In a previous article I talked about a little boy who sent a postcard across the Pacific Ocean. With a leap of faith, I related that to ocean swell propagation. (See the article HERE). This time, imagine that the little boy sends two postcards at the same time; one first-class and the other second-class. The first-class service uses a faster boat.

As both boats travel across the ocean, the first-class one progressively outpaces the second-class one, the distance between them increasing all the time. When they arrive at the destination, the time between the arrival of the first one and the arrival of the second one depends on how far they have travelled. If the cards were sent from not very far away they would arrive almost at the same time; but if they were sent from a long way away, the second one would arrive a long time after the first one.

**See the direction of a swell, HERE.**

Now, the occupier of the address where the cards were sent doesn’t know where they came from. But she/he does know the speeds of first and second-class postal boats, and they also know the time between the arrival of the two cards. They can use this knowledge to help guess where they came from.

This can be directly related to wave propagation. Longer period waves travel faster than shorter period ones, so the long period waves arrive at the destination first, followed by the shorter period ones. And, just like the two postcards, the time between the arrival of waves of different periods depends on how far the swell travelled. If the swell came from a storm close-by, the waves of different periods arrive almost at the same time; but if the swell came from a distant storm, the short-period waves would arrive a long time after the long-period ones.

**Forecast: UK + Ireland**

At the coast, an observer will see a decrease in wave period from when the swell arrives until it disappears. The rate at which the period decreases depends on how far the swell has travelled; it will decrease quickly if the swell came from a close-by storm, and it will decrease more gradually if the storm was further away.

So, let’s compare a couple of real swells. In that previous article I used the ones that hit Cornwall on Sunday 19 and Monday 20 January. So we’ll use them here too. To show the two swells arriving at the coast we’ll use the MSW forecast. Don’t get confused – this is a forecast, not a real-time observation, but it illustrates the principle well enough.

The first swell arrives at 00:00 on Sunday and lasts until 18:00h on Monday. The second one arrives at 06:00 on Monday and disappears at 09:00 on Friday. You can see this on the MSW forecast I have stitched together (below).

On the forecast you can see that the second swell lasts a lot longer than the first one, and the period decreases much more gradually. Therefore, the first swell probably came from a relatively close-by storm and the second one from a more distant storm.

To visualise this a bit better I have plotted a graph of period against time for the two swells, where you can see them both decreasing over time. For the first swell, the period took 42 hrs to decrease from its initial value of 17 secs down to its final value of 8 secs just before it disappeared. For the second swell, the period took a staggering 99 hrs to decrease from its initial value of 23 secs down to its final value of 9 secs. The rate of decrease of the second swell is slower, which confirms that it came from a storm further away.

But plotting period against time still doesn’t give us a really clear way of ascertaining the rate of period decrease. That’s because the period decreases nonlinearly; it goes down quickly at first then gradually levels off. It would be much easier if the period decreased in a uniform fashion. But it doesn’t.

So what we need to do is to convert the period into frequency. Frequency is the inverse of period. It is measured in cycles per second, or Hertz (Hz). For example, a wave that has a period of 10 secs has a frequency of 0.1 Hz. If you were in the water and a 10-sec wave went past, you would see a tenth of a wave going past every sec. The wave has a frequency of a tenth of a wave per sec, or a tenth of a Hz.

I plotted a second graph; frequency instead of period against time. Here you can see that the frequency increases for both swells, and for the first swell it increases more quickly. But the increase in frequency is linear (the lines I’ve drawn through the points are straight), which makes life much easier. Now we can determine the exact rate of frequency increase for both swells. From the graph we can work out that, for Swell 01 the frequency increases at a rate of 0.0016 Hz per hr; and for Swell 02 the frequency increases at 0.0006 Hz per hr (much slower).

Now, if you are still with me, here is the clever bit. By using a simple formula we can use the rate of frequency increase to estimate how far away the storm was; the distance in km from the storm to the coast is equal to 2.8 divided by the rate of change of frequency in Hz per hr.

Where did that formula come from? Well, let’s just say it was derived from the principles of wave dispersion. It works surprisingly well even though it is highly simplified. It assumes that the swell came from a single, stationary point on the ocean, which is rarely the case.

Anyway, if we apply the formula to our two swells, it tells us that Storm 01 was about 1,800 km away and Storm 02 was about 4,000 km away. That more or less corresponds to the two storms that were in the North Atlantic a couple of days before the swells arrived at the coast (see chart).

*Cover shot by Ian Mitchinson.*