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Finite-depth gravity-wave dispersion — worked interpretation

Example

Take $h = 1 m$ , $g = \frac{9}{4} m s^{- 2}$ , $k = 1 m^{- 1}$ , and $a = \frac{1}{10} m$ . Write $x_{m} = x / (1 m)$ , $z_{m} = z / (1 m)$ , and $t_{s} = t / (1 s)$ for the numerical coordinate values in SI units. Then $s = k h = 1$ , $ω = \frac{3}{2} \sqrt{\tanh (1)} s^{- 1}$ , and the theorem's mode is

$θ = x_{m} - \frac{3}{2} \sqrt{\tanh (1)} t_{s},$

$η = \frac{1}{10} m C (θ), ϕ = \frac{3}{20 \sqrt{\tanh (1)}} m^{2} s^{- 1} \frac{\cosh (z_{m} + 1)}{\cosh (1)} S (θ) .$

Both comparison frequencies equal $ω_{s h} = ω_{d e e p} = \frac{3}{2} s^{- 1}$ . The exact relative errors are

$\frac{ω_{s h}^{2} - ω^{2}}{ω_{s h}^{2}} = 1 - \tanh (1), 1 - \frac{ω}{ω_{s h}} = 1 - \sqrt{\tanh (1)}, \frac{ω_{d e e p}^{2} - ω^{2}}{ω_{d e e p}^{2}} = 1 - \tanh (1) .$

They satisfy $0 \leq 1 - \tanh (1) \leq 1 / 3$ , $0 \leq 1 - \sqrt{\tanh (1)} \leq 1 / 3$ , and $0 \leq 1 - \tanh (1) \leq 2 e^{- 2}$ by the finite-depth theorem. The values are a formal parameter choice, not measured wave data.

Example references

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

Facts & Assumptions

Given: The positive parameter values, dimensional coordinate convention, and linear wave model in the Example.

[F1]

For $h, g, k > 0$ , the frequency $ω = \sqrt{g k \tanh (k h)}$ gives the displayed exact linear potential mode and the shallow and deep relative-error bounds (Finite-depth gravity-wave dispersion with quantitative limits).

Verification

1.1givenF1algebra

The assignments give $k h = 1$ and $g k = (9 / 4) s^{- 2}$ , hence $ω = \sqrt{(9 / 4) \tanh (1)} = (3 / 2) \sqrt{\tanh (1)} s^{- 1}$ . The theorem's profile factor is $\cosh (k (z + h)) / \cosh (k h) = \cosh (z_{m} + 1) / \cosh (1)$ , and $a g / ω = (3 / (20 \sqrt{\tanh (1)})) m^{2} s^{- 1}$ . Thus [F1] specializes to the stated $η$ and $ϕ$ .

2.1givenF1algebra∎

The comparison definitions give $ω_{s h} = k \sqrt{g h} = (3 / 2) s^{- 1}$ and $ω_{d e e p} = \sqrt{g k} = (3 / 2) s^{- 1}$ . Dividing the dispersion identity $ω^{2} = g k \tanh (1)$ by either squared comparison frequency yields the exact relative-error expressions in the Example. Applying the bounds in [F1] at $k h = 1$ gives the three stated inequalities, with no decimal approximation or empirical inference.

Depends on

Finite-depth gravity-wave dispersion with quantitative limits