Monday, 8 July 2013

Falnes's diagram: just a bunch of old maths?

What does Falnes's well-known diagram depicting the principle 'to absorb a wave you must make a wave' (Fig 1) actually show? Is this achievable or just a mathematical slight of hand, a bit of physics-sudoku with no relevance to the real world?

Correct and not at all correct interpretations of this drawing

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

First it is necessary to clear up a potential ambiguity. This diagram (Fig 1) lends itself to an incorrect interpretation. The first few parts of Fig 1 are clear:

Fig 1, parts (a) - (c) show a body being mechanically driven; we are putting power into the system to cause motion, and this power is carried away from the body in the form of a radiated wave.

Fig 1 (d) shows the incident wave; nature is putting power into the system; all that power passes through the system.

The not at all correct conclusion that could be drawn: Fig 1(e) shows (c) + (d): if you put power into the system to create motion, and nature then puts even more power into the system in the form of the incident wave, you've magically extracted energy from this wave.

The intended interpretation: Indeed, (c) shows the wave radiated when the body is motored. However, when Fig1 (e) shows (c) combined with (d), in this case (c) is the wave radiated when the body is excited by the incident wave, and when the resulting motion is the same as the driving motion required to radiate the wave shown in (c).

So (c) represents either a wave radiated due to motion resulting from driving the power take off system in reverse, or due to an equivalent motion resulting from excitation by the incident wave.

The modelling context

Different mathematical models suit different purposes. Falnes's 'wave absorber = wave maker' diagram is a model whose purpose was to communicate the basic principles of wave absorption in the simplest manner possible. I have chosen to use this diagram for different purposes, such as to illustrate the difference between large and small absorbers, or to describe the types of radiated waves that do not contribute to power absorption. However, Falnes intended this diagram to depict the amplitude and phase conditions for optimal absorption only.

Another model that is useful for gaining an overview of WECs is a linear mass-spring-damper model. A quick recap of mass-spring-damper models reminds us that at the natural frequency of the system (\(f_n\)), the response amplitude is highest, and the response phase is zero (with respect to the excitation phase). The inverse of the \(f_n\), the natural period (\(T_n\)), suits our convention of describing ocean waves in terms of period. A system that is being excited at its natural period is described as resonant.

Falnes's diagram shows a resonant system

A key insight into the wave absorber = wave maker diagram is that this describes a resonant system. How do we know that? Fig 1 depicts 100% absorption of the wave, and from linear wave theory we know that this occurs when the response (velocity) is in phase with the wave excitation. As we have just noted, the frequency at which the response is in phase with the excitation is the natural frequency.

Small systems have high natural frequencies

We know in general that small things have high \(f_n\) (low \(T_n\)) and big things have low \(f_n\) (high \(T_n\)). Hence violins resonate at high notes and cellos at low notes. The same general principle holds for all resonant systems, including wave absorbers. We can do our back-of-envelope check by recalling that our mass-spring-damper system has a natural frequency of \(f_n=\frac{1}{2\pi}\sqrt{k/m})\); a natural period of  \(T_n  = 2\pi\sqrt{m/k}\), where \(m\) is the mass and \(k\) is the spring. Mass increases with the volume, and buoyancy spring with the area, so an overall size increase leads to a longer natural period.

Falnes's diagram shows a body too small to be resonant

I'm going on the assumption that a body with dimensions much smaller than the wavelength has a natural period shorter than the wave period. This is a bit of a risky assumption because I've not derived the equations or considered a broad selection of case studies, and I would be interested to hear from anyone who knows more about this area. Nevertheless, the few examples I have considered suggest to me that this is a sound assumption in general.

In a previous article I noted that Falnes's diagram describes a body with dimensions smaller than the wavelength of the incident wave. We know this because the diagram shows that the incident wave does not appear to be impacted by the presence of the stationary body (in particular, there is no reflection). We know from linear wave theory that small (with respect to the waves) stationary bodies do not have a big impact on the incident waves.

So how is Falnes's diagram both resonant and too small to be resonant?

Fig 1 was not intended to represent dimensionally transcendental engineering. Falnes used his diagram to demonstrate a concept that he was researching at the time, often called complex conjugate control, or reactive control. It's a very clever idea whereby you force the system into resonance by doing clever things with the power take off (PTO) force.

The theory behind tricking the system into resonance

The theory of complex conjugate control strays off topic somewhat, but is an interesting subject no less, so there is a separate article dedicated to it. It highlights several difficulties with putting this theory to practice. In particular, this theory relies on the assumption that there are no limits to the amplitude of motion. When known amplitude constraints are applied, the cases where this theory is still valid are greatly reduced, and this excludes devices that are small with respect to wavelength. Very small devices need to make very large excursions in order to radiate waves with the same amplitude as the ocean waves that are of economic interest. This theory is also based on the assumption of an exciting wave with only one frequency component, and requires a 100% efficient power train that could operate as either a motor and a generator, and which typically operates at a fraction of the nameplate rated power.

So this diagram shows something that is physically impossible?

If this diagram is interpreted as a representation of something physically possible, rather than as a model, then it is indeed contradictory. It 'shows' behaviour that is physically impossible in useful applications: the amplitudes of motion required from a body so small that it experiences no diffraction (no reflection in Fig 1), would probably be larger than the proportions of the body allow.

If the body was large enough to cause diffraction, different motion would be required (see Fig 4 in this link): the aim being to radiate waves that interfere destructively with those diffracted waves. With the higher natural period you'd expect from a larger body, the system would be closer to resonance, and the benefit of the more expensive PTO system would be less valuable.

So what does this diagram tell us?

This diagram was never intended to suggest a solution; its purpose was to frame the problem. It communicates the concept that to absorb a wave, you must create a wave. This one diagram sums up the fundamental mechanism for capturing waves ,along with the requirements for efficient operation. These concepts are key to achieving our goal of economic wave energy. In my opinion, it is vital that everyone involved in wave energy development has access to these ideas.

I have gone looking for contradictions arising from literal interpretation of this model (Fig 1) in the same spirit that I have looked at art by Botticelli and Piero di Cosimo. These paintings and Falnes's 'wave absorber = wave maker' diagram have this in common: to communicate beautiful ideas to a non-specialist audience requires a simplicity that omits details of interest to the specialist audience.

Image credits:

Mathmatoduck belongs to Michael J. Doré :
Photo by Holly Falconer:
Musical ducks from :