Shock fronts: literature and surfing

By Marcus Wilson 08/02/2013

One of baby Benjamin’s books has a storyline that goes like this. (Not wishing to fall foul of the Copyright Act I shall not quote directly from it – any sensible quote is about 10% or greater of the work!)

There’s an animal that’s being chased by another animal. This second animal is being chased by a third. The third by a fourth. And so on. The seventh animal is being chased by the first… One can picture the scene of seven animals trouping round in a circle. And that’s it. Enough to entertain a seven-month old. (Though I think it’s the colours, Daddy’s voice and the fact that he can chew it that leads to most of the entertainment, rather than the nuances of the text and storyline.)

Anyway, the book reminded me of a lecture I had as an undergraduate looking at characteristics in differential equations (don’t worry about those) and shock fronts. The lecturer wanted to use an example of a physical system where velocity increases with density of the things doing the moving. The example he chose? (and he admitted this was a bit naff): A infinite line of people interspersed with tigers. When there is a large separation of the members of this line, it moves slowly. But when the tigers are close at the heels of the people, everyone moves quickly. It’s the opposite of more common physical systems where things slow down at high density, such as with traffic flow. High traffic density causes traffic jams, not high speed driving.

The tigers and people line has an interesting property of being able to generate a shock-front. Here there is a discontinuous change in the density of the system. A more reasonable example that does something similar is a hydraulic jump. When flowing water experiences a sudden change in depth (and therefore velocity), there can be shock-fronts formed.  I remember experiencing a strange example of this (or something similar) when sailing along the south coast of England several years ago. We sailed over a sand bank that had a strong tidal current flowing over it – and at the edge where the depth changed quickly, the water transitioned from very calm to choppy. The change was extremely sudden. Right at the edge of the front, there was a set of very well formed standing waves. It would have been good to have had a video camera to record the action.

I had a look on Youtube for some movies of hydraulic jumps. Here’s a nice one for you surfers. Fancy being a designer on this – it would make your engineering degree well worth while.