Swell Arrival Time from a Distant Storm

Swell does not carry its energy at the speed of its crests. The energy of a wave train travels at the group velocity, and in deep water that speed depends on the period: $c_g = gT/(4\pi)$, so the travel time over a distance $D$ is $t = D/c_g = 4\pi D/(gT)$. Longer periods travel faster, which is why the longest swell arrives first and why the period at a buoy shortens as the swell passes. This is derived physics, from the dispersion of a wave train in deep water (Dean and Dalrymple, chapter 3), not a fitted curve.

Each symbol has a plain meaning. $t$ is the time the swell energy takes to arrive, $D$ is the distance along the path from the storm to the site, and $T$ is the peak period of the swell, in seconds. Because $T$ sits in the denominator, a longer period always means a shorter trip. The formula can be solved for any of its quantities. Given $D$ and $T$ it gives the arrival time. Given the period at the first arrival and the time since the storm, $D = gTt/(4\pi)$ gives the storm's distance, and the first arrival is the right one to use because it is the longest swell present. Given $D$ and $t$ it gives the period.

Take a 14 s swell from a storm 1,000 nautical miles away. A hand rule for this case is $t = 2D/(3.03\,P)$ hours, with $D$ in nautical miles and $P$ the period in seconds, which gives 2000 / 42.42 = 47.15 h. The page gives 47.09 h, 0.13 % lower, so the two agree closely. Run backward, a 14 s swell first seen 47.15 h after the storm came from 1,000 nautical miles, which is 1,852 km.

The speed used is the deep-water group velocity, so the water must be deep enough for it to hold; the deep-water wavelength page gives the length of a swell of that period, which sets how deep that is. The result is a travel time for the swell's energy, and the distance $D$ is measured along the path from the storm. The haversine great-circle page gives that distance between a storm and a buoy from their latitudes and longitudes.