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Finite-depth gravity-wave dispersion with quantitative limits

Statement

Let $h, g, k > 0$ , let $a \in R$ , and set $s = k h$ and $ω = \sqrt{g k \tanh (s)}$ , using the positive square root. Here $h$ is depth, $g$ a constant acceleration parameter, and $k$ a positive horizontal wave number; $s$ is dimensionless and $a$ is a real elevation amplitude. The strict positivity of $\tanh (s)$ for $s > 0$ makes $ω$ well-defined and positive. Let $x, y \in R$ be horizontal Cartesian coordinates and $z \in (- h, 0)$ the vertical coordinate, positive upward with the flat mean surface at $z = 0$ and bottom at $z = - h$ ; time $t \in R$ . Write $θ = k x - ω t$ . A surface elevation is the scalar field $η (t, x, y)$ , and a velocity potential is a scalar field $ϕ (t, x, y, z)$ . Define the finite-depth linear potential-wave problem by

$Δ ϕ = 0 (- h < z < 0), ϕ_{z} (t, x, y, - h) = 0,$

$η_{t} (t, x, y) = ϕ_{z} (t, x, y, 0), ϕ_{t} (t, x, y, 0) + g η (t, x, y) = 0,$

where $Δ = \partial_{x}^{2} + \partial_{y}^{2} + \partial_{z}^{2}$ . Let $S, C$ be the real sine and cosine functions defined on this page. Then

$η (t, x, y) = a C (θ), ϕ (t, x, y, z) = \frac{a g}{ω} \frac{\cosh (k (z + h))}{\cosh (k h)} S (θ)$

are fields that are $C^{2}$ in the spatial variables and $C^{1}$ in time, with the displayed derivatives extending to $z = - h$ and $z = 0$ , and they satisfy this system exactly. Define the shallow and deep comparison frequencies by $ω_{s h} = k \sqrt{g h}$ and $ω_{d e e p} = \sqrt{g k}$ . They obey

$0 \leq \frac{ω_{s h}^{2} - ω^{2}}{ω_{s h}^{2}} \leq \frac{(k h)^{2}}{3}, 0 \leq 1 - \frac{ω}{ω_{s h}} \leq \frac{(k h)^{2}}{3}, 0 \leq \frac{ω_{d e e p}^{2} - ω^{2}}{ω_{d e e p}^{2}} \leq 2 e^{- 2 k h} .$

These are exact statements about the displayed linear boundary problem and its dispersion relation. They assert no nonlinear free-surface solution or error bound.

In the optional SI interpretation, $[h] = m$ , $[g] = m s^{- 2}$ , $[k] = m^{- 1}$ , $[a] = [η] = m$ , $[ω] = s^{- 1}$ , and $[ϕ] = m^{2} s^{- 1}$ .

Statement references

Finite-depth linear surface-gravity-wave equations and dispersion relation: Feddersen, Linear Surface Gravity Waves 1, §§2.1–2.4, PDF pp. 1–6.

Independent finite-depth formulation and shallow/deep limits: Firing, OCN 620, chapter 19, §§19.1–19.7.

Cartesian Laplacian convention: Formal parameterized fluid systems and observables.

Real sine and cosine as real restrictions of complex trigonometric functions

Derivative identities for the real restrictions of complex sine and cosine

The six hyperbolic functions and their natural domains

Addition formulas, identities, parity, and derivatives of the hyperbolic functions

Quantitative hyperbolic-tangent bounds: Quantitative shallow and deep bounds for the hyperbolic tangent.

Facts & Assumptions

Given: The positive constants $h, g, k$ , amplitude $a$ , the fields and equations in the Statement, and $s = k h > 0$ .

[A1]

The target equations are the harmonic-potential equation, impermeable bottom condition, and the stated linear kinematic and dynamic surface conditions as defined in the Statement.

[F1]

The scalar Laplacian in Cartesian coordinates is the sum of the pure second partial derivatives (Formal parameterized fluid systems and observables).

[F2]

The page functions $S, C$ are real-valued and satisfy $S^{'} = C$ and $C^{'} = - S$ (Real sine and cosine as real restrictions of complex trigonometric functions, Derivative identities for the real restrictions of complex sine and cosine).

[F3]

For real $r$ , $\cosh r = (e^{r} + e^{- r}) / 2 > 0$ , $\tanh r = \sinh r / \cosh r$ , and $sech r = 1 / \cosh r$ (The six hyperbolic functions and their natural domains).

[F4]

$\cosh^{'} = \sinh$ , $\sinh^{'} = \cosh$ , $\tanh (0) = 0$ , and $\tanh$ is strictly increasing on $R$ (Addition formulas, identities, parity, and derivatives of the hyperbolic functions).

[F5]

Differentiable compositions obey the real chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[F6]

Every positive real number has a unique positive square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F7]

For every $r \geq 0$ , $0 \leq r - \tanh r \leq r^{3} / 3$ and $0 \leq 1 - \tanh r \leq 2 e^{- 2 r}$ (Quantitative shallow and deep bounds for the hyperbolic tangent).

Proof

1.1givenF3F4F5F6algebra

Since $s > 0$ , [F4] gives $\tanh (s) > \tanh (0) = 0$ , so [F6] makes $ω$ a positive real number. Set $G (z) = \cosh (k (z + h)) / \cosh (k h)$ . By [F3]–[F5], its denominator is positive, $G^{''} (z) = k^{2} G (z)$ , $G^{'} (- h) = 0$ , $G (0) = 1$ , and $G^{'} (0) = k \tanh (k h)$ .

1.2givenF7algebra

Since $ω_{s h}^{2} = g h k^{2} = g k s$ and $ω^{2} = g k \tanh (s)$ , division by the positive $g k s$ gives $(ω_{s h}^{2} - ω^{2}) / ω_{s h}^{2} = (s - \tanh (s)) / s$ . Applying [F7] and dividing by $s > 0$ proves the first shallow bound.

2.1givenA1F1F2F5step 1.1algebra

The identities in [F2] and the chain rule [F5] imply $S^{''} = - S$ . The field $ϕ$ is independent of $y$ , and therefore [F1] gives $Δ ϕ = (a g / ω) (G^{''} - k^{2} G) S (θ) = 0$ . Also $ϕ_{z} = (a g / ω) G^{'} (z) S (θ)$ , so $G^{'} (- h) = 0$ proves the bottom condition.

2.2givenA1F2F5step 1.1algebra

By [F2] and [F5], $η_{t} = a ω S (θ)$ and $ϕ_{t} (t, x, y, 0) = - a g C (θ)$ , using $G (0) = 1$ . The surface derivative is $ϕ_{z} (t, x, y, 0) = (a g k \tanh (s) / ω) S (θ) = a ω S (θ)$ because $ω^{2} = g k \tanh (s)$ ; also $ϕ_{t} (t, x, y, 0) + g η = - a g C (θ) + a g C (θ) = 0$ . Thus both surface conditions in [A1] hold.

2.3givenstep 1.1step 1.2algebra

Put $r = ω / ω_{s h} > 0$ . step 1.2 gives $0 \leq 1 - r^{2} \leq s^{2} / 3$ , and $1 - r = (1 - r^{2}) / (1 + r)$ with $1 + r \geq 1$ . Hence $0 \leq 1 - r \leq 1 - r^{2} \leq s^{2} / 3$ , proving the shallow frequency bound.

3.1givenF7algebra∎

Since $ω_{d e e p}^{2} = g k$ , the dispersion identity gives $(ω_{d e e p}^{2} - ω^{2}) / ω_{d e e p}^{2} = 1 - \tanh (s)$ . The second inequality of [F7] yields the stated deep bound, with its lower bound as in [F7].

Source notes

Real trigonometric functions and derivatives: Real sine and cosine as real restrictions of complex trigonometric functions and Derivative identities for the real restrictions of complex sine and cosine.

Hyperbolic functions and positivity: The six hyperbolic functions and their natural domains and Addition formulas, identities, parity, and derivatives of the hyperbolic functions.

Finite-depth gravity-wave dispersion with quantitative limits

Statement

Statement references

Facts & Assumptions

Proof

Source notes

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