Linear capillary-gravity wave dispersion

Let $I\subseteq \mathbb{R}$ be a nonempty open interval, with absolute time $t\in I$ measured in seconds, and let $E$ be an oriented three-dimensional Euclidean affine space: its translation vectors form an oriented three-dimensional Euclidean vector space and no origin is distinguished. The Newtonian spacetime is $I\times E$. Define $\mathrm{S}\mathrm{O}(3):=\{Q\in {\mathbb{R}}^{3\times 3}:Q^{\mathsf{T}}Q=I_3,\det Q=1\}$. A Cartesian inertial chart is the event-coordinate map $(t,X)\mapsto (t,{\mathbf{r}}_t(X))$ for which, at some fixed $t_{\mathrm{r}\mathrm{e}\mathrm{f}}\in I$, ${\mathbf{r}}_{t_{\mathrm{r}\mathrm{e}\mathrm{f}}}:E\to {\mathbb{R}}^3$ is an orientation-preserving affine isometry whose linear part lies in $\mathrm{S}\mathrm{O}(3)$ and ${\mathbf{r}}_t(X)={\mathbf{r}}_{t_{\mathrm{r}\mathrm{e}\mathrm{f}}}(X)+w(t-t_{\mathrm{r}\mathrm{e}\mathrm{f}})$ for a constant $w\in {\mathbb{R}}^3$ in metres per second. Thus each fixed-time spatial map is an orientation-preserving affine isometry; inertial charts are related by $t^{\ast }=t+t_0$ and ${\mathbf{r}}^{\ast }=Q\mathbf{r}+vt+a$, with constant $t_0$ in seconds, $Q\in \mathrm{S}\mathrm{O}(3)$, $v\in {\mathbb{R}}^3$ in metres per second, and $a\in {\mathbb{R}}^3$ in metres. In the selected chart, $(x_1,x_2,z)$ are Cartesian coordinates in metres, with $x=(x_1,x_2)$ horizontal and $z$ upward; $e_z$ is the upward unit vector.

Fix depth $h>0$ metres and horizontal periods $L_1,L_2>0$ metres, and define ${\mathbb{T}}_L^2:={\mathbb{R}}^2\mathrm{/}(L_1\mathbb{Z}\times L_2\mathbb{Z})$. A field on this torus is interpreted through its periodic lift $\tilde{f}(\tilde{x})=f([\tilde{x}])$. For fields on products of this torus with $I$ and $[-h,0]$, $C^r$ means that the lifted field has continuous ordinary coordinate partial derivatives through total order $r$, with continuous one-sided extensions at $z=-h,0$. Define ${\mathrm{\nabla }}_h:=({\mathrm{\partial }}_{x_1},{\mathrm{\partial }}_{x_2})$ and ${\mathrm{\Delta }}_h:={\mathrm{\partial }}_{x_1}^2+{\mathrm{\partial }}_{x_2}^2$ on the lifts. The real first-order displacement coefficient is $\eta :{\mathbb{T}}_L^2\times I\to \mathbb{R}$, in metres, and the real velocity-potential coefficient is $\varphi :{\mathbb{T}}_L^2\times [-h,0]\times I\to \mathbb{R}$, in metres squared per second; assume $\eta$ is at least $C^2$ and $\varphi$ at least $C^3$ in this sense. Their coefficients are formal and unrestricted in amplitude. Define the fluid-velocity field $u:=\mathrm{\nabla }\varphi =({\mathrm{\partial }}_{x_1}\varphi ,{\mathrm{\partial }}_{x_2}\varphi ,{\mathrm{\partial }}_z\varphi ):{\mathbb{T}}_L^2\times [-h,0]\times I\to {\mathbb{R}}^3$, in metres per second, with no added uniform velocity component. Take constant density $\rho >0$ kilograms per cubic metre, downward gravitational acceleration $-g{e}_z$ with $g>0$ metres per second squared, and surface tension $\gamma >0$ newtons per metre. The selected model is the first-order capillary free-surface model of the Postulate titled “Linear capillary free-surface model”: its equations on the flat reference slab ${\mathbb{T}}_L^2\times (-h,0)$ are

${\mathrm{\Delta }}_h\varphi +{\mathrm{\partial }}_z^2\varphi =0(-h<z<0),{\mathrm{\partial }}_z\varphi (x,-h,t)=0,$

${\mathrm{\partial }}_t\eta (x,t)={\mathrm{\partial }}_z\varphi (x,0,t),{\mathrm{\partial }}_t\varphi (x,0,t)+g\eta (x,t)-\frac{\gamma }{\rho }{\mathrm{\Delta }}_h\eta (x,t)=0.$

Let $\mathbb{C}:=\mathbb{R}[X]\mathrm{/}(X^2+1)$, with $i:=[X]$ and $i^2=-1$; this quotient is the complex field. Let $\exp :\mathbb{C}\to \mathbb{C}$ be the everywhere-convergent series-defined complex exponential $\exp w:={\sum }_{n=0}^{\mathrm{\infty }}{w}^n\mathrm{/}n!$. Let $\pi$ be twice the least positive zero of cosine. For $m=(m_1,m_2)\in {\mathbb{Z}}^2\setminus \{(0,0)\}$ define $\mathbf{k}:=2\pi (m_1\mathrm{/}{L}_1,m_2\mathrm{/}{L}_2)$ in inverse metres and $k:=\mathrm{\mid }\mathbf{k}\mathrm{\mid }>0$. For $\omega \in \mathbb{C}$ in inverse seconds, define on the quotient torus $q_m([x],t):=\exp (i(\mathbf{k}\cdot \tilde{x}-\omega t)),$ where $\tilde{x}$ is any representative of $[x]$. Changing representatives changes the exponent by $2\pi iN$ for an integer $N$; the exponential addition law and $\exp (i\pi )=-1$ give $\exp (2\pi iN)=1$, so $q_m$ is well-defined. Its lift is smooth and satisfies ${\mathrm{\Delta }}_h{q}_m=-k^2{q}_m$.

A nonzero complex mode of angular frequency $\omega$ and index $m$ is a pair ${\eta }_c:{\mathbb{T}}_L^2\times I\to \mathbb{C}$ 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 ${\varphi }_c([x],z,t)=F(z)q_m([x],t)$, satisfying the complex-linear extension of the displayed equations, where $H\in \mathbb{C}\setminus \{0\}$ is in metres and $F\in C^2([-h,0],\mathbb{C})$ is in metres squared per second. Here $F$ is continuous on the closed interval, twice continuously differentiable on $(-h,0)$, and its first two derivatives extend continuously to both endpoints. The real and imaginary parts of every such mode solve the real linear model. Such a nonzero mode exists exactly when ${\omega }^2=(gk+\frac{\gamma }{\rho }{k}^3)\tanh (kh)\mathrm{.}$ For these hypotheses the right-hand side is positive because $k,h,g,\gamma ,\rho >0$ and $\tanh (kh)>0$. Thus each $m\mathrm{\ne }(0,0)$ admits modes at the two real angular frequencies $\omega =\pm \sqrt{(gk+(\gamma \mathrm{/}\rho )k^3)\tanh (kh)}$, with the positive real square root. This is a conditional prediction of the stated first-order model, not a measurement, a finite-amplitude approximation claim, or an assertion that an actual fluid satisfies every assumption.

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

Georgi, The Physics of Waves, §§11.5.1–11.5.2, printed pp. 285–291

Fitzpatrick, Fluid Mechanics, “Gravity Waves,” equations (11.17)–(11.21)

Fitzpatrick, Fluid Mechanics, “Surface Tension,” equations (11.116)–(11.121)