ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Capillary dispersion reduces to gravity waves

Example

Fix $h, g, ρ, k > 0$ , an amplitude $a \in R$ , and $σ \geq 0$ . Let

$ω_{σ} = \sqrt{(g k + \frac{σ}{ρ} k^{3}) \tanh (k h)}, ω_{0} = \sqrt{g k \tanh (k h)}, G (z) = \frac{\cosh (k (z + h))}{\cosh (k h)} .$

Then $ω_{σ} = ω_{0}$ when $σ = 0$ , and the capillary-gravity mode in the theorem specializes to

$η_{0} = a C (k x - ω_{0} t), ϕ_{0} = \frac{a g}{ω_{0}} G (z) S (k x - ω_{0} t),$

the finite-depth gravity-wave mode. Moreover, with $h, g, ρ, k$ fixed,

$\lim_{σ ↓ 0} ω_{σ} = ω_{0} .$

This is an exact parameter specialization and one-sided limit within the linear mathematical model, not a nonlinear-wave result or an experimental observation. In the optional SI interpretation, $[h] = m$ , $[g] = m s^{- 2}$ , $[ρ] = k g m^{- 3}$ , $[k] = m^{- 1}$ , $[σ] = N m^{- 1}$ , $[a] = m$ , and $[ω_{σ}] = s^{- 1}$ .

Example references

Capillary-gravity dispersion and its exact mode: Capillary-gravity dispersion and conserved linear surface energy.

Finite-depth gravity-wave mode: Finite-depth gravity-wave dispersion with quantitative limits.

Facts & Assumptions

Given: The positive parameters, amplitude, and frequency definitions in the Example.

[F1]

Capillary-gravity dispersion and conserved linear surface energy: for $h, g, ρ, k > 0$ and $σ \geq 0$ , $ω_{σ}^{2} = (g k + σ k^{3} / ρ) \tanh (k h)$ and the positive-frequency mode has potential coefficient $a ω_{σ} / (k \tanh (k h))$ .

[F2]

Finite-depth gravity-wave dispersion with quantitative limits: for $h, g, k > 0$ , $ω_{0} = \sqrt{g k \tanh (k h)} > 0$ and the finite-depth gravity mode has potential coefficient $a g / ω_{0}$ .

Verification

1.1givenF1F2algebra

Setting $σ = 0$ in [F1] gives $ω_{σ}^{2} = g k \tanh (k h) = ω_{0}^{2}$ . Both frequencies are positive by [F1] and [F2], so $ω_{σ} = ω_{0}$ . The capillary mode's potential coefficient satisfies $a ω_{0} / (k \tanh (k h)) = a g / ω_{0}$ because $ω_{0}^{2} = g k \tanh (k h)$ and $k \tanh (k h) > 0$ . Thus both the elevation and potential specialize exactly as stated; no division by $a$ is used.

2.1givenF1F2algebra∎

For $σ \geq 0$ , $ω_{σ}^{2} - ω_{0}^{2} = σ k^{3} \tanh (k h) / ρ \geq 0$ , so the positive roots satisfy $ω_{σ} \geq ω_{0}$ . For $σ > 0$ , rationalizing gives $ω_{σ} - ω_{0} = (ω_{σ}^{2} - ω_{0}^{2}) / (ω_{σ} + ω_{0}) = (σ k^{3} \tanh (k h) / ρ) / (ω_{σ} + ω_{0})$ . The numerator tends to zero as $σ ↓ 0$ , while $ω_{σ} + ω_{0} \geq ω_{0} > 0$ . Hence the quotient tends to zero and $ω_{σ} \to ω_{0}$ .

Capillary dispersion reduces to gravity waves

Example

Example references

Facts & Assumptions

Verification

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