} -\!\alpha\hat{\varPhi}+\frac{\partial \hat{\varPhi}}{\partial z}\left[1-\frac{2s}{B}\frac{1-i}{\sqrt{2}}\right]=0, \end{equation}

yielding the solution for $c_m^{(1)}$

(3.46a)\begin{gather} c_m^{(1)}=\frac{1}{-\alpha+\alpha_m^v}\frac{1}{\|\varPhi_m\|^2}\left\langle \left[-\alpha\varPhi_0+\frac{\partial \varPhi_0}{\partial z}\left(1-\frac{2s}{B}\frac{1- i}{\sqrt{2}}\right)\right],\varPhi_m\right\rangle, \end{gather}

(3.46b)\begin{gather}\alpha_m^v=\alpha_m\left[1-\frac{2s}{B}\frac{1-i}{\sqrt{2}}\right]. \end{gather}

The phase of the resonant eigenmode is equal to the argument of $c_n^{(1)}$, it is strongly dependent on $\alpha$ and changes from ${\rm \pi}$ to $0$ in the vicinity of $\alpha _n$.

Figure 6.

Inverse amplitude of the dominant coefficient $|c_n^{(1)}|$ for case I and different assumptions adopted for dissipation.

3.2.3. Surface elevation

The surface elevation amplitude is found from (3.30) as

(3.47)\begin{equation} \hat{\eta}={-}{\rm i}\alpha\left(\hat{\varPhi}+s\frac{(1-i)}{\sqrt{2}}\hat{\varPhi}^v\right), \end{equation}

which, under assumption (iii), is simplified to

(3.48)\begin{equation} \hat{\eta}={-}{\rm i}\alpha\hat{\varPhi}. \end{equation}

The decomposition of the surface shapes onto modes (3.17) reads

(3.49a,b)\begin{equation} \hat{\eta}={\rm i}+\sum_{m=1}^M b_m^{(1)}\cos\left(\frac{m}{n}(x+{\rm \pi})\right),\quad b_m^{(1)}=|b_m^{(1)}|\exp\left[{\rm i}\left(\theta_m+\frac{\rm \pi}{2}\right)\right], \end{equation}

where $|b_m^{(1)}|$ and $\theta _m$ are the amplitudes and the phases of the $m$th mode. The shift of ${\rm \pi} /2$ is introduced to define the phase $\theta _m$ relative to the wavemaker's absolute displacement that is proportional to $\sin t$.

For any $\alpha$, the surface shape is a superposition of an infinite number of spatial modes, including those with wavenumbers lower than $n$ ($m=1,\ 2$ in case of $n=3$). Figure 7 presents the dependence of the spatial modes’ amplitudes on the detuning $\alpha -1$. Note that, for $(\alpha -1)\ll \kappa$, the linear theory predicts a cosinusoidal free-surface shape with a relatively small amplitude. For the resonant mode ($m=n$), the surface elevation above the wavemaker is in anti-phase with the wavemaker below the resonance and in-phase above it. In the framework of inviscid theory, those phases are either $0$ or ${\rm \pi}$, but the jump is smoothened by viscous dissipation. The phases of non-resonant modes ($m\neq n$) change both at the resonance and at zero detuning.

Figure 7.

Amplitudes (a,c) and phases relative to the wavemaker absolute displacement (b,d) of major spatial modes for the cases I (a,b) and II (c,d) described in table 1; solid and dashed lines correspond to full Stokes layer dissipation and the inviscid model.

3.3.1. Solution structure

To study the effect of finite wave amplitude, the dimensionless wave steepness $\delta =\tilde {k}_n\tilde {a}$ is introduced, where $\tilde {a}$ is the dimensional scale of the surface elevation amplitude. The Moiseev (Reference Moiseev1958) matching condition between the cubic nonlinearity and the forcing yields the following relation between $\delta$, the dimensionless wavemaker amplitude $\varepsilon$ and the transmitting factor $\kappa$, introduced in (3.22):

(3.50)\begin{equation} \delta^3=\varepsilon \kappa. \end{equation}

The dimensional velocity amplitude at the surface scales as $\tilde {\omega }\tilde {a}$, the spatial scale $\tilde {k}_n^{-1}$ defines the new scaling for potential

(3.51)\begin{equation} \tilde{\varPhi}=\tilde{k}_n^{{-}1}\tilde{\omega}\tilde{a}\varphi, \end{equation}

ensuring $\varphi \sim 1$. From (3.51) and (3.6), $\tilde {a}\varphi =\tilde {A}\varPhi$ and

(3.52)\begin{equation} \delta \varphi=\varepsilon \varPhi. \end{equation}

The boundary condition at the free surface (3.13) is rewritten as

(3.53a)\begin{align} &\frac{\partial^{2} \varphi}{\partial t^{2}}+(\alpha^{{-}1}+\varepsilon \sin t)\frac{\partial \varphi}{\partial z}+2\delta\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\frac{\partial \varphi}{\partial t} +\frac{\delta^2}{2}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla} (\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi)\nonumber\\ &\quad -\frac{\varepsilon \cos t}{\alpha^{{-}1}+\varepsilon \sin t}\left[ \frac{\partial \varphi}{\partial t}+\frac{\delta}{2}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi\right] =0\quad \text{at } z=\varepsilon \eta, \end{align}

(3.53b)\begin{equation} \eta={-}\frac{1}{\alpha^{{-}1}+\varepsilon \sin t}\frac{\delta}{\varepsilon} \left[ \frac{\partial \varphi}{\partial t}+\frac{\delta}{2}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla} \varphi\right] \quad \text{at } z=\varepsilon \eta. \end{equation}

And at $z\to -\infty$

(3.54)\begin{equation} \boldsymbol{\nabla}\varphi\to-\frac{\varepsilon}{\delta}\boldsymbol{e}_z\cos t. \end{equation}

In this study, the values of the independent parameters $\varepsilon \sim 10^{-2}$ and $\delta \sim 10^{-1}$; adopting Moiseyev's approach suggests retaining in the Taylor expansion of the boundary conditions (3.53) the terms up to $O(\delta ^2)$. Neglecting higher order in $\delta$ terms allows us to keep the terms $O(\varepsilon )$ only, yielding

(3.55a) \begin{align} &\frac{\partial^{2} \varphi}{\partial t^{2}}+\alpha^{{-}1}\frac{\partial \varphi}{\partial z} +\delta\left[2\boldsymbol{\nabla} \varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\frac{\partial \varphi}{\partial t}-\alpha\frac{\partial \varphi}{\partial t}\frac{\partial }{\partial z} \left(\frac{\partial^{2} \varphi}{\partial t^{2}}+\alpha^{{-}1}\frac{\partial \varphi}{\partial z}\right)\right]\nonumber\\ &\quad +\delta^2\left[\frac{1}{2}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla} \left(\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi\right) -2\alpha\frac{\partial \varphi}{\partial t}\frac{\partial }{\partial z}\left(\boldsymbol{\nabla }\varphi\boldsymbol{\cdot} \boldsymbol{\nabla}\frac{\partial \varphi}{\partial t} \right)\right.\nonumber\\ &\quad +\alpha\left(\frac{\partial \varphi}{\partial t}\frac{\partial^2\varphi}{\partial t\partial z} -\frac{1}{2}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi\right) \frac{\partial }{\partial z}\left(\frac{\partial^{2} \varphi}{\partial t^{2}}+\alpha^{{-}1}\frac{\partial \varphi}{\partial z}\right) \nonumber\\ &\quad \left. +\,\frac{\alpha^2}{2}\left(\frac{\partial \varphi}{\partial t}\right)^2\frac{\partial^{2} }{\partial z^{2}} \left(\frac{\partial^{2} \varphi}{\partial t^{2}}+\alpha^{{-}1}\frac{\partial \varphi}{\partial z}\right) \right]\nonumber\\ &\quad +\varepsilon\left[\sin t \frac{\partial \varphi}{\partial z}-\cos t \frac{\partial \varphi}{\partial t} \right]+ O(\delta^3,\delta\varepsilon,\varepsilon^2) =0, \end{align}

(3.55b) \begin{align} \eta&={-}\frac{\delta}{\varepsilon}\alpha\left\{ \frac{\partial \varphi}{\partial t}+\delta\left[\frac{1}{2}\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi-\alpha\frac{\partial \varphi}{\partial t}\frac{\partial^2\varphi}{\partial t\partial z} \right]\right.\nonumber\\ &\quad \left. +\,\delta^2\left[-\frac{\alpha}{2}(\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi) \frac{\partial^2\varphi}{\partial z\partial t} -\frac{\alpha}{2}\frac{\partial \varphi}{\partial t}\frac{\partial^{} }{\partial z^{}}(\boldsymbol{\nabla}\varphi\boldsymbol{\cdot}\boldsymbol{\nabla}\varphi) +\frac{\alpha^2}{2}\left(\frac{\partial \varphi}{\partial t}\right)^2 \frac{\partial^3\varphi}{\partial z^2\partial t} \right]\right.\nonumber\\ &\quad -\left.\varepsilon\alpha\frac{\partial \varphi}{\partial t}\sin t +O(\delta^3,\delta \varepsilon,\varepsilon^2)\right\}, \end{align}

where all functions and their derivatives are evaluated at $z=0$.

In view of the modified (3.55), it is convenient to use in solution (3.15) the potential $\varphi _0 =O(1)$ that corresponds to the vanishing velocity at $z\to - \infty$, instead of the rescaled forcing potential $\varPhi _0$ defined by (3.18)

(3.56)\begin{equation} \frac{\varepsilon}{\delta}\varPhi_0=\frac{\varepsilon}{\delta}z +\frac{\varepsilon\kappa}{\delta}\varphi_0. \end{equation}

The time-dependent coefficients $c_m(t)$ in (3.16) are represented as a sum of temporal harmonics with integer frequencies and slowly varying amplitudes. The slow time of $\delta ^2 t$ is defined by the scales of the forcing potential. As in Moiseev (Reference Moiseev1958), the analysis is limited to the two lowest harmonics

(3.57)\begin{align} \varphi(t,x,z)&=\frac{1}{2}\left\{ \left[-\frac{\varepsilon}{\delta}z-\delta^2\varphi_0 (x,z)+\frac{\varepsilon}{\delta}\sum_{m=1}^\infty c_{m}^{(1)}(\delta^2 t)\varPhi_m(x,z)\right] {\rm e}^{{\rm i}t}\right.\nonumber\\ &\quad \left. +\,\frac{\varepsilon}{\delta}\sum_{m=1}^\infty c_{m}^{(2)}(\delta^2 t)\varPhi_m(x,z) \,{\rm e}^{{\rm i}2t}\right\}+ \text{c.c.} +\text{higher temporal harmonics}. \end{align}

The eigenmode with $m=n$ at the first harmonic is in resonance and makes the major contribution to the potential. By definition of the scale for $\varphi$ (3.51), it has an amplitude of the order of unity. For brevity, denote

(3.58)\begin{equation} C_n=\frac{\varepsilon}{\delta}c_{n}^{(1)}\sim 1, \end{equation}

and introduce the functions $F_1$ and $F_2$ of the order of unity for non-resonant eigenmodes at the first and of the second harmonics. The former has the amplitude of $\delta ^2$, the same as forcing; the latter is, generally, of the order of $\delta$

(3.59a)\begin{gather} \delta^2 F_1(\delta^2 t, x,z)=\frac{\varepsilon}{\delta}\sum_{m=1, m\neq n}^\infty c_{m}^{(1)}(\delta^2 t)\varPhi_m(x,z), \end{gather}

(3.59b)\begin{gather}\delta F_2(\delta^2 t,x,z)=\frac{\varepsilon}{\delta}\sum_{m=1}^\infty c_{m}^{(2)}(\delta^2 t)\varPhi_m(x,z). \end{gather}

The function $F_1$ is orthogonal to $\varPhi _n$, since $\varPhi _m$ is orthogonal to $\varPhi _n$ for any $m\neq n$. The transformed form of the solution reads

(3.60)\begin{align} \varphi(t,x,z)&=\frac{1}{2}\left\{\left[-\frac{\varepsilon}{\delta}z-\delta^2\varphi_0 (x,z)+ C_{n}(\delta^2 t)\varPhi_n(x,z)+\delta^2 F_1(\delta^2 t,x,z) \right] {\rm e}^{{\rm i}t}\right.\nonumber\\ &\quad \left.+\,\delta F_2(\delta^2 t, x,z) \,{\rm e}^{{\rm i}2t} \vphantom{\frac{\varepsilon}{\delta}}\right\}+ \text{c.c.} +\text{higher temporal harmonics}. \end{align}

The solution is substituted to the free-surface boundary condition (3.55). Considering the leading terms in (3.55) at the first harmonic only, one obtains

(3.61)\begin{equation} C_n\varPhi_n\left[{-}1+\frac{\alpha_n}{\alpha}\right]+O(\delta^2)=0. \end{equation}

The existence of the non-trivial solutions limits the range of the detuning values considered

(3.62)\begin{equation} \alpha-\alpha_n\sim \delta^2. \end{equation}

The solution process continues as follows. First, considering the terms of $O(\delta )$ that correspond to the second harmonic defines the connection between the resonant eigenmode and the second harmonic. Then, the nonlinear effects at the first harmonic are accounted for, leading to a nonlinear equation for $C_n$. The amplitudes of the non-resonant eigenmodes at the first harmonic are found to finalize the solution of the problem for the potential. The surface elevation is then found.

3.3.2. Potential

The equation at the second harmonic is

(3.63)\begin{align} &\left({-}4F_2+\alpha^{{-}1}\frac{\partial F_2}{\partial z}\right)-\frac{i}{2} C_{n}^2\left[{-}2\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n+\alpha \varPhi_n\left(-\frac{\partial \varPhi_n}{\partial z}+\alpha^{{-}1}\frac{\partial^{2} \varPhi_n}{\partial z^{2}}\right)\right]\nonumber\\ &\quad -3{\rm i}C_n\frac{\varepsilon}{\delta}\varPhi_n+ O\left(\delta^2,\varepsilon,\frac{\varepsilon^2}{\delta}\right)=0. \end{align}

This approximation does not contain time derivatives of $F_2$; the second harmonic depends on slow time $\delta ^2 t$ parametrically via $C_{n}$. If $4\alpha$ is not an eigenvalue, this equation has a unique solution

(3.64a)\begin{gather} F_2=\frac{i}{2} C_{n}^2\sum_{m=1}^{\infty}\frac{1}{-4+\alpha_m/\alpha}\frac{\langle P,\varPhi_m\rangle }{\|\varPhi_m\|^2}\varPhi_m -{\rm i}C_n\frac{\varepsilon}{\delta}\varPhi_n+ O\left(\delta,\frac{\varepsilon}{\delta}\right), \end{gather}

(3.64b)\begin{gather}P={-}2\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n+\alpha \varPhi_n\left[-\frac{\partial \varPhi_n}{\partial z}+\alpha^{{-}1}\frac{\partial^{2} \varPhi_n}{\partial z^{2}}\right]. \end{gather}

For brevity, the normalized amplitude of the second harmonic $f_2$ is introduced

(3.65)\begin{equation} F_2=\frac{i}{2}C_n^2\,f_2-{\rm i}C_n\frac{\varepsilon}{\delta}\varPhi_n. \end{equation}

A non-trivial expression for the second harmonic distinguishes the present problem with a finite wavemaker from the pure rectangular cavity case (Faltinsen Reference Faltinsen1974). In the latter case, the first term in $P$ is constant, yielding zero contribution in $F_2$, while the second one is proportional to $-1+\alpha /\alpha _n\sim \delta ^2$, which is below the considered order of magnitude. The function $f_2$ is of the order of unity. The numerator $\langle P, \varPhi _m\rangle$ is of the order of $\kappa$ for any $m$, since the deviation of the eigenmodes $\varPhi _m$ from those for the pure rectangular cavity is caused by the presence of the wavemaker. From (3.27), the denominator for $m=4n$ is also of the order of $\kappa$. Therefore, the major contribution to the surface shape at the second harmonic comes from the eigenmode that is in resonance defined by the deep-water dispersion relation.

In the framework of the inviscid model, the solution does not exist for $\alpha =\alpha _{4n}/4$; the amplitude of the second temporal harmonic is then infinite. In the vicinity of the resonance for the double frequency, the basic assumption of the model that

(3.66)\begin{equation} \|F_2\|\ll \|C_{n}\varPhi_n\|, \end{equation}

is violated. The dissipation due to the Stokes layers is negligible at the considered order of magnitude; however, accounting for this mechanism regularizes the solution and does not introduce a significant error in the domain of the model validity.

The terms of the order of $\delta ^2$ at the first harmonic give an equation for $C_{n}$ and the expression for $F_1$, defining the solution

(3.67) \begin{align} & 2{\rm i}C_{n}'\varPhi_n+ \delta^{{-}2}C_{n}\left[-\varPhi_n+\alpha^{{-}1}\frac{\partial \varPhi_n}{\partial z}\right]+ \left[{-}F_1+\alpha^{{-}1}\frac{\partial F_1}{\partial z}\right]-\left(-\varphi_0+\alpha^{{-}1}\frac{\partial \varphi_0}{\partial z}\right)\nonumber\\ &\quad +|C_{10}^2|C_{10}\left\{-\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}f_2 +\alpha\left[f_2\left(-\frac{\partial \varPhi_n}{\partial z}+\alpha^{{-}1}\frac{\partial^{2} \varPhi_n}{\partial z^{2}}\right)\right.\right.\nonumber\\ & \quad-\left.\left.\frac{1}{2}\varPhi_n\left({-}4\frac{\partial f_2}{\partial z}+\alpha^{{-}1}\frac{\partial^{2} f_2}{\partial z^{2}}\right)\right] +\frac{3}{8}\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla} \left(\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n\right) \right.\nonumber\\ &\quad - \left.\frac{1}{2}\alpha\varPhi_n \frac{\partial }{\partial z}\left(\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot} \boldsymbol{\nabla}\varPhi_n\right) +\frac{1}{4}\alpha\varPhi_n\frac{\partial \varPhi_n}{\partial z}\frac{\partial }{\partial z} \left(-\varPhi_n+\alpha^{{-}1}\frac{\partial \varPhi_n}{\partial z}\right) \right.\nonumber\\ &\quad -\left.\frac{3\alpha}{8}\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla} \varPhi_n\frac{\partial }{\partial z}\left(-\varPhi_n+\alpha^{{-}1}\frac{\partial \varPhi_n}{\partial z} \right) +\frac{\alpha^2}{8}\varPhi_n^2\frac{\partial^{2} }{\partial z^{2}}\left(-\varPhi_n+\alpha^{{-}1}\frac{\partial \varPhi_n}{\partial z} \right)\right\}\nonumber\\ &\quad+ O\left(\delta,\frac{\varepsilon}{\delta}\right)=0, \end{align}

where primes denote derivative with respect to the slow time $\delta ^2 t$. The inertia force as well as the terms corresponding to the wavemaker displacement are negligible at this order.

In the two-step solution, the scalar product of the equation with $\varPhi _n$ first determines $C_{n}$; then, the function $F_1$ is found in the second step from considering the scalar products with $\varPhi _m$, $m\neq n$. The qualitative properties of the solution are solely defined by the value of $C_{n}$. Denoting the normalized detuning $\beta$ as

(3.68)\begin{equation} \beta={-}\delta^{{-}2}\left({-}1+\frac{\alpha_n}{\alpha}\right), \end{equation}

one obtains the equation for $C_{n}$

(3.69)\begin{equation} 2{\rm i}C_{n}' -\beta C_{n}+\frac{1}{\|\varPhi_n\|^2} \left\langle \left(-\varphi_0+\alpha^{{-}1}\frac{\partial \varphi_0}{\partial z}\right),\varPhi_n\right\rangle +|C_{n}|^2C_{n}\frac{\langle Q, \varPhi_n\rangle}{\|\varPhi_n\|^2}=0, \end{equation}

where $Q$ stands for terms with $|C_{n}|^2C_{n}$ in (3.67). Accounting for the presence of the Stokes layers at the front and back walls replaces the coefficient $\alpha ^{-1}$ of the $\partial \varPhi _n/\partial z$ in the linear term by

(3.70)\begin{equation} {\alpha}^{{-}1}\left[1- \frac{2s}{B} \frac{1-i}{\sqrt{2}}\right], \end{equation}

introducing the imaginary part of the order of $\delta ^{-2} s/B\sim 10^{-1}$. After correction of $\beta$ due to dissipation, the equation reads

(3.71a)\begin{gather} 2{\rm i}C_{n}' -\beta_d C_{n}+\frac{1}{\|\varPhi_n\|^2}\left\langle \left(-\varphi_0+\alpha^{{-}1}\frac{\partial \varphi_0}{\partial z}\right),\varPhi_n\right\rangle +|C_{n}|^2C_{n}\frac{\langle Q, \varPhi_n\rangle}{\|\varPhi_n\|^2}=0, \end{gather}

(3.71b)\begin{gather}\beta_d={-}\delta^{{-}2}\left({-}1 +\frac{\alpha_n}{\alpha}\left[1- \frac{2s}{B} \frac{1- i}{\sqrt{2}}\right]\right). \end{gather}

Equation (3.71) is the Duffing equation simplified for the case of the sinusoidal forcing at a small detuning (McCartin Reference McCartin1992).

3.3.3. Qualitative analysis

For the steady-state regime, the terms with slow-time derivatives in (3.69), (3.71) vanish, and the equations become algebraic. Note a limit transition for $\delta \to 0$: in this case, normalized detuning affected by viscosity $\beta _d$ is at least of the order of $\delta ^{-2}s/B\gg 1$. Large $\beta _d$ implies $|C_{n}|\ll 1$ and a negligible contribution of the nonlinear terms. The value of $C_{n}$ coincides with $c_n^{(1)}$ given by (3.30) rescaled by (3.58). Physically, this means that for weak forcing, the amplitude at the resonance is mostly defined by the viscous dissipation rather than by the nonlinear effects.

For the inviscid model, the amplitude of the resonant mode is governed by (3.69) that has real coefficients that change weakly for $\alpha$ in the vicinity of $\alpha _n$. For qualitative analysis, the equation with the coefficients calculated at $\alpha =\alpha _n$ is used. Equation (3.69) has only real solutions, the number of roots varies from one to three, depending on the value of the normalized detuning $\beta$; absolute values of the roots are plotted in figure 8. If three roots exist, only solutions with maximum and minimum absolute values are stable (McCartin Reference McCartin1992). A system that starts from rest reaches the steady state with the smallest possible amplitude (Stoker Reference Stoker1992). The transition from single multiple solutions corresponds to the maximum amplitude for the steady-state regime attained from rest (effective resonance). The transition occurs at the value of $\beta =\beta _{eff}$ that depends on the geometry of the system and defines the variation of the effective resonance frequency $\alpha _{eff}$ with the forcing amplitude

(3.72)\begin{equation} \alpha_{eff}=\alpha_n(1-\beta_{eff}\delta^2)=\alpha_n(1-\beta_{eff}\kappa^{2/3} \varepsilon^{2/3}). \end{equation}

The maximum eigenmode amplitude $|C_{n\,eff}|$ corresponding to $\beta =\beta _{eff}+0$ is presented in table 2 along with the normalized detuning values $\beta _{eff}$ for cases considered in the experiments.

Figure 8.

Amplitude of the resonant eigenmode for the inviscid model as a function of normalized detuning for cases I, II and III (table 1).

Table 2.

Effective resonance parameters.

Equation (3.71) that accounts for the dissipation differs from the inviscid (3.69) by a forcing amplitude-dependent complex coefficient. However, the dependence of the resonant eigenmode amplitude $|C_{n}|$ on the forcing frequency $\alpha$ remains qualitatively the same (figure 9). The position of the effective resonance (maximum amplitude) shifts toward lower frequencies when the forcing amplitude increases. If nonlinearity prevails over dissipation, the response curve becomes asymmetric; multiple solutions appear in a certain range of frequencies, while the phase dependence on frequency becomes S-shaped. The transition from one stable branch to another implies a jump-like change in phase.

Figure 9.

The amplitude (a) and the phase (b) of the resonant eigenmode as a function of detuning, $C_{n}=|C_{n}|\exp ({\rm i}\gamma )$. Black, red and blue lines correspond to subcases Ia, Ib and Ic. Solid, dashed and dotted lines are as in figure 8. Thin lines show the results of the inviscid model; thick ones correspond to Stokes layer dissipation with the value of coefficient adopted from the experiments. Gaps in the thin lines denote the region where the inviscid model ceases to be applicable.

When dissipation is accounted for, for forcing amplitude $\varepsilon$ exceeding the critical level that depends on dissipation, a frequency range appears where multiple solutions exist (figure 10). With the increase in the forcing amplitude, this frequency range expands and shifts downwards. Its upper limit changes slowly, while the lower one quickly leaves the vicinity of the resonance. The increase of dissipation increases the critical value of the amplitude corresponding to multiple solutions and shifts the frequency diapason downwards. The dependence of the effective resonance frequency on the forcing amplitude has three segments. At infinitely small amplitudes, it is downshifted from the eigenfrequency $\alpha _n$ due to dissipation and weakly depends on the amplitude. At the next stage, the effective resonance is defined by the nonlinear effects only; the results for the models with dissipation and without it coincide. For larger $\varepsilon$, the effective resonance frequency depends on the forcing amplitude similar to the inviscid case (3.72), but the coefficient is slightly different.

Figure 10.

The multiple solution existence domain for parameters of case I. Dashed lines show parameters of the maximum amplitude, $s_1$ denotes the values of $s$ in table 1.

3.3.4. Surface elevation

The surface elevation given by (3.55) consists of four terms: shift due to the movement of frame of reference, time-independent offset, the first and second harmonics

(3.73a) \begin{gather} \eta(x)=\frac{1}{2}[{\rm ie}^{{\rm i}t}+\eta_0(x)+\eta_1(x)\,{\rm e}^{{\rm i}t}+\eta_2(x)\,{\rm e}^{{\rm i}2t}]+\text{c.c.} +\text{higher temporal harmonics}, \end{gather}

(3.73b)\begin{gather}\eta_0={-}\frac{\delta^2}{\varepsilon}\alpha\left\{\frac{|C_n|^2}{2} \left[\frac{1}{2}\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n -\alpha\varPhi_n\frac{\partial \varPhi_n}{\partial z}\right]+O\left(\delta,\frac{\varepsilon}{\delta}\right)\right\}, \end{gather}

(3.73c) \begin{align} \eta_1&={-}\frac{{\rm i}\delta}{\varepsilon}\alpha\left\{C_n\varPhi_n+\delta^2\left[ F_{1} -\varphi_0 + \frac{|C_{n}|^2C_{n}}{4}\boldsymbol{\nabla}f_2 \boldsymbol{\cdot} \boldsymbol{\nabla}\varPhi_n -\frac{\alpha |C_{n}|^2|C_n}{2} \frac{\partial f_2\varPhi_n}{\partial z}\right.\right.\nonumber\\ &\quad \left.+\,\frac{|C_n|^2C_n}{4}\left(-\frac{\alpha}{2}\varPhi_n\frac{\partial }{\partial z} (\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n) -\alpha(\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n) \frac{\partial \varPhi_n}{\partial z}+\frac{3\alpha^2}{2}\varPhi_n^2\frac{\partial^{2} \varPhi_n}{\partial z^{2}} \right) \right]\nonumber\\ &\quad \left.+\,O(\delta^3,\delta\varepsilon) \vphantom{\frac{|C_{n}|^2C_{n}}{4}}\right\}, \end{align}

(3.73d)\begin{equation} \eta_2={-}\frac{\delta^2}{\varepsilon}\alpha\left\{2{\rm i} F_2+\frac{C_n^2}{2}\left[\frac{1}{2}(\boldsymbol{\nabla}\varPhi_n\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi_n) +\alpha\varPhi_n\frac{\partial \varPhi_n}{\partial z}\right] +{\rm i}\frac{\varepsilon}{\delta} C_n \alpha \varPhi_n +O(\delta,\varepsilon)\right\}. \end{equation}

The spatial modes are

(3.74a,b)\begin{gather} \eta_0=\sum_{m=1}^{\infty} b_m^{(0)}\cos\left(\frac{m}{n}(x+{\rm \pi})\right),\quad \eta_1=\sum_{m=1}^{\infty} b_m^{(1)}\cos\left(\frac{m}{n}(x+{\rm \pi})\right), \end{gather}

(3.74c)\begin{gather}\eta_2=\sum_{m=1}^{\infty} b_m^{(2)}\cos\left(\frac{m}{n}(x+{\rm \pi})\right), \end{gather}

(3.74d,e)\begin{gather}b_m^{(1)}=| b_m^{(1)}|\exp \left[{\rm i}\left(\theta_m^{(1)}+\frac{\rm \pi}{2}\right)\right],\quad b_m^{(2)}=| b_m^{(2)}|\exp {\rm i}\theta_m^{(2)}. \end{gather}

The shift in defining the phases for the first harmonic is consistent with the linear theory.

The leading term defining the surface elevation amplitude at the antinode at the first harmonic is $\delta |C_n|/\varepsilon$. At the effective resonance, it is $\delta |C_{n\text { eff}}|/\varepsilon$ and scales as $\delta ^{-2}$, see (3.50). The resonant eigenmode constitutes the major contribution to the first harmonic. The $2n$th and $4n$th eigenmodes dominate the second harmonic. The $2n$th eigenmode is also present for the nonlinear standing waves in the pure rectangular cavity (Stiassnie & Shemer Reference Stiassnie and Shemer1984); the $4n$th eigenmode arises due to deviation of the eigenfunctions from sinusoidal shape caused by the finite size of the wavemaker. The relative displacement of the wavemaker manifests itself in the last term of the second harmonic only.