Monday 7 January 2013

Wrong turns on the way through the 96% capture maze

The photo below (Fig 1) was used to show that 96% of the energy in the wave coming from the right had been absorbed by the duck. In a previous post I showed how we can draw this conclusion from the photo alone. Like navigating through a maze, my first attempts at solving this problem took me up several dead ends. It is worth discussing these, as I'm sure I'm not the only one to have made these mistakes?

First attempt at solving this conundrum

 Fig1: Duck in action, by Jamie Taylor, the University of Edinburgh

The wave travelling from right to left has only 1/5th of the wave height after encountering the duck. As energy transport is proportional to the square of the wave height, the wave only contains 1/25th of the energy after passing the duck. If only 4% of the energy remains then the remaining 96% must have been captured by the duck. Q.E.D.?

This reasoning is very tempting, but would lead one to conclude that a wave reflecting off a brick wall was being completely absorbed by the wall. It is clear that waves are being reflected by the duck because there are standing waves in front of it.

Second attempt at solving this conundrum
I was clearly missing something, so I decided to compare this photo to Falnes's well-known diagram that demonstrates the principle that "a good wave absorber needs to be a good wave maker" (Fig 2). I had assumed that it applied to all types of wave energy converters. Note I have flipped the original around for easy comparison with the duck photo above (Fig 1).

 Fig 2: wave absorber = wave maker, after Falnes.

To recap, the wave absorber = wave maker picture (Fig 2) shows how a symmetrical wave (a) combined with an asymmetrical wave (b)  gives a wave radiating in one direction only (c). The incoming wave (d) combined with this radiated wave (e) can, in theory (with numerous caveats, which I will address in future articles), capture 100% of the incoming waves.

The superposition of the incoming and radiated waves (e) appears similar to the duck photo, but don't be tempted to think that Fig 2e describes what is going on in Fig 1!

In Fig 2 the wave is radiated in the same direction as the incident wave. However, we know that the duck was designed to generate waves at the 'beak' and for the rounded side to disturb the water as little as possible. The duck generates waves towards the incoming waves.

Third time lucky
Why did Falnes' wave absorber = wave maker diagram not match up with the duck photo? The key to answering this is the distinction between the behaviour of bodies that are small compared to the waves and bodies that are a similar size as the waves, or bigger. When waves encounter a small stationary body, they are transmitted. When waves encounter a large stationary body, diffraction occurs,  causing some or all of the incident wave to be reflected.

The wave absorber = wave maker diagram (Fig 2) is specific to bodies that are small in relation to the waves because the incoming wave (d) experiences no reflection; it is completely transmitted.

 Fig 3: wave absorber = wave maker, for the case where all the incoming wave is reflected.

The duck photo (Fig 1) is better described by considering a version of Falnes's diagram (Fig 3) that describes the case where all the incoming waves (a) are reflected (b). Again, a combination of symmetrical and asymmetrical waves can create a wave radiated in one direction (c), but this time it is necessary to radiate waves towards the incoming waves. A time lapse photograph of Fig 3 (d) would show the standing wave to the right of the body as a series of nodes and anti-nodes. This standing wave can be seen on the right of the duck in Fig 1.

Image credits: Screenshot from Hungry Ducks game, Phil Kingsley.

Other posts in the wave absorber = wave maker series:
Falnes's diagram: just a bunch of maths?
Complete absorption with 1 DoF