Relative Depth of a Wave (Depth over Wavelength) and Which Formula Holds

Wave formulas come in families, and the first question is which family applies. The answer is one ratio, the water depth $d$ over the wavelength $L$: $R = d/L$. If the water is deeper than half a wavelength ($R$ above 1/2), the wave does not feel the bottom, its speed depends only on its period, and the deep-water formulas hold. If the water is shallower than a twentieth of a wavelength ($R$ below 1/20), the wave feels nothing but the bottom, its speed depends only on the depth, and the shallow-water formulas hold. Between the two, the full dispersion relation is needed.

Use the wavelength the wave has in that water. The full dispersion relation gives it and holds in all three regimes. A 14 s swell has a deep-water wavelength $gT^{2}/2\pi = 9.80665\times196/6.283185 = 305.906$ m, and the ratio reaches 0.5 at half of that, 152.953 m. So the swell is in shallow-water trouble in anything under 15 m and is not in deep water until the water is 153 m deep. A 4 s wind sea, 25 m long, is in deep water at 12.5 m.

The page's example is the deep-water boundary itself: 50 m of water under a 100 m wave gives $R = 0.5$ exactly, and 5 m under a 100 m wave gives 0.05, the shallow-water boundary. A case between them is 10 m of water under a 92.36 m wave, $R = 0.108272$, which is neither deep nor shallow, so the full relation is needed. The page runs backwards as well: $L = d/R$ shows that a wave just in the shallow-water limit in 5 m of water is $5/0.05 = 100$ m long, and $d = R\,L$ gives the 152.953 m above from $R = 0.5$ and $L = 305.906$ m.

The source is the linear wave theory in Dean and Dalrymple (1991), which sets out the deep, shallow and intermediate regimes. The ratio only picks the regime within that theory. Whether linear theory itself is good enough is a separate question, answered by the Ursell number, a ratio of height, wavelength and depth.