Monday, 15 April 2013

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.

Conservation of energy from the device perspective
Consider the boundary between a WEC and the sea. In order for the WEC to absorb energy from within that boundary, the energy leaving must be less than that entering. The only mechanism that ensures that the energy entering the system (wave excited oscillation) is less than the energy leaving the system (oscillation induced radiation), is to damp this oscillation (with a power take off system). Therefore, for energy capture, the oscillations and the resulting radiated waves must be less than when no energy is captured.

The paradox being...
Herein lies the radiation paradox: from the conservation of energy perspective, radiation of waves is necessary for power capture; the bigger the gap between the energy entering and leaving the far field and device boundaries, the more energy is available for capture. Yet to create these destructive radiated waves, it is necessary to radiate waves containing less power than when energy is not being captured. Although we can say 'it is necessary to radiate waves to capture waves', an oscillating body will radiate waves even when it is not capturing any energy. So it is equally valid to say 'it is necessary to radiate fewer waves to capture waves' (radiated waves need to be the right kind of course!)

Making sense of the paradox
The only way to make sense of this is to consider Falne's standard formulation for power capture: absorbed power is the difference between excited power and radiated power:
Pa = Pe- Pr.

Pe is proportional to oscillation (velocity, for a primary wave activated body with an oscillating position), and Pr is proportional to the square of the oscillation. Pa plotted against oscillation gives a parabola, with zero power at oscillations that are too large or small. This means that radiation of waves too large or too small will lead to ineffective power capture.

Control over the size of radiated wave is exercised through choice of the amount of damping applied. Unsurprisingly, the most effective value lies between the extremes of zero damping (Pe= Pr = excitation and radiation power for undamped response), and the damping required to reduce the oscillations to zero (Pe= Pr= 0).

Image credits:
Rubber ducky X-ray copyright of Andrew Liepzig


  1. Hi Ally,

    Just wanted to add another point. The Radiated waves also need to be in the same useful direction. An Axis-symmetric body will radiate waves in all direction, hence depending on the comparisons, should account for some dissipated energy if not more, even when resonating in harmony with the incident waves.

  2. hi Vernash

    Indeed the radiated waves do need to be in the right direction. Directionality is an important part of what makes a radiated wave 'the right kind'. Fig 6 of shows a simplified 2D illustration of what happens when the waves are in the wrong direction - the ideas apply to the 3D case.