I don't have an answer to this simple question right now. I think it needs further research to justify a valid definition. Currently the reason everyone uses this definition is primarily because, well, everyone uses this definition. May be for its simpleness and thus it is convenient to use. But curiously, no one seems care enough to wonder or asking "WHY?"

Anyone can define things where no other definitions have been adopted so far. Someone must have defined it as such and others followed it. This is the usual way, there is no further reasoning for taking twice the significant wave height as rogue (freak) wave height. The significant wave height definition is as arbitrary as the one for the rogue wave height. Once it is adopted it stays forever unless some "committee" makes a re-definition (e.g. Pluto not being a planet any more).

Dear Paul,

The definition is totally irrational. I avoid using it because I find it wholly misleading. Using Hs as a reference is also irrational but less so since nearly everybody seems to believe it with religious zeal, and it has become an unfortunate unavoidable fixture in oceanography and ocean engineering that is impossible to get rid of and replace with the rms surface displacements as the more logical choice. To an outsider well informed in statistics and stochastic processes, this will look a rather nonsensical parameter for scaling various features of the sea-surface elevations.

There are more rational ways for describing wave occurrences that do not seem to follow expectations defined in terms of a reliable wave-height model. These are basically outliers that can be defined on a logical basis as a joint event that a wave height several times larger than the surface rms occurs, and it does so more frequently than what we predict from a well-established wave-height frequency distribution. In other words, the definition must include a threshold representing a relatively large magnitude as well as some higher rate of occurrence. Otherwise, wave-heights satisfying thresholds such as H//Hs > 2-3 or more are not necessarily abnormal since they are predicted by the Gaussian model, obviously with very low frequencies of occurrence.

An apparent difficulty with the alternative definition implied here is that how one decides on a relative threshold magnitude that would be of importance in theory and practice. My experience is that even the so-called abnormal waves do not violate the Miche-Stokes type upper bounds. Relatively low or small waves do, but we are not interested in these in the present context. Given the spectrum, the largest wave that we might observe based on the Gaussian or some non-linear theory is that implied by this upper bound. If one uses the spectral average period (T01) associated with the largest waves in a sea state, the period of the largest ones seem to converge to a value 1-1.25 * T01 approximately, depending on the frequency bandwidth. The corresponding upper limit wave height then follows immediately from the Miche-Stokes bound. That is one possible clue to the threshold magnitude needed. Consequently, the relative frequency of that threshold is easily  estimated from the theoretical wave-height frequency curve. If the wave height and associated occurrence frequency of a large wave actually observed exceeds the two thresholds both noticeably, say about 5-10% or so, then one can contemplate if the observed wave is an abnormal occurrence. 

The bigger dilemma is that there is presently no way for experimentally verifying the existence of these "abnormal waves" under oceanic conditions. They seem to be isolated occurrences observed as outliers in some relatively short wave record, but it is ignored that the wave field where that record comes from actually contains zillions of other waves. Fixed-point observations are archaic and simply misleading for extremes to begin with in realistic seas characterized with a directional wide-band spectral content. So, what is needed is much larger spatial-temporal observations of the wind-disturbed sea-surface over long intervals of time. That has not become a routine yet. Numerical and experimental simulations in wave tanks do not explain what might be happening in the ocean. There is a lot more to discuss on the general topic, but enough now.

Best regards...Aziz

Just one thought...may be because the maximum wave height  is about 1.6~2.0 times the significant wave height? Using the equation in Goda's book, the mean ratio of the maximum wave height and significant wave height is related to the number of waves in one wave record (base on Rayleigh distribution) and would be less than 2.2 even the number of waves is 10,000 (several hours).

Dear Paul,

Another simple thought...

Statistics is one thing. Physics another one. If you could define a freak wave in terms of processes for its origin and physical characteristics, then the question is solved. However we need some simple criteria to identify such waves independently of the processes at works, or else in terms of statistics

Regards,

G.C.

My sincere thanks to all the helpful conducive answers to my question, I am gratified. Allow me to answer chronologically:

To Peter Schluessel: Interesting that you make the Pluto analogy and the community overpowering. I am not certain the freaque waves community is as well organized as the Astronomy. At least no one dictate us to use a definition yet!

To Mehmet Aziz Tayfun: Dear Aziz, I am overly delighted to read your comment. Reading your comments is as enjoyable as listening to your lecture, full of heuristic enlightenment and wisdom. It is gratifying for me to see an old friend spending time to offer these opinions. I am particularly thrilled to see your point that “what is needed is much larger spatial-temporal observations”, I couldn’t agree more. That’s in fact the very point I have been trying advocating in recent years. Now I think what we need is a viable definition for freaque waves. A freaque wave is characterized by its size and its unexpectedness. So far we have only pay attention to the size part, totally ignored the unexpectedness of a freaque wave’s occurrence. That’s another major difficulty we face – how to define unexpec tedness? A storm wave field can have wave higher than 2*Hs but that’s not unexpected. Large storm waves are clearly not necessarily freaque waves.

To Leng-Hsuan Tseng: You are quite right to indicate that the connection with Rayleigh distribution provided the bases for the current definition. I think the use of 2 or 2.2 times for the definition is rather arbitrary, some freaque waves can be much higher than that.

To Adrian Constantin: I guess we are basically seeking what could be a realistic convention!

To G. Caniaux: My problem is that I have no idea what a wave condition of one wave height greater than 2*Hs really looks like, what really makes it a freaque wave?

 My way of thinking as an operational marine forecaster that is supporting an operational wave model: The difference between the ordinary wind wave and freak wave must be qualitative rather than quantitative.  A real life example: one or two years ago the people from a small coastal town complained that our wave forecast failed and some wooden cottages on the coast were destroyed by the waves and there was no warning. The model prediction was for 1.5 m significant wave height near shore. So 2x significant wave height was 3m for the maximum wave height.  Even if a single wave was say 3x significant wave height- 4.5m  this still will agree with our prediction- such low probability event may happen simply because it is possible and even such individual wave is not qualitatively different from a 3m wave- it cannot cause damages. Even a 6m wave cannot.  In all our coastal stations for that date there were no high waves and strong wind. Later we realized that the local people are speaking for a damage due to a single wave and it was not related to any seismic event.  So this was an event that was in principle not predictable and it is an event with a qualitatively higher potential impact. Wave that is 2.5x SWH is a low probability event but it still agrees with the SWH- it is a part of the same distribution and normal part of the spectrum and You don't need any different theory to explain it. So freak event= event with a qualitatively higher potential impact that was in principle not predictable. If it is predictable it cannot be freak.  

Hey I really like your last comment: "If it is predictable it cannot be freak."  Yes, indeed, A freaque wave in the ocean is unexpected and unpredictable. It happens all the time out there, even when no one's around to encounter it. But we don't know where, when, how, or why. Now what can sciences do to help?

Interesting discussion.  

The answer has been given using maybe too many words, but here is how I will explain it to my u/g students.

Hrms = 4*sigma (sigma - st. dev. of sea surface variation)

Hsig = sqrt(2)*Hrms = sqrt(2)*4*sigma

Hfreak >= 2*Hsig thus >= 2*sqrt(2)*4*sigma = 11.36*sigma

Ampl_freak = Hfreak/2 >= 4*sqrt(2)*sigma = 5.68*sigma

Now for a normal distribution (if we assume waves are normally distributed) a freak wave corresponds to a sea surface variability that is greater than 2*sigma and we know that 2*sigma captures 95% of the variability, 3*sigma captures 99.7% of the variability. So 5.68*sigma captures approx. 99.99999% (someone can do the exact calculation)

If we look now on time-series analysis people have been using the 3-Edit rule to identify spurious data. That rule classifies data points as “normal” (in the region mean±3σ ) or “suspicious” (outside the range mean±3σ). This suspicious area is what someone can call "freak" in the waves world.

So combining the two ideas Hfreak could be defined as > 6*sigma and not as 11.36*sigma. As you can see the 11.36*sigma (5.68*sigma in terms of amplitude) is much higher the area of "suspicious" and if yoru instrument work well and it is real, then is in the "freak" (i.e., extremely unlikely) region. Again note that the definition is relative to sea state (see variance) and not to an absolute value.

Not sure if I offered anything new, at least it made me think and find a way to explain this to my students (if they ask me!)

Thank you, George, for this very admirably clear elucidation.  So you basically define freaque waves by their likely-ness or unlikely-ness. A minor new twist. But the most dangerous character of a freaque wave overall is still its unexpectedness. Is it possible to squeeze out some notion of unexpectedness through your "suspiciousness", "likely-ness" or "unlikely-ness" ?

The meaning of the term ‘freak’ (or ‘rogue’) waves in science and life is quite different. According to the marine folklore freak waves are described as the ‘monster’ waves appearing like high ‘walls of water’ with deep ‘holes in the sea’ around them. In science the ‘freak’ wave is defined as a wave whose trough-to-crest height exceeds doubled significant wave height . ( is some typical wave height above the mean level). So, if a significant wave height is equal to one meter, then all the waves with a trough-to-crest height exceeding two meters should be referred to as belonging to the category of ‘freak’ waves. It is hard to imagine that such waves can be characterized as ‘monster’ waves even for a small vessel. On the other side, if a steady West wind with a speed of 20 m/s in the South ocean generates a wave with the height of around 20 m and length of around 0.5 km (according to the reports of oceanographers sailing in those areas such waves are not rare), then such a wave would just lift and drop a vessel, the only damage incurred, being yet another attack of seasickness among the vessel crew. What seems remarkable is that the sea folklore provides a better description of the freak wave properties focusing on their shape and assuming, of course, that such waves are very big. The term ‘vertical walls’ definitely indicates a wave surge in front of observer and undergoes an active phase of breaking. Evidently, the current scientific definition of the term ‘freak wave’ is imperfect. However, the scientific definition of freak waves has a solid theoretical background, since it is based on the fundamental properties of the adiabatic motion described by Euler equations, i.e., these equations are self-similar, because, being transformed into a nondimensional form they do not contain a nondimensional parameter. Hence, the nondimensional equations describe the whole class of motions. Thus, for obtaining specific dimensional results it is enough to multiply the solution by an appropriate length scale. It can be still concluded that for nondimensional equations the scientific definition of freak waves is justified.

Thanks Dmitry, for the very heuristic explanation on the theoretical justification of the widely used popular  "scientific" definition of freaque waves -- more than two times of significant wave height. My problem is two folds, on the one hand I feel a freaque wave is unexpected and unpredictable, the scientific definition is clearly not able to define the unexpectedness, which is more important characterization of freaque wave than size. On the other hand, as a data analyst I have seen many multitudes of conventional wave data recordings and they all show typically galaxies of points with Hmax/Hs greater than 2 -- almost like common occurrences of the daily wave world. Then why it is called freaque?  I am also wondering since we talk about monster waves and we also talk about greater then 2*Hs freaque waves are they the same thing?

Dear Paul, Surely the number 2 is conventional, some people prefer 2.1.

Considering practical application of the rare wave theory, we can also come to conclusion that a strict unconditional ‘definition’ of freak waves is not required at all. For better use of the research recommendations, it would be more efficient to define categories of freak waves, as it has been done, for examples, for tropical storms. A reasonable warning on appearance of such waves should sound like: ‘from 6 am today until 6 am tomorrow in a specific area of 100x100 km a breaking wave with a height of 10 m (category three) shall be one of waves, a breaking wave with a height of 15 m (category five) shall be one of waves,… etc’. For unbreaking waves probability of such waves is somewhat higher. The probability of coming across a freak wave is convenient to express in terms of expectance time for waves of different categories. A set of the most important dynamic characteristics of such waves can be also provided. Potential customers can decide themselves, whether it is a real ‘freak’ wave, and either modify their route or degree of preparedness accordingly. Similar recommendations can be developed for ship designing, sea constructions and insurance purposes. Naturally, extreme wave is a phenomenon, which manifests itself in a direct contact with an object. Such cases can be relatively frequent in uncomfortable areas with high winds and low intensity of navigation (for example, in middle and high latitudes of the South Ocean) and therefore, remain unnoticed. On the contrary, in the areas of recommended routes (southern Africa), even a single catastrophic event may create a freak wave of publications. If the probability of extreme waves could be connected with the more or less standard oceanographic characteristics (for example, data on wind and wave climate), the estimations of climatology of dangerous waves of different categories might be very useful for industry, navigation, ship design and, of course, for insurance purposes. The reliable data on direct registration of freak waves events are sketchy, and operational monitoring of extreme waves from satellites is the most important, though perhaps, not resolved problem.

Much more surprisingly, that simple calculations show, that freak wave is not something special, since superposition of linear waves with empirical spectrum and random phases gives absolutely the same probability of extreme waves. So, it is not dynamics, it is just geometry, and no any special theories are needed for their explanations. Unfortunately, I also fell in serious discussion of nonlinear nature of freak waves: Phys of Fluid, 2009,21, 076602. Right now I am writing a paper, proving that such non-linear problem does not exist at all.

By the way nondimensional freak wave are not too rare. To calculate the probability of freak height is nesessary to multiply the probability of nondimensional wave by probability of significant wave height.

Hope it will be useful ,

Regards,

DCh