Ursell Number (wave theory regime)

Every wave calculation begins with a choice of theory, and this is the number that makes the choice defensible. Fritz Ursell set it out in 1953: the shallow-water expansion has two small parameters, and the theory you may use depends on which of them is smaller.

The first is nonlinearity, the wave height against the water depth, $H/d$. The second is dispersion, the depth against the wavelength, $(d/L)^2$. Linear Airy theory keeps only first-order terms in the first while making no approximation in the second — it is a small-amplitude, fully dispersive theory. The shallow-water and cnoidal theories do the reverse. The Ursell number is the ratio of the two parameters:

\[\mathrm{Ur} = \frac{H/d}{(d/L)^2} = \frac{H L^2}{d^3}\]

and it says which effect is winning. Small Ursell number means dispersion outweighs nonlinearity, and linear theory is the right tool. Large Ursell number means nonlinearity outweighs dispersion, the wave has a peaked crest and a long flat trough, and cnoidal theory — or, at the extreme, solitary-wave theory — is the right tool.

The cube on the depth is what makes the number move so violently as a wave shoals. A swell coming into a beach has its wavelength shortened, which pushes Ur down through $L^2$, and its height raised by shoaling, which pushes Ur up; but the depth falling as the cube overwhelms both. Every wave becomes nonlinear on the way in. This is not a subtlety of the theory, it is the reason surf looks nothing like a sine wave.

Now the honesty, and it is the point of the page. The threshold usually quoted against this form is $\mathrm{Ur} \approx 32$, and it is worth saying clearly what that figure is and is not.

It is not a switch. There is no Ursell value at which Airy theory stops working and cnoidal theory starts. Both are asymptotic expansions in small parameters, and an asymptotic expansion does not fail abruptly — its error grows steadily, and at some point you decide the error is more than you want. The number 32 marks roughly where the two expansions have comparable error, which is exactly the region where neither is comfortable. Stream-function theory, fitted numerically instead of expanded analytically, exists to cover that gap, and Dean's 1970 comparison of theories against measured waves is the standard reference for which to use where.

Nor is 32 a universal number, because the group is not universally normalized. Different authors define the Ursell parameter with the wave amplitude instead of the height (a factor of 2), with $k = 2\pi/L$ instead of $L$ (a factor of $4\pi^2$), or normalized by $32\pi^2/3$ so that the threshold becomes 1. Thresholds of 25, 26, 40 and 100 all appear in the literature for what their authors describe as the same physical boundary. These are not disputes about the ocean; they are the same physics divided by different constants. If you carry a threshold in from a book, carry its definition too.

The coastal shard's own warning — that Airy's derivation assumes a wave small compared with both the depth and its own length, and that the design wave is precisely the wave where that assumption is weakest — is what this page puts a number on. It should be read as a number with a soft edge. A wave at Ur = 20 is not "linear" and one at Ur = 45 is not "cnoidal"; one is a wave where linear results are probably good to a few percent and the other is a wave where they are probably good to tens of percent, and the word "probably" is doing real work in both.

Two practical notes. Use the local wavelength from the full dispersion relation, not the deep-water one — at intermediate depth they differ by tens of percent and the difference is squared. And in deep water the Ursell number becomes small automatically and stops being informative; there the question is steepness, $H/L$, and its limiting value of about one seventh.