Positive gravity and capillary modal restoring terms

Use $\mathbb{C}:=\mathbb{R}[X]\mathrm{/}(X^2+1)$ with $i:=[X]$, the series-defined complex exponential $\exp w:={\sum }_{n=0}^{\mathrm{\infty }}{w}^n\mathrm{/}n!$, and $\pi$ defined as twice the least positive zero of cosine.

Fix periods $L_1,L_2>0$ metres and let ${\mathbb{T}}_L^2:={\mathbb{R}}^2\mathrm{/}(L_1\mathbb{Z}\times L_2\mathbb{Z})$. Define horizontal derivatives by differentiating the $L$-periodic lift to ${\mathbb{R}}^2$, and set ${\mathrm{\Delta }}_h:={\mathrm{\partial }}_{x_1}^2+{\mathrm{\partial }}_{x_2}^2$. Fix an integer lattice mode $m=(m_1,m_2)\in {\mathbb{Z}}^2\setminus \{(0,0)\}$ and define its wavevector and magnitude by $\mathbf{k}:=2\pi (m_1\mathrm{/}{L}_1,m_2\mathrm{/}{L}_2)$ in inverse metres and $k:=\mathrm{\mid }\mathbf{k}\mathrm{\mid }>0$. For any representative $\tilde{x}\in {\mathbb{R}}^2$ of $[x]\in {\mathbb{T}}_L^2$, define the horizontal mode factor $q_m^h([x]):=\exp (i\mathbf{k}\cdot \tilde{x})$. A change of representative by $(L_1{n}_1,L_2{n}_2)$ changes its exponent by $2\pi iN$, where $N=m_1{n}_1+m_2{n}_2\in \mathbb{Z}$; the exponential addition law and $\exp (i\pi )=-1$ give $\exp (2\pi iN)=1$ for every integer $N$ (by repeated addition for $N\ge 0$ and $\exp (w)\exp (-w)=1$ for $N<0$), so $q_m^h$ is well-defined. The identity $\exp (i\mathbf{k}\cdot \tilde{x})\exp (-i\mathbf{k}\cdot \tilde{x})=\exp (0)=1$ also shows $q_m^h$ never vanishes. Its periodic lift has ${\mathrm{\Delta }}_h{q}_m^h=-k^2{q}_m^h$. Thus, for any unrestricted nonzero formal coefficient amplitude $H\in \mathbb{C}$ in metres, the nonzero horizontal mode ${\eta }_m:=H{q}_m^h$ satisfies ${\mathrm{\Delta }}_h{\eta }_m=-k^2{\eta }_m$. For real $s$, write $e^s:=\exp (s)$ and define $\sinh s:=(e^s-e^{-s})\mathrm{/}2$, $\cosh s:=(e^s+e^{-s})\mathrm{/}2$, and $\tanh s:=\sinh s\mathrm{/}\cosh s$. Fix $h,\rho ,g,\gamma >0$, where $h$ is depth in metres, $\rho$ is constant liquid mass density in kilograms per cubic metre, $g$ is the magnitude of the downward gravitational acceleration $-g{e}_z$ in metres per second squared, and $\gamma$ is the constant surface-tension coefficient in newtons per metre; $e_z$ is the upward unit vector. The vertical coordinate $z$ is upward, with the flat reference surface at $z=0$ and the fixed bed at $z=-h$. Let $I\subseteq \mathbb{R}$ be the model's nonempty open time interval in seconds. The adopted first-order model is the one specified in the Postulate titled “Linear capillary free-surface model.” Its real coefficient fields are $\eta :{\mathbb{T}}_L^2\times I\to \mathbb{R}$ in metres and $\varphi :{\mathbb{T}}_L^2\times [-h,0]\times I\to \mathbb{R}$ in metres squared per second; in periodic-lift coordinates they have continuous partial derivatives through total orders two and three, respectively, with one-sided continuous extensions at $z=-h,0$. Its equations in the selected periodic cell are ${\mathrm{\Delta }}_h\varphi +{\mathrm{\partial }}_z^2\varphi =0$ for $-h<z<0$, ${\mathrm{\partial }}_z\varphi (x,-h,t)=0$, ${\mathrm{\partial }}_t\eta (x,t)={\mathrm{\partial }}_z\varphi (x,0,t)$, and ${\mathrm{\partial }}_t\varphi (x,0,t)+g\eta (x,t)-\frac{\gamma }{\rho }{\mathrm{\Delta }}_h\eta (x,t)=0$. For $\omega \in \mathbb{C}$ in inverse seconds define $q_m([x],t):=q_m^h([x])\exp (-i\omega t)$ for $t\in I$. A nonzero complex mode at index $m$ and angular frequency $\omega$ is a pair of fields ${\eta }_c:{\mathbb{T}}_L^2\times I\to \mathbb{C}$ and ${\varphi }_c:{\mathbb{T}}_L^2\times [-h,0]\times I\to \mathbb{C}$ given by ${\eta }_c([x],t)=H{q}_m([x],t)$ and ${\varphi }_c([x],z,t)=F(z)q_m([x],t)$ and satisfying the complex-linear extension of those equations, with $H\in \mathbb{C}\setminus \{0\}$ in metres and $F\in C^2([-h,0],\mathbb{C})$ in metres squared per second; here $F$ is continuous on $[-h,0]$, twice continuously differentiable on $(-h,0)$, and its first two derivatives extend continuously to the endpoints. The mode definition and dispersion result are those of the theorem titled “Linear capillary-gravity wave dispersion.” Its frequency relation is ${\omega }^2=(gk+(\gamma \mathrm{/}\rho )k^3)\tanh (kh)>0$, since $h,k,g,\gamma ,\rho >0$ and $\tanh (kh)>0$. Thus the gravity coefficient $g$ and capillary coefficient $\gamma k^2\mathrm{/}\rho$ are both positive, and their contributions enter additively in the modal factor $g+\gamma k^2\mathrm{/}\rho$. No finite-displacement constraint is imposed on the formal coefficient: optional geometric displacements are scaled by a sufficiently small dimensionless factor on compact time windows as in the postulate. This is a symbolic conditional model example; it reports no observation or numerical measurement.

MIT OpenCourseWare 18.354, Nonlinear Dynamics II: Continuum Systems, §23.1, printed pp. 104–105

Georgi, The Physics of Waves, §11.5.2, printed pp. 290–291

Fitzpatrick, Fluid Mechanics, “Surface Tension,” equations (11.119)–(11.121)