Deep-Water Wavelength from Period

This is the first number to compute about any sea state, and it is startling how much it contains. Give a wave its period and, provided the water is deep, its length follows — with no reference to the wind that raised it, the height it reached, or the depth beneath it. A 0.3 m ripple and a 12 m storm wave of the same 10-second period are both about 156 m long. That independence is a direct consequence of Airy's small-amplitude assumption, and it is one of the results that make linear wave theory worth learning even though the waves you care about violate its assumptions.

Where it comes from is the general dispersion relation $\omega^{2} = gk\tanh(kd)$. In water deeper than about half a wavelength, $kd$ exceeds $\pi$, and $\tanh(\pi) = 0.9963$ — within four parts in a thousand of unity. The hyperbolic tangent simply disappears, the depth goes with it, and what is left rearranges to $L_0 = gT^{2}/2\pi$. The subscript zero is not decoration; throughout coastal engineering it means "the deep-water value", and keeping it visible is how you avoid the commonest error in the subject.

Which brings us to what "deep water" actually means, because this is where most readers go wrong. Deep water is not a depth. It is a ratio: $d/L > 0.5$. Fifty metres of water is deep for a 6-second wind wave, whose length is 56 m, and it is emphatically not deep for an 18-second Southern Ocean swell, whose deep-water length is 506 m and which is already shoaling in 50 m. The same water, the same day, two waves, two answers. When people say a formula "only works in deep water" and then reach for a chart datum to check, they have already lost the thread. Compute the length first, then compare.

The arithmetic shortcut is worth memorising because it makes the ratio checkable in your head. $g/2\pi = 1.56$, so the wavelength in metres is $1.56\,T^{2}$ with $T$ in seconds; in feet it is $5.12\,T^{2}$. An 8-second wave is 100 m long, a 10-second wave 156 m, a 14-second wave 306 m. Once those are in your head, the deep-water test — is the depth more than half that? — takes no calculation at all.

Period is the invariant of a wave's whole life, and this is the fact to hold onto. A wave born under a storm in the Southern Ocean arrives on a beach in California with a different height, a different length, a different direction and a different speed — but the same period. Height changes with shoaling and refraction, length shortens with depth, direction bends towards the shore normal, and the period does not move. That is why every buoy record, every forecast and every design sea state is indexed by period, and it is why running this equation backwards from a measured wavelength has a trap in it: if the water under your measurement was not deep, the length you measured belongs to a longer period than this returns.

One boundary to respect. This is a relation for surface GRAVITY waves in deep water, and it stops applying at both ends of the period range. Below about 0.5 s, surface tension takes over from gravity as the restoring force, the waves are capillary ripples, and short waves travel FASTER than long ones — the dispersion runs the other way. At the long end, anything with a period of minutes or hours is a shallow-water wave everywhere on earth: a tsunami at 10 to 60 minutes and a tide at 12.4 hours both have wavelengths so enormous that the whole ocean is shallow to them, and neither is described by this equation at all.