# Faraday Waves in a Low-Viscosity Fluid Covered with a Floating Elastic Sheet

Vahideh Sardari<EMAIL_ADDRESS>Department of Physics, Institute for
Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran  Leila
Bahmani<EMAIL_ADDRESS>Department of Physics, Institute for Advanced
Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran  Maniya Maleki
<EMAIL_ADDRESS>Department of Physics, Institute for Advanced Studies in
Basic Sciences (IASBS), Zanjan 45137-66731, Iran

###### Abstract

The standing surface waves in a rectangular vertically oscillating vessel
filled with water (Faraday waves) in the presence of a floating elastic sheet
are studied experimentally and theoretically . The threshold amplitude of the
instability and the wavelength of the patterns are measured as a function of
the frequency. A theoretical model based on Hamiltonian method is used to
describe the system. Using the experimental attenuation coefficients, we see a
very good agreement between the theory and the experiment for threshold
amplitude. Also the dispersion relation obtained from the theory is consistent
with experiments.

††preprint: APS/123-QED

##  I Introduction

When a vessel containing a liquid is subjected to vertical oscillation,
nonlinear standing waves form on the liquid interface, which were first
reported by Michael Faraday [1] and are called ”Faraday Waves” after him. Many
studies have been performed on Faraday waves, investigating their patterns [2,
3, 4, 5, 6], threshold amplitude for their formation, and their wavelength [7,
3, 8]. Other studies have investigated the effect of different parameters like
the viscosity [9, 10, 11] and surface tension [12, 13] of the liquid, the
filling depth, and the shape of the container [9, 14, 15, 16, 17, 18]. Several
theories have been developed for the formation mechanism of the Faraday waves,
namely the linear theory of an ideal fluid by Benjamin and Urcell [19], the
linear theory of a viscous liquid by Kumar and Tuckerman [20], and the
Lagrangian method of John Miles [21, 22, 23, 24, 25]. Experimental studies
have been also performed on the subject [23, 9, 26, 27, 28].

In this paper, we have studied Faraday waves in a rectangular vessel filled
with water, to which we added a floating thin elastic sheet. This eliminates
the effect of surface tension and introduces elastic effects instead. Waves
forming on floating elastic sheets have been studied in different researches,
and have applications in stability analysis of ocean ice sheets or floating
constructions [29, 30, 31, 32, 33]. We have studied the standing waves on a
floating elastic sheet, which covers almost completely the water surface but
is not pinned to the vessel. The presence of the sheet affects the dynamic of
the system and changes the wavelength and threshold amplitude of the waves. We
first propose a theoretical model to predict the wavelength and amplitude of
the waves as a function of driving frequency. The we present the experimental
results on the same problem and compare them with theory.

##  II Theory

###  II.1 Lagrangian

For modeling the system, we use the Lagrangian method used by John Miles [23,
24]. Considering a vorticity-free incompressible flow, the velocity field
$\vec{v}$ with respect to the oscillating liquid container can be written as
the gradient of a scalar field $\phi(x,y,z;t)$ , where $x$, $y$ and $z$ are
the Cartesian coordinates and $t$ is time. $\phi$ is defined in the volume of
the liquid, i. e.

$\vec{r}=(x,y)\in S,\\\ -d<z<\eta,$ (1)

where $S$ is the container cross section (a rectangle in our case), and
$\eta(\vec{r};t)$ gives the vertical displacement $z$ of the free surface
points with respect to the plane of the level surface. $\phi$ should satisfy

$\displaystyle\vec{v}=\vec{\nabla}\phi,$ (2) $\displaystyle\nabla^{2}\phi=0.$
(3)

For standing waves, we can separate the local and temporal variables of $\eta$
and $\phi$ as follows

$\displaystyle\eta(\vec{r};t)$ $\displaystyle=$
$\displaystyle\sum_{n}\eta_{n}(t)\psi_{n}(\vec{r}),$ (4)
$\displaystyle\phi(\vec{r},z;t)$ $\displaystyle=$
$\displaystyle\sum_{n}\phi_{n}(t)\chi_{n}(\vec{r},z),$ (5)

where $\eta_{n}$ and $\phi_{n}$ are generalized coordinates, and $\psi_{n}$
and $\chi_{n}$ are orthogonal eigenfunctions in the linear approximation.
[23].

The normal velocity on the bottom and walls of the container must be zero,
thus we have the boundary condition

$\hat{n}.\vec{\nabla}\phi=0$ (6)

on this area. The dynamic boundary condition on the free surface reads

$v_{z}=\frac{\partial\phi}{\partial z}=\frac{\partial\eta}{\partial
t}+\vec{v}.\vec{\nabla}\eta.$ (7)

In the linear approximation, the last term is negligible and we can write the
equation to the leading term as

$\frac{\partial\phi}{\partial z}\approx\frac{\partial\eta}{\partial
t},\;\;\;\left(z=\eta(\vec{r})\approx 0\right).$ (8)

Considering the container to have a rectangular cross section with walls at
$x=0,x=L_{x}$; $y=0,y=L_{y}$ and $z=-d$, one can write $\phi$ as a Fourier
series and obtain the from of functions $\chi_{n}$ as

$\displaystyle\begin{split}\chi_{n}&=A_{n}\cos(k_{nx}x)\cos(k_{ny}y)\times\\\
&\text{ sech}(k_{n}d)\cosh k_{n}(z+d),\\\
\vec{k}_{n}&=(k_{nx},k_{ny})=(\frac{n\pi}{L_{x}},\frac{n\pi}{L_{y}}).\end{split}$
(9)

Substituting equation 9 in equation 8, we will have

$(\nabla^{2}+k_{n}^{2})\psi_{n}=0,$ (10)

Using the variational principle, one can write

$\phi_{m}=\sum_{j,l}\kappa^{-1}_{ml}d_{jl}\dot{\eta}_{j},$ (11)

where

$\displaystyle d_{mn}$ $\displaystyle=$ $\displaystyle S^{-1}\iint
dS(\chi_{n})_{z=\eta}\psi_{m},$ (12) $\displaystyle\kappa_{mn}$
$\displaystyle=$ $\displaystyle S^{-1}\iint dS\int
dz\vec{\nabla}\chi_{m}.\vec{\nabla}\chi_{n}.$ (13)

Then we can calculate the Lagrangian of the system as a function of
generalized coordinates $\eta_{n}$ by writing $T_{0}$, the kinetic energy and
$V_{0}$, the potential energy as [21]

$\displaystyle T_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{m,n}\rho
Sa_{mn}\dot{\eta}_{m}\dot{\eta}_{n},$ (14) $\displaystyle V_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{n}\rho
Sg^{\prime}\eta_{n}^{2},$ (15)

where $\rho$ is the fluid density, $g^{\prime}$ is the sum of gravitational
acceleration $g$ and container acceleration $\ddot{z}_{0}$, and $\pmb{a}$ is a
symmetric matrix defined as

$\pmb{a}=\pmb{d}\pmb{\kappa^{-1}}\pmb{d^{\prime}},$ (16)

The elements of $\pmb{a}$ can be expanded as taylor series of $\eta_{n}$.

$a_{mn}=\delta_{mn}a_{m}+\sum_{l}a_{lmn}\eta_{l}+\frac{1}{2}\sum_{j,l}a_{jlmn}\eta_{j}\eta_{l}+...,$
(17)

where

$a_{n}=k_{n}^{-1}\coth k_{n}d\equiv\frac{g}{\omega_{n}^{2}},$ (18)

and other series coefficients are related to correlation integrals $C$ and $D$

$\displaystyle a_{lmn}$ $\displaystyle=$ $\displaystyle
C_{lmn}-D_{lmn}a_{m}a_{n},$ (19) $\displaystyle a_{jlmn}$ $\displaystyle=$
$\displaystyle-D_{jjlmn}(a_{m}+a_{n})+2D_{jmi}D_{lni}a_{l}a_{m}a_{n},$ (20)

which are defined as follow [23].

$\displaystyle C_{lmn}$ $\displaystyle=$ $\displaystyle S^{-1}\iint
dS\psi_{l}\psi_{m}\psi_{n}dS,$ (21) $\displaystyle C_{jlmn}$ $\displaystyle=$
$\displaystyle S^{-1}\iint dS\psi_{j}\psi_{l}\psi_{m}\psi_{n}dS,$ (22)
$\displaystyle D_{lmn}$ $\displaystyle=$ $\displaystyle S^{-1}\iint
dS\psi_{l}\vec{\nabla}\psi_{m}.\vec{\nabla}\psi_{n}dS,$ (23) $\displaystyle
D_{jlmn}$ $\displaystyle=$ $\displaystyle S^{-1}\iint
dS\psi_{j}\psi_{l}\vec{\nabla}\psi_{m}.\vec{\nabla}\psi_{n}dS.$ (24)

Based on these equation, the Lagrangian is calculated as

$\mathcal{L_{0}}=\frac{1}{2}\sum_{n}a_{n}(\dot{\eta}_{n}^{2}-\omega_{n}^{2}\eta_{n}^{2}-\ddot{z}_{0}\eta_{n}^{2})+\frac{1}{2}\sum_{l,m,n}a_{lmn}\eta_{l}\dot{\eta}_{m}\dot{\eta}_{n}\\\
+\frac{1}{4}\sum_{j,l,m,n}a_{jlmn}\eta_{j}\eta_{l}\dot{\eta}_{m}\dot{\eta}_{n}.$
(25)

Now, we should incorporate the terms related to the elastic floating sheet
sheet into the Lagrangian. The elastic potential of the sheet reads

$\begin{split}V_{el}&=\frac{1}{2}\iint K[\nabla^{2}\eta]^{2}dS\\\
&=\frac{1}{2}\sum_{m,n}\iint
K[\nabla^{2}\psi_{m}][\nabla^{2}\psi_{n}]\eta_{m}(t)\eta_{n}(t)dS\\\
&=\frac{1}{2}\sum_{n}Kk_{n}^{4}\eta_{n}^{2}S,\end{split}$ (26)

where $K$ is the bending rigidity of the sheet.

The potential energy due to the vertical displacement of the sheet can be
written as

$\begin{split}V_{g}&=\frac{1}{2}\iint\sigma[1+\lvert\vec{\nabla}\eta\rvert^{2}]^{1/2}\eta
g^{\prime}dS\\\ &=\frac{\sigma}{2}\sum_{l,m,n}\iint
g^{\prime}\vec{\nabla}\psi_{m}.\vec{\nabla}\psi_{n}\eta_{m}(t)\eta_{n}(t)\psi_{l}(t)\eta_{l}(t)S\\\
&=\frac{\sigma}{2}\sum_{l,m,n}g^{\prime}D_{lmn}\eta_{l}(t)\eta_{m}(t)\eta_{n}(t),\end{split}$
(27)

where $\sigma$ is the mass per unit area of the sheet.

The Kinetic energy of the sheet is given by

$\begin{split}T_{s}&=\frac{\sigma}{2}\iint[1+\lvert\vec{\nabla}\eta\rvert^{2}]^{1/2}(\vec{\nabla}\phi)^{2}_{z=\eta}dS\\\
&=\frac{\sigma}{2}\sum_{m,n}\iint\vec{\nabla}\chi_{m}.\vec{\nabla}\chi_{n}dS+\frac{\sigma}{4}\sum_{j,l,m,n}\iint\vec{\nabla}\chi_{m}.\vec{\nabla}\chi_{n}\times\\\
&\vec{\nabla}\psi_{j}.\vec{\nabla}\psi_{l}\eta_{j}(t)\eta_{l}(t)\phi_{m}(t)\phi_{n}(t)dS\\\
&=\frac{\sigma}{2}\sum_{m,n}S\phi_{m}\phi_{n}\kappa^{\prime}_{mn}+\frac{\sigma}{4}\sum_{j,l,m,n}SM_{jlmn}\phi_{m}\phi_{n}\eta_{j}\eta_{l},\end{split}$
(28)

where

$M_{jlmn}=k_{n}^{2}k_{l}^{2}\eta_{j}\eta_{l}C_{jlmn}+D_{mnjl}\kappa_{m}\kappa_{n}.$
(29)

The elements of $\pmb{\kappa^{\prime}}$ are defined as

$\kappa^{\prime}_{mn}=\frac{1}{S}\iint\vec{\nabla}\chi_{m}.\vec{\nabla}\chi_{n}dS,$
(30)

and be expanded in power series of $\eta$ as follows

$\displaystyle\kappa^{\prime}_{mn}$ $\displaystyle=$
$\displaystyle\delta_{mn}(k_{m}^{2}+\kappa_{m}^{2})+\sum_{l}[D_{lmn}(\kappa_{m}+\kappa_{n})$
(31) $\displaystyle+$ $\displaystyle
C_{lmn}(k_{m}^{2}\kappa_{n}+k_{n}^{2}\kappa_{m})]\eta_{l}+\frac{1}{2}\sum_{j,l}[D_{jlmn}(k_{m}+\kappa_{n}T_{m})$
$\displaystyle+$ $\displaystyle
C_{jlmn}k_{n}(k_{m}^{2}+\kappa_{m}\kappa_{n})]\eta_{j}\eta_{l}+...,$
$\displaystyle\kappa_{m}$ $\displaystyle=$ $\displaystyle
a_{m}^{-1}=k_{m}\tanh k_{m}d\equiv k_{m}T_{m}.$ (32)

Using equation 11, $T_{s}$ can be written as

$T_{s}=\frac{\sigma}{2}\sum_{m,n}b_{mn}\dot{\eta}_{m}\dot{\eta}_{n}+\frac{\sigma}{4}\sum_{j,l,m,n}r_{jlmn}\eta_{j}\eta_{l}\dot{\eta}_{m}\dot{\eta}_{n}.$
(33)

The matrix $\pmb{b}$ and the coefficient $r$ are defined as follows

$\displaystyle b_{mn}$ $\displaystyle=$
$\displaystyle\sum_{j,l}\kappa^{\prime}_{mj}\kappa^{-1}_{jl}a_{ln},$ (34)
$\displaystyle r_{jlmn}$ $\displaystyle=$
$\displaystyle\kappa^{-1}_{mn}M_{jlmn}a_{mn}.$ (35)

The elements of $\pmb{b}$ can also be expanded as

$\displaystyle b_{mn}$ $\displaystyle=$
$\displaystyle(k_{n}^{2}a_{n}^{2}+1)\delta_{mn}+\sum_{l}b_{lmn}\eta_{l}$ (36)
$\displaystyle+$ $\displaystyle\sum_{j,l}b_{jlmn}\eta_{j}\eta_{l},$
$\displaystyle b_{lmn}$ $\displaystyle=$
$\displaystyle(k_{n}^{2}a_{n}^{2}+1)(a_{lmn}\kappa_{n}-1)$ (37)
$\displaystyle+$ $\displaystyle 2(D_{lmn}+C_{lmn}k_{n}^{2}),$ $\displaystyle
b_{jlmn}$ $\displaystyle=$
$\displaystyle\frac{1}{2}a_{jlmn}(\kappa_{n}+k_{n}^{2}a_{n})+D_{jlmn}a_{n}(k_{n}a_{n}+T_{n})$
(38) $\displaystyle+$ $\displaystyle C_{jlmn}k_{n}(k_{m}^{2}a_{n}^{2}+1).$

Since in equation 33 the last term has is $r_{jlmn}\times O(\eta^{4})$,we keep
only the leading term of $r_{jlmn}$, which is

$r_{jlmn}=2(D_{jlmn}+C_{mlj}k_{n}^{2}k_{j}^{2}a_{n}^{2})\delta_{mn}.$ (39)

Defining $\gamma\equiv\sigma/\rho$, the Lagrangian of the system with the
elastic sheet divided by $\rho S$ is hence given by

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\frac{1}{\rho
S}(T-V)$ (40) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j,l,m,n}\left(a_{mn}+\gamma
b_{mn}+\frac{\gamma}{2}r_{jlmn}\eta_{j}\eta_{l}\right)\dot{\eta}_{j}\dot{\eta}_{l}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{m,n}\left((g^{\prime}\delta_{mn}+\frac{Kk_{n}^{4}}{\rho}\right)\eta_{m}\eta_{n}.$

Substituting the expansions and keeping up to the second order, we have

$\begin{split}\mathcal{L}&=\frac{1}{2}\sum_{n}\frac{a_{n}}{2}\left[\left(1+\gamma
k_{n}^{2}a_{n}\right)\dot{\eta}_{n}^{2}-\omega_{n}^{2}\eta_{n}^{2}\right]\\\
&-\frac{1}{2}\sum_{n}\left[\left(\ddot{z}_{0}+\frac{Kk_{n}^{4}}{\rho}\right)\eta_{n}^{2}+\gamma\dot{\eta}_{n}^{2}\right]\end{split}$
(41)

###  II.2 Threshold Amplitude

Miles showed that if the reservoir oscillates vertically with the angular
velocity $2\omega$ and amplitude $a_{0}$

$z_{0}=a_{0}\cos 2\omega t,$ (42)

for low amplitudes $\omega^{2}|a_{0}|\ll g$ in the nonlinear regime, only the
harmonic and sub-harmonic responses are excited, with slowly varying
amplitudes [25]. Hence, each mode $\eta_{n}$ can be written as

$\begin{split}\eta_{n}&=\delta_{1n}\lambda\left[p(\tau)\cos\omega
t+q(\tau)\sin\omega t\right]\\\
&+\frac{\lambda^{2}}{a_{1}}\left[A_{n}(\tau)\cos 2\omega t+B_{n}(\tau)\sin
2\omega t+C_{n}(\tau)\right],\end{split}$ (43)

where $p$, $q$, $A_{n}$, $B_{n}$ and $C_{n}$ are the slowly varying amplitudes
and

$\displaystyle\epsilon=\frac{a_{0}}{a_{1}},\quad 0<\epsilon\leqslant 1,$ (44)
$\displaystyle\tau=\epsilon\omega t,$ (45)

and $\lambda=O(\epsilon^{1/2}a_{1})$ is a length scale. Putting this in the
Lagrangian, we can write the action integral in one period and divide it to
the period to find

$\displaystyle\langle\mathcal{L}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon
g\lambda^{2}\bigg{[}\beta(p^{2}+q^{2})+\left(1+\gamma(a_{1}k_{1}^{2}+a_{1}^{-1})\right)$
$\displaystyle\times$
$\displaystyle(\dot{p}q-\dot{q}p)+\left(p^{2}-q^{2}\right)\bigg{]}$
$\displaystyle+$
$\displaystyle\frac{g\lambda^{4}}{2a_{1}^{2}}\bigg{[}\frac{1}{16}a_{1}(p^{2}+q^{2})^{2}(a_{1111}+\gamma
b_{1111}+\frac{\gamma}{4}r_{1111})$ $\displaystyle+$
$\displaystyle\sum_{n}\bigg{\\{}\frac{\Omega_{n}}{2}(A_{n}^{2}+B_{n}^{2})-C_{n}^{2}(1+k_{n}^{4}l_{*}^{2})$
$\displaystyle+$
$\displaystyle\left(a_{11n}+\frac{\gamma}{2}(b_{11n}+b_{1n1})-\frac{1}{4}(a_{n11}+\gamma
b_{n11})\right)$ $\displaystyle\times$
$\displaystyle\left(A_{n}(p^{2}-q^{2})+2B_{n}pq\right)+\frac{C_{n}}{2}(a_{n11}+\gamma
b_{n11})(p^{2}+q^{2})\bigg{]}.$

In the above equation, dot means the derivative with respect to $\tau$ and
parameters $\beta$, $l_{*}$ and $\Omega_{n}$ are defined as follows.

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{1}{2\epsilon}\left[\frac{\left(1+\gamma(a_{1}k_{1}^{2}+a_{1}^{-1})\right)\omega^{2}-\omega_{1}^{2}}{\omega_{1}^{2}}-k_{1}^{4}l_{*}^{4}\right],$
(47) $\displaystyle\Omega_{n}$ $\displaystyle=$ $\displaystyle
4\frac{a_{n}}{a_{1}}\left(1+\gamma(a_{n}k_{n}^{2}+a_{n}^{-1})\right)-k_{n}^{4}l_{*}^{4}-1,$
(48) $\displaystyle l_{*}$ $\displaystyle=$ $\displaystyle\left(\frac{K}{\rho
g}\right)^{1/2},$ (49)

Using the principle of stationary action, $\langle\mathcal{L}\rangle$ should
not change with variation of amplitudes $p$, $q$, $A_{n}$, $B_{n}$ and $C_{n}$
[25]. This condition gives the amplitudes as

$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\frac{-(p^{2}-q^{2})\left(a_{11n}+\frac{\gamma}{2}(b_{11n}+b_{1n1})-\frac{1}{4}(a_{n11}+\gamma
b_{n11})\right)}{\Omega_{n}},$ $\displaystyle B_{n}$ $\displaystyle=$
$\displaystyle\frac{-2pq\left(a_{11n}+\frac{\gamma}{2}(b_{11n}+b_{1n1})-\frac{1}{4}(a_{n11}+\gamma
b_{n11})\right)}{\Omega_{n}},$ $\displaystyle C_{n}$ $\displaystyle=$
$\displaystyle\frac{(a_{n11}+\gamma
b_{n11})(p^{2}+q^{2})}{4(1+k_{n}^{4}l_{*}^{2})}.$ (50)

Substitution of the amplitudes in the average Lagrangian, we obtain

$\displaystyle\langle\mathcal{L}\rangle$ $\displaystyle=$
$\displaystyle\epsilon
g\ell^{2}\bigg{\\{}\frac{1}{2}(\dot{p}q-p\dot{q})\left[1+\gamma(a_{1}k_{1}^{2}+a_{1}^{-1}\right]+H(p,q)\bigg{\\}}$
(51) $\displaystyle+$ $\displaystyle O(\epsilon^{2}),$

where the length scale $\ell$ and the Hamiltonian $H$ are defined as follows

$\displaystyle\ell$
$\displaystyle=2\left(\frac{\epsilon}{|f|}\right)^{1/2}k_{1}^{-1}\tanh
k_{1}d,$ (52) $\displaystyle H$
$\displaystyle(p,q)=\frac{1}{2}(\beta+1)p^{2}+\frac{1}{2}(\beta-1)q^{2}+\frac{s}{4}(p^{2}+q^{2})^{2},$
(53)

and $s$ is

$\displaystyle s$ $\displaystyle=$ $\displaystyle\text{sgn}(f),$ (54)
$\displaystyle f$ $\displaystyle=$
$\displaystyle\frac{a_{1}\tanh^{4}k_{1}d}{2}\Bigg{\\{}a_{1111}+\gamma
b_{1111}+\frac{\gamma}{4}r_{1111}$ $\displaystyle+$
$\displaystyle\frac{(a_{n11}+\gamma b_{n11})^{2}}{1+k_{n}^{4}l_{*}^{2}}$
$\displaystyle+$
$\displaystyle\frac{\left[4a_{11n}+8\gamma(b_{11n}+b_{1n1})-(a_{n11}+\gamma
b_{n11})\right]^{2}}{8a_{n}\left[a_{1}(1+k_{n}^{4}l_{8}^{2})-1-\gamma(a_{1}k_{1}^{2}+a_{1}^{1})\right]}\Bigg{\\}}.$

In order to take into account the viscous dissipation in the system, we have
to add a dissipation function $D$ to the Hamiltonian

$D=\frac{1}{2}\alpha(p^{2}+q^{2}),\quad\alpha=\frac{\delta}{\epsilon},$ (56)

in which $\delta$ is the damping ratio of a free wave of frequency $\omega$
[25]. The evolution equations are thus obtained as

$\displaystyle\dot{p}$ $\displaystyle+\alpha p+(\beta-1+p^{2}+q^{2})q=0,$ (57)
$\displaystyle\dot{q}$ $\displaystyle+\alpha q-(\beta+1+p^{2}+q^{2})p=0.$ (58)

For $\alpha>0$, the only fixed point of the above equations, is the point
$p=q=0$, which corresponds to the nonlinear response of the system as a sub-
harmonic internal resonance [25]. To determine whether the fixed point is
stable or not, we apply a small disturbance $\propto\exp{\sigma\tau}$ and find
the stability condition from the characteristic equation

$\Delta=\begin{vmatrix}\sigma+\alpha+2pq&\beta-1+p^{2}+3q^{2}\\\
-(\beta+1+q^{2}+3p^{2})&\sigma+\alpha-2pq\end{vmatrix}=0.$ (59)

For $p=q=0$, the solution of the characteristic equation is

$\sigma=-\alpha+(1-\beta^{2})^{1/2},$ (60)

and $\sigma=0$ gives the neutral stability condition

$\beta=\pm(1-\alpha^{2})^{1/2}.$ (61)

This condition gives us the threshold amplitude $a_{0}$ for the Faraday waves.
substituting equation 18, 45, 48 and 49, we find

$a_{0}=(k\text{ tanh }kd)^{-1}\left\\{\delta^{2}+\left[\frac{(1+2\gamma
k\text{ coth }2kd)\omega^{2}-\omega^{\prime
2}}{2\omega^{2}}-\frac{k^{4}K}{2\rho g}\right]^{2}\right\\}^{1/2},$ (62)

where

$\omega^{\prime}=\omega(1-\delta).$ (63)

This equation becomes equivalent to the threshold for Faraday waves without
the sheet in the limit of $\sigma\rightarrow 0$ and $K\rightarrow 0$ [23]

$a_{0}=(k\text{ tanh
}kd)^{-1}\left\\{\delta^{2}+\left[\frac{\omega^{2}-\omega^{\prime
2}}{2\omega^{2}}\right]^{2}\right\\}^{1/2}$ (64)

As we expect, the threshold is higher in the presence of the elastic sheet,
due to the effects of the weight and the elasticity of the sheet.

###  II.3 Dispersion Relation

To find the dispersion relation of the waves, we first write the kinetic and
potential energies of the system. Using equations 14,15, 26, 27 and 28 we can
write

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int\rho
v^{2}dV+\frac{1}{2}\int\sigma v^{2}dA,$ (65) $\displaystyle V$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int\left[\rho
g\eta^{2}+K(\nabla^{2}\eta)^{2}+2\sigma
g\eta\left(1+|\vec{\nabla}\eta|^{2}\right)^{1/2}\right]dS,$

where $dV$ is the volume element inside the liquid and $dA$ is the surface
element on the sheet. Using equation 2, we have

$T=\frac{\rho}{2}\int|\vec{\nabla}\phi|^{2}dV+\frac{\sigma}{2}\int\left[1+(\vec{\nabla}\eta)^{2}\right]^{1/2}|\vec{\nabla}\phi|_{z=\eta}^{2}dS.$
(66)

Then, we use the Fourier transforms of the functions $\eta(\vec{r};t)$ and
$\phi(\vec{r},z;t)$, which are the continuous forms of equations 4 and 5.

$\eta(\vec{r};t)=\int\frac{d\vec{k}}{(2\pi)^{2}}\;\tilde{\eta}(\vec{k};t)e^{i\vec{k}.\vec{r}},$
(67)

and using 9 we have

$\phi(\vec{r},z;t)=\int\frac{d\vec{k}}{(2\pi)^{2}}\;\tilde{\phi}(\vec{k};t)\cosh
k(z+d)\text{ sech }kd\;e^{i\vec{k}.\vec{r}}.$ (68)

Substituting the above transforms into the boundary condition 8, we obtain a
relation between the Fourier amplitudes as

$\tilde{\phi}(\vec{r};t)=\frac{\tilde{\eta}_{t}(\vec{r};t)}{k\text{ tanh
}kd},$ (69)

where $\tilde{\eta}_{t}$ is the partial derivative of $\tilde{\eta}$ with
respect to $t$. Writing the kinetic energy $T$ in terms of $\tilde{\eta}$ up
to the second order in $\eta$, we derive

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{\rho}{2}\int\frac{d\vec{k}}{(2\pi)^{2}}\frac{|\tilde{\eta}_{t}(\vec{k};t)|^{2}}{k\text{
tanh }kd}$ $\displaystyle+$
$\displaystyle\frac{\sigma}{2}\int\frac{d\vec{k}}{(2\pi)^{2}}|\tilde{\eta}_{t}(\vec{k};t)|^{2}(1+\text{coth}^{2}\;kd).$

Likewise, the potential energy $V$ is calculated up to the second order as

$V=\frac{1}{2}\int\frac{d\vec{k}}{(2\pi)^{2}}(\rho
g+Kk^{4})|\tilde{\eta}(\vec{k};t)|^{2}$ (71)

Now, we form the Lagrangian with the constraint that the total surface area of
the sheet is constant, using a Lagrange multiplier $\Lambda$.

$L=T-V-\Lambda\int dA,$ (72)

where

$\int
dA\simeq\int\left(1+\frac{|\vec{\nabla}\eta|^{2}}{2}\right)dS=\int\frac{d\vec{k}}{(2\pi)^{2}}\frac{k^{2}}{2}|\tilde{\eta}(\vec{k};t)|^{2}$
(73)

Thus the Lagrangian can be written as a Fourier transform as follows

$\displaystyle L$ $\displaystyle=$
$\displaystyle\int\frac{d\vec{k}}{(2\pi)^{2}}\left\\{\left[\frac{\rho}{2k\text{
tanh }kd}+\frac{\sigma(1+\text{coth
}kd)}{2}\right]|\tilde{\eta}_{t}(\vec{k};t)|^{2}-\frac{\rho g+Kk^{4}+\Lambda
k^{2}}{2}|\tilde{\eta}(\vec{k};t)|^{2}\right\\}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d\vec{k}}{(2\pi)^{2}}\left(l_{1}(k)|\tilde{\eta}_{t}(\vec{k};t)|^{2}+l_{2}(k)|\tilde{\eta}(\vec{k};t)|^{2}\right).$

The Euler-Lagrange equation for each Fourier mode then is written as

$\frac{\partial L}{\partial\tilde{\eta}(\vec{k};t)}-\frac{d}{dt}\frac{\partial
L}{\partial\tilde{\eta}_{t}(\vec{k};t)}=0,$ (75)

which gives

$\tilde{\eta}_{tt}(\vec{k},t)=\frac{l_{2}(k)}{l_{1}(k)}\tilde{\eta}(\vec{k},t).$
(76)

Considering $\tilde{\eta}(\vec{k},t)$ to be a product of two functions of $t$
and $k$

$\tilde{\eta}(\vec{k},t)=Q(t)R(k),$ (77)

and supposing $Q(t)$ to be a periodic function

$Q(t)=e^{i\omega t},$ (78)

we obtain

$\omega^{2}=\frac{-l_{2}(q)}{l_{1}(q)}.$ (79)

Thus the dispersion relation reads

$\omega=\sqrt{\frac{\rho gk+\Lambda k^{3}+Kk^{5}}{\rho\text{ tanh }kd+2\sigma
k\frac{1+e^{4kd}}{(1+e^{2kd})^{2}}}}.$ (80)

The above equation gives the relation for the non-dissipative case. In order
to take into account the viscous dissipation, we should replace $\omega$ with
$\omega^{\prime}$ [23].

We should also note that equation 79 [23] is written in the continuous form,
although we have quantized values for $(k_{x},k_{y})$ as seen in equation 9
for a bounded rectangle reservoir.

Figure 1: Schematic .

##  III Experimental Data

We performed experiments to validate our theoretical predictions of threshold
amplitude and wavelength. We used a rectangular vessel with dimensions of 33.5
cm $\times$ 14.1 cm and a depth of 2 cm. The vessel was filled with water and
a floating rectangular silicone sheet with the same dimensions and the
thickness of 0.25 mm was put on the water. The vessel was placed on top of a
horizontal speaker. We made oscillations with different frequencies and
amplitudes using a function generator and an amplifier. The oscillation
amplitude was measured accurately with the moire technique. We used two
similar gratings, one attached to the vessel and another one parallel to it
outside the speaker, with a small angle between their grating lines (Fig. 1).
By filming the moire patterns with a fast CCD camera and tracing them with a
Matlab code, we were able to measure the vertical displacement of the vessel
during the time with an accuracy of 50 microns, which gave us the oscillation
amplitude.

To measure the threshold amplitude, we first fixed the oscillation frequency
and then increased the amplitude gradually, until we observed the Faraday
waves. We used a stroboscope to measure the frequency of the waves in each
experiment. We changed the stroboscope frequency until we saw a fixed pattern
on the surface. We observed that the frequency of the Faraday waves was always
half the oscillation frequency. The experimental values of the threshold
amplitude as a function of the driving frequency are shown in Fig. 2.

In order to compare the experimental results with the theory, we must know the
experimental value for $K$. Knowing the Young modulus $E$ and the Poisson
ratio $\nu$ of the sheet, we can calculate the bending rigidity of a sheet
with the thickness $h$ using the formula [34]

Figure 2: The threshold amplitude of the waves vs. the angular frequency.
Black diamonds: experimental data, and blue circles: theoretical relation.

$K=\frac{Eh^{3}}{12(1-\nu^{2})}$ (81)

For elastomers, the Poisson ratio is about $1/2$, which results from the
incompressibility of these materials [35]. To measure the Young modulus, we
used a simple longitudinal stress-strain experiment, which gave us $E=(1.8\pm
0.1)\times 10^{6}$ Pa.

Figure 3: The projection of the laser beam reflected from the surface for (a)
an amplitude less than the threshold and no Faraday waves, and (b) an
amplitude just above the threshold.

Another parameter we need to evaluate is the damping ratio $\delta$, defined
as $\delta=\Gamma/\omega$, $\Gamma$ being the damping rate of the waves [22].
Henderson and Miles [23] showed that all the theoretical works give an
unrealistic value for the damping rate, thus it is better to use the
experimental value in the theoretical threshold formula. We used an almost
vertical laser beam reflected from the surface, and projected the reflection
on a vertical wall using a mirror. As shown in Figure 2, when there is no
Faraday wave formed, a light spot is seen on the wall, while a Lissajous-like
curve forms in the presence of the waves. This enabled us to determine the
exact threshold. In order to determine the damping rate, we filmed the
Lissajous curve with a high enough frame rate (300 or 600 fr/s) to locate the
laser spot in different frames. After the Faraday waves were formed, we turned
off the speaker and let the waves damp. During the damping, the Lissajous
curve became smaller. We plotted the vertical position of the laser spot as a
function of time. As shown in Figure 3, first we have a sinusoidal curve (a),
and after turning off the oscillations, it starts to decay exponentially (b).
Fitting a function $y=A\sin(\omega t+\phi)\exp(-\Gamma t)$ to the experimental
values, we found the damping rate $\Gamma$ for each frequency and hence
calculated $\delta$ in equation 62.

Figure 4: A typical plot of the $y$ coordinate of the laser spot on the wall
as a function of time. (a) the sinusoidal variation with time when the speaker
is on; (b) the sinusoidal function times an exponentially decaying factor.
Points represent the experimental values, and the curve is the fit.

For measuring the wave number $k$, we used the method of Douady and Fauve
[36]. We used the stroboscope as the light source to get a sharp picture and
took pictures of the waves from above. For low frequencies, the patterns were
regular and the wave numbers could be found easily( Figure 5(a)). For high
frequencies, the patterns became irregular and the wavenumber was measured
using the Fourier analysis (Figure 5(b)). Having all the necessary parameters,
we can calculate theoretical amplitudes for each frequency. The results are
shown in Figure 2. There is a very good agreement between the theory and the
experiment, except for forcing frequencies more than 18 Hz. In this range, the
formation of capillary waves at the interface menisci between the water and
the vessel, and between water and the sheet leads to wave formation in very
small amplitudes [37].

Figure 5: The Faraday wave patterns for two different frequencies: (a) 8 Hz
and (b) 25 Hz . The scale bars are 2 cm long.

Figure 6: The wavenumber of the Faraday patterns as a function of the angular
frequency. Black diamonds: experimental data, and blue curve: theoretical
relation.

In figure 6 we have compared the theoretical dispersion relation with the
experiments. The inverse of relation 80 has been plotted as the theoretical
curve, and has been compared to the experimental data derived from the pattern
pictures. We see an overall good agreement. The difference between the theory
and experiments is mainly due to the fact that the theoretical dispersion
relation is obtained for an unbounded system and the modes have been
considered as a continuous spectrum, while in the experiments we have the
quantization of the modes due to the boundary conditions on the vessel sides.

##  IV conclusion

In this paper, we studied the Faraday waves forming in a system containing a
thin elastic sheet floating on water in a rectangular vessel. The threshold
amplitude was calculated theoretically by adding elastic terms to the John
Miles theory. The theory was compared with experimental data and a very good
agreement was observed between them. The dispersion relation of the waves was
obtained theoretically and compared to the experiments, showing a very good
agreement.

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