Who doesn’t love a good comic? They are funny and informative. Well, here is part of one from some time ago (meaning before the recent East-Coast Quake).

Click the image to read the rest at xkcd.

So, the premise is that people use twitter to post things all the time. Things like what kind of sandwich they are eating or things like “hey that was an earthquake”. Of course, a tweet will travel faster than the seismic waves from the Earthquake but the Earthquake gets a head start (because people don’t tweet instantly – and on top of that they still have to at least type “earthquake”). However, at some point, the tweet wave will pass the seismic wave.

A first estimate

What we have here is a constant velocity problem with two things traveling at different speeds. Sort of like the “train A leaves Chicago…” problem, but way cooler.

We need some starting info (or assumptions). Suppose there is an earthquake, then (with estimates from xkcd):

• v_s is the speed of the seismic waves. xkcd lists this at 3-5 km/s.
• There is a time delay between the earthquake and the tweet response. Let me call this t_t – perhaps around 20-30 seconds.
• How fast does the tweet move? This will be v_t and has a value of about 200,000 km/s.

Now for the math. If I want to represent the position of the leading edge of the seismic wave and the “tweet wave”, I can write (oh, I will assume the earthquake starts at time t = 0 seconds):

Of course, the position function for x_t is zero until the time greater than t = t_t. This also assumes that the tweeter and the earthquake are at the same location. Let me go ahead and plot this using the numbers from xkcd.

And here you can see the problem. The tweet-speed is so much faster than the seismic wave speed, it really only matters how far the seismic wave has traveled in time delay. Anyway, with this model, the seismic wave travels 80 km during that delay. For the recent quake centered in Mineral, VA – this is what that would look like.

Assuming the values from xkcd, anyone outside this red circle has a chance of getting a tweet before getting a shake.

What about the speed of a seismic wave?

I don’t know much about typical seismic wave speeds. However, I don’t need to know much. I can just look at the recent Virginia quake and use the speed of the waves from that event. I saw this nice animation posted on Bad Astronomy showing how different detectors across the USA detected the event. Here is that video-animation:

Using Tracker Video analysis (I know, I use this all the time) and the above video, I can get the position as a function of time for the leading edge of the wave. Here is what that looks like. Oh, note that the video is not in real time. Using the time stamp at the bottom, I did adjust the frame rate in the data to correspond to real seconds.

This gives a wave speed of 528 km/s. This is significantly higher than the estimation of 3-5 km/s.

UPDATE: Ok, somehow I am an idiot.  In this video above the frame rate is 4 real-world seconds per frame.  Here is what that graph should really look like:

This gives a wave speed of about 7 km/s.  Much better.   Ok, if that is true AND if this is a periodic wave, then I can estimate the frequency. Here is a picture of something that looks like a wavelength.

From this, I would say the wavelength is about 450 km. If this is actually the wavelength (not sure because it looks like there could be a much shorter wavelength in there) then the following would be true.

Here, ? is the wavelength. Putting in my values for the velocity and the wavelength, I get a frequency of 0.016 Hz. I haven’t been in an earthquake and and I certainly not a quakeologist, so I guess that is ok?

Ok, I briefly looked around. It seems 15 km/s is the maximum number that comes up on the internet. I guess 7.3 km/s isn’t too bad.

The speed of a tweet

So, xkcd estimates that tweets are pretty darn fast. That may be correct, but there is nothing like a simple experiment to test this. Here was my idea: I would post a note on twitter and see how fast the responses were. Here is that tweet.

So, the idea was that by looking at the difference in time of tweet and the time the person viewed the tweet I could get the speed. Oh, since the responders gave me a location I could use distance divided by time. I had many great (and quick) responses, but there was a problem. I sent the tweet at 1:48 PM Central, most of the replies said they received the post at 1:48 PM central. Some even got it at 1:47 PM. I suspect the times are reported according to the other user’s computer time. Really, the only thing I can say is that most of these tweets arrived in some time less than 1 minute. But some of the responders were quite far away. I had one from Germany (estimated 8200 km) and one from South Africa (13,000 km). This would put a lower bound on the speed at:

There was one other possibly useful tweet. Jenn Cutter (@jenncutter) responded with a time of 25 seconds and a distance of 2300 km. This would give a tweet speed of:

But still, this 25 seconds is probably the time that Jenn’s computer got the tweet. Ok, then another experiment. This time I coerced my brother to measure the time between when I press “post” and he sees the tweet in his feed. He recored at time of 39 seconds at a distance of 1384 km. This gives a speed of 35 km/s.

Ok, so I am getting twitter speed values much less than what xkcd had. 217 km/s is significantly less than 200,000 km/s (in case you couldn’t tell). If the speed is as low as 35 km/s, then it could be slower than the earthquake wave (could be – because I am still not sure about my calculated seismic wave speed).

The speed of the tweet doesn’t really matter. The information has to get to the person. So, how long does it take a person to see a tweet? (I know someone, somewhere dies a little bit inside every time I say ‘tweet’) Luckily, when people responded to my twitter experiment they also reported the time they saw my post. Here are some of the responders times.

So, it looks like a good portion of the people could get the message in under 10 minutes.

How does this data change the problem?

If I use the earthquake speed of about 500 km/s, then there seems to be no realistic twitter warning. In just 20 seconds, the wave would travel 10,000 km. This is about one fourth the way around the Earth. Even Jenn, the fastest twitter responder, wouldn’t have the warning in time.

The actual speed of a tweet wave

Check this out. This is an animated map of geotagged tweets about the earthquake (I first saw this on Mashable).

Of course I had to analyze this. So, using Tracker video again I recorded the leading edge of the tweet wave in the direction from the earthquake towards New York. Eric Fischer (the creator of the video) was thoughtful enough to include the real frame rate in his comments. So, here is what I get.

From the linear fit, this looks like a tweet-wave speed of 2.7 km/s. Ok, let me assume these are people that are tweeting about experience with an actual earthquake and not people responding to other posts about the earthquake. If so, then this could also give an estimate of the speed of an earthquake. If I assume that all of these people have about the same delay time from experience to posting about it (you know, you have to get your phone out of your pocket and stuff) then this would be an estimate for the speed of the seismic wave.

This brings me back to the speed of the earthquake I calculated before. I am pretty sure that isn’t a seismic wave.

Back to the original problem once again

Let me re-do the original problem. I will use the following:

• Earthquake speed (v_s) of 3 km/s.
• Tweet speed (v_t) of about 100 km/s.
• Reaction time for tweet readers = 1 min. I know this is leaving out a lot of responders, but this way some people will at least have a chance.

Ok, now re-creating my first plot from above (same equations, different numbers).

Here the blue line is the position of the leading edge of the seismic wave and the green line is the leading edge of the received tweet wave. The point where these two are the same is at around 186 km from the center of the earthquake. Here is a map showing that area of people with no hope of a twitter warning.

The problem here is that the size of this circle is close to the area of effect of the earthquake. Sure, people much farther away than 180 km felt the earthquake – but they probably didn’t need a warning.