The radiation paradox

Ya cannae get aroond conservation o' energy, ya reekie oxtered sassenachs yae. (Translation from the Scots: You can't get around conservation of energy).

Conservation of energy from a far field perspective

Consider an arbitrary boundary around a big bit of sea. In order for a wave energy converter (WEC) to absorb energy from within that boundary, the waves leaving the boundary must have less energy than the waves entering the boundary. The only mechanism to capture useful energy while ensuring less energy in the waves leaving the boundary, is to create waves that destructively interfere with waves that would leave the system in the absence of energy capture. To create waves while absorbing energy involves the radiation of waves by damped (by a power take off system that is) wave excited motion.

Does Falnes's wave maker diagram apply to all WECs ?

Fig 1: Falnes's wave absorber = wave maker diagram

Falnes's assertion that 'a good wave absorber must be a good wavemaker', and his diagram depicting the amplitude and phase conditions for optimum absorption have been around since the early days of wave energy theory. I'd always thought that the wave absorber = wave maker diagram (Fig 1) was general, and applied to all wave energy converters. Would you be surprised if I told you that it applied to point absorbers only? (by point absorber I mean a body that is at least an order of magnitude smaller than the wavelength of the incident waves)

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?

Can a photo demonstrate percentage of wave power absorbed?

Photo of Duck in action, by Jamie Taylor, the University of Edinburgh

This conundrum puzzled me for a while, so I'd like to share it with you. The question is whether it is possible to determine what percentage of wave power was being absorbed by the Duck in the time lapse photo.

 

About this photo 

This photo has an interesting history behind it: it was presented as proof to the British public (and funding bodies) that the Edinburgh Duck was capable of absorbing 96% of the power in a wave. Yet the story told by this photo is not as simple as it first seems.

Standing wave visualisation

Code showing standing wave due to terminator absorbing wave power

I've discussed how reflection results in a standing wave. It can be useful to visualise how complete and partial reflection look.

At some point I'll make this into something more fun and user-friendly. For now, if you are really curious to visualise how a standing wave looks when 96% of the energy has been absorbed, then paste the following into Octave (Matlab will do), and change Ar to 0.04:

clear;
close;
Ai = 1; %incident amplitude
Ar = 0.9; %amplitude of combined radiated and reflected waves - try varying between 0 (full absorption) and 1(no absorption).
NumWaves = 2;
wtmax = NumWaves*2*pi; %maximum time to plot to give NumWaves # of cycles

kx= 1:100;
wt = 1:wtmax/(20*NumWaves):wtmax %discretisation: 20 points per wave
for ii = 1:length(wt)
     wi(ii,:) = Ai*sin(wt(ii)+kx);
     wr(ii,:) = Ar*sin(wt(ii)-kx);
end

ws = wi+wr;
plot(ws)