Linear Wave Dispersion Relation

Everything else on this shard is a special case of this one equation. $\omega^{2} = gk\tanh(kd)$ comes straight out of Airy's 1845 solution of the linearised free-surface problem, and it holds at every depth. The two familiar formulas are its limits and nothing more: as $kd$ grows past about $\pi$, $\tanh(kd) \to 1$ and you get the deep-water relation; as $kd$ becomes small, $\tanh(kd) \to kd$ and you get $c = \sqrt{gd}$. The hyperbolic tangent is the whole story — it is the function that switches the depth on and off.

Written in the units an engineer works in, it says $L = \dfrac{gT^{2}}{2\pi}\tanh\!\left(\dfrac{2\pi d}{L}\right)$, and the thing to notice is that $L$ appears on both sides. This equation cannot be rearranged for the wavelength. It is transcendental in $k$, and no amount of algebra will make it otherwise — the same way $x = \cos x$ has a perfectly definite solution that no closed form expresses.

What it does have is exactly one positive root, and that is what makes it tractable. $F(k) = gk\tanh(kd)$ is zero at $k = 0$, rises steadily, and grows without bound, so it crosses any given $\omega^{2}$ once and only once. Both limiting formulas underestimate $k$, because $\tanh x \le \min(1, x)$, so they give a guaranteed lower bracket; expand upward until $F$ overshoots and you have the root trapped. This page then bisects that bracket 120 times, which drives it below the spacing of adjacent floating-point numbers. The answer is converged to about sixteen significant figures — to the precision of the arithmetic itself rather than to any stated engineering tolerance.

There is a whole literature of explicit approximations for people who did not want to iterate: Eckart's 1952 formula, Hunt's nine-term rational polynomial, Guo's 2002 logarithmic fit, Fenton and McKee's. The good ones are accurate to a fraction of a percent, and they earned their place in an era when this calculation was done on a slide rule or in an inner loop that ran millions of times. There is no reason to accept an approximation when the exact root costs a few dozen machine operations, so this page does not.

The other two directions are closed-form, and it is worth seeing why. Given the length and the depth, the period follows directly by evaluating the right-hand side — no iteration at all. Given the length and the period, the depth follows too, because $\tanh(kd)$ equals $L/L_0$ exactly, so $d = (L/2\pi)\operatorname{artanh}(L/L_0)$. That identity — the local wavelength divided by the deep-water wavelength for the same period IS the hyperbolic tangent of $kd$ — is the tidiest way to hold the whole relation in your head.

The depth direction has a genuine use and a genuine limit. It is the basis of estimating nearshore bathymetry from wavelengths visible in aerial or satellite imagery, which works because a wave in intermediate depth is a depth gauge. But as $L$ approaches $L_0$ the inverse hyperbolic tangent runs away to infinity: a wavelength read half a percent long throws the depth out by tens of percent, and at exactly the deep-water length it returns infinity. That is not a numerical defect. It is the physical statement that a deep-water wave has no contact with the bottom and therefore no way of telling you where it is.

Two limits, one relation, and a boundary region between them. Depth ratios above 0.5 are deep, below 0.05 are shallow, and the band between — which is where harbours, beaches, breakwaters, cable routes and wave-energy sites overwhelmingly live — is the region this equation exists for. If there is one habit to take away from this shard, it is to compute $d/L$ before reaching for a formula, every time.