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94-473: After being hit by a rogue wave during a storm, the trimaran capsized, trapping the crew inside. After cutting an escape hatch, they set the emergency position-indicating radiobeacon (EPIRB), convinced that they would be rescued a few days later. The water tanks, which contained 140 liters of fresh water, slowly emptied themselves unbeknown to the crew. The EPIRB, which had a radius of one hundred nautical miles , stopped working on June 13 after 8 days. They made

188-485: A polynomial equation such as x 2 + x − 1 = 0. {\displaystyle x^{2}+x-1=0.} The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or the real roots; see real-root isolation . Solving systems of polynomial equations , that

282-469: A laser rangefinder wave recorder on the platform's underside. At 3 pm on 1 January 1995, the device recorded a rogue wave with a maximum wave height of 25.6 m (84 ft). Peak elevation above still water level was 18.5 m (61 ft). The reading was confirmed by the other sensors. The platform sustained minor damage in the event. In the area, the SWH at the time was about 12 m (39 ft), so

376-467: A linear map (or linear function ) f ( x ) {\displaystyle f(x)} is one which satisfies both of the following properties: Additivity implies homogeneity for any rational α , and, for continuous functions , for any real α . For a complex α , homogeneity does not follow from additivity. For example, an antilinear map is additive but not homogeneous. The conditions of additivity and homogeneity are often combined in

470-524: A nonlinear system (or a non-linear system ) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers , biologists , physicists , mathematicians , and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems , describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems . Typically,

564-510: A "normal" wave begins to draw energy from other nearby waves, and briefly becomes very large. Such phenomena are not limited to water and are also studied in liquid helium, nonlinear optics, and microwave cavities. A 2012 study reported that in addition to the Peregrine soliton reaching up to about three times the height of the surrounding sea, a hierarchy of higher order wave solutions could also exist having progressively larger sizes and demonstrated

658-456: A 29 m (95 ft) wave off the coast of Scotland near Rockall . This was a scientific research vessel fitted with high-quality instruments. Subsequent analysis determined that under severe gale-force conditions with wind speeds averaging 21 metres per second (41 kn), a ship-borne wave recorder measured individual waves up to 29.1 m (95.5 ft) from crest to trough, and a maximum SWH of 18.5 m (60.7 ft). These were some of

752-785: A breaking wave of 15 t/m , a rogue wave can dwarf both of these figures with a breaking force far exceeding 100 t/m . Smith presented calculations using the International Association of Classification Societies (IACS) Common Structural Rules for a typical bulk carrier. Peter Challenor, a scientist from the National Oceanography Centre in the United Kingdom, was quoted in Casey 's book in 2010 as saying: "We don’t have that random messy theory for nonlinear waves. At all." He added, "People have been working actively on this for

846-423: A frictionless pendulum under the influence of gravity . Using Lagrangian mechanics , it may be shown that the motion of a pendulum can be described by the dimensionless nonlinear equation where gravity points "downwards" and θ {\displaystyle \theta } is the angle the pendulum forms with its rest position, as shown in the figure at right. One approach to "solving" this equation

940-422: A fully nonlinear model by R. H. Gibbs (2005) brings this mode into question, as it is shown that a typical wave group focuses in such a way as to produce a significant wall of water at the cost of a reduced height. A rogue wave, and the deep trough commonly seen before and after it, may last only for some minutes before either breaking or reducing in size again. Apart from a single one, the rogue wave may be part of

1034-557: A limited selection of notable incidents. The loss of the MS ; München in 1978 provided some of the first physical evidence of the existence of rogue waves. München was a state-of-the-art cargo ship with multiple water-tight compartments and an expert crew. She was lost with all crew, and the wreck has never been found. The only evidence found was the starboard lifeboat recovered from floating wreckage sometime later. The lifeboats hung from forward and aft blocks 20 m (66 ft) above

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1128-457: A massive displacement of water, often resulting from sudden movements of the ocean floor , after which they propagate at high speed over a wide area. They are nearly unnoticeable in deep water and only become dangerous as they approach the shoreline and the ocean floor becomes shallower; therefore, tsunamis do not present a threat to shipping at sea (e.g., the only ships lost in the 2004 Asian tsunami were in port.). These are also different from

1222-591: A paper in 2000, collating evidence that freak waves were not the rare realizations of a typical or slightly non-gaussian sea surface population ( classical extreme waves) but were the typical realizations of a rare and strongly non-gaussian sea surface population of waves ( freak extreme waves). A workshop of leading researchers in the world attended the first Rogue Waves 2000 workshop held in Brest in November 2000. In 2000, British oceanographic vessel RRS Discovery recorded

1316-424: A rain water collecting device by splitting lengths of plastic pipe. After about 2–3 months, Barnacles and mollusks began to grow on the hulls, making fishing easier. Their story is told in the 2015 New Zealand television film Abandoned , starring Dominic Purcell and Peter Feeney, along with Owen Black and Greg Johnson. It was directed by John Laing . This article about a specific civilian ship or boat

1410-470: A recorded maximum wave height of 25.6 m (84 ft) and peak elevation of 18.5 m (61 ft). During that event, minor damage was inflicted on the platform far above sea level, confirming the accuracy of the wave-height reading made by a downwards pointing laser sensor. The existence of rogue waves has since been confirmed by video and photographs, satellite imagery , radar of the ocean surface, stereo wave imaging systems, pressure transducers on

1504-469: A report of an 11-metre (36 ft) wave in a significant wave height of 1.9 metres (6 ft 3 in), but the authors cast doubt on that claim. A paper written by Craig B. Smith in 2007 reported on an incident in the North Atlantic, in which the submarine Grouper was hit by a 30-meter wave in calm seas. Because the phenomenon of rogue waves is still a matter of active research, clearly stating what

1598-502: A scientific model (and also ship design method) to describe the waves encountered did not exist. This finding was widely reported in the press, which reported that "according to all of the theoretical models at the time under this particular set of weather conditions, waves of this size should not have existed". In 2004, the ESA MaxWave project identified more than 10 individual giant waves above 25 m (82 ft) in height during

1692-510: A short phase of relative order". In 2019, researchers succeeded in producing a wave with similar characteristics to the Draupner wave (steepness and breaking), and proportionately greater height, using multiple wavetrains meeting at an angle of 120°. Previous research had strongly suggested that the wave resulted from an interaction between waves from different directions ("crossing seas"). Their research also highlighted that wave-breaking behavior

1786-488: A short survey period of three weeks in a limited area of the South Atlantic. By 2007, it was further proven via satellite radar studies that waves with crest-to-trough heights of 20 to 30 m (66 to 98 ft) occur far more frequently than previously thought. Rogue waves are now known to occur in all of the world's oceans many times each day. Rogue waves are now accepted as a common phenomenon. Professor Akhmediev of

1880-467: A single large wave. Recent research suggests sea state crest-trough correlation leading to linear superposition may be a dominant factor in predicting the frequency of rogue waves. Among other causes, studies of nonlinear waves such as the Peregrine soliton , and waves modeled by the nonlinear Schrödinger equation (NLS), suggest that modulational instability can create an unusual sea state where

1974-492: A wave packet consisting of a few rogue waves. Such rogue wave groups have been observed in nature. A number of research programmes are currently underway or have concluded whose focus is/was on rogue waves, including: Researchers at UCLA observed rogue-wave phenomena in microstructured optical fibers near the threshold of soliton supercontinuum generation and characterized the initial conditions for generating rogue waves in any medium. Research in optics has pointed out

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2068-637: Is a simple harmonic oscillator corresponding to oscillations of the pendulum near the bottom of its path. Another linearization would be at θ = π {\displaystyle \theta =\pi } , corresponding to the pendulum being straight up: since sin ⁡ ( θ ) ≈ π − θ {\displaystyle \sin(\theta )\approx \pi -\theta } for θ ≈ π {\displaystyle \theta \approx \pi } . The solution to this problem involves hyperbolic sinusoids , and note that unlike

2162-488: Is a stub . You can help Misplaced Pages by expanding it . Rogue wave Rogue waves (also known as freak waves or killer waves ) are large and unpredictable surface waves that can be extremely dangerous to ships and isolated structures such as lighthouses . They are distinct from tsunamis , which are long wavelength waves, often almost unnoticeable in deep waters and are caused by the displacement of water due to other phenomena (such as earthquakes ). A rogue wave at

2256-528: Is a natural ocean phenomenon that is not caused by land movement, only lasts briefly, occurs in a limited location, and most often happens far out at sea. Rogue waves are considered rare, but potentially very dangerous, since they can involve the spontaneous formation of massive waves far beyond the usual expectations of ship designers , and can overwhelm the usual capabilities of ocean-going vessels which are not designed for such encounters. Rogue waves are, therefore, distinct from tsunamis . Tsunamis are caused by

2350-568: Is a rogue hole recording from an oil platform in the North Sea, revealed in Kharif et al. The same source also reveals a recording of what is known as the 'Three Sisters'. Rogue waves are waves in open water that are much larger than surrounding waves. More precisely, rogue waves have a height which is more than twice the significant wave height ( H s or SWH). They can be caused when currents or winds cause waves to travel at different speeds, and

2444-431: Is always useful whether or not the resulting ordinary differential equation(s) is solvable. Another common (though less mathematical) tactic, often exploited in fluid and heat mechanics, is to use scale analysis to simplify a general, natural equation in a certain specific boundary value problem . For example, the (very) nonlinear Navier-Stokes equations can be simplified into one linear partial differential equation in

2538-666: Is exposed to the Atlantic Ocean. The peak pressure recorded by a shore-mounted transducer was 745 kPa (7.45 bar; 108.1 psi). This pressure far exceeds almost any design criteria for modern ships, and this wave would have destroyed almost any merchant vessel. In November 1997, the International Maritime Organization (IMO) adopted new rules covering survivability and structural requirements for bulk carriers of 150 m (490 ft) and upwards. The bulkhead and double bottom must be strong enough to allow

2632-432: Is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborated algorithms have been designed, such as Gröbner base algorithms. For the general case of system of equations formed by equating to zero several differentiable functions , the main method is Newton's method and its variants. Generally they may provide a solution, but do not provide any information on

2726-454: Is in effect 20 m (66 ft) of seawater (possibly a super rogue wave) flowing over the vessel. The deck cargo hatches on the Derbyshire were determined to be the key point of failure when the rogue wave washed over the ship. The design of the hatches only allowed for a static pressure less than 2 m (6.6 ft) of water or 17.1 kPa (0.171 bar; 2.48 psi), meaning that

2820-552: Is one-dimensional heat transport with Dirichlet boundary conditions , the solution of which can be written as a time-dependent linear combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the construction of new solutions. First order ordinary differential equations are often exactly solvable by separation of variables , especially for autonomous equations. For example,

2914-544: Is sufficient pressure to lead to brittle fracture of mild steel. Evidence of failure by this mechanism was also found on the Derbyshire . Smith documented scenarios where hydrodynamic pressure up to 5,650 kPa (56.5 bar; 819 psi) or over 500 metric tonnes/m could occur. In 2004, an extreme wave was recorded impacting the Alderney Breakwater, Alderney , in the Channel Islands . This breakwater

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3008-480: Is to use d θ / d t {\displaystyle d\theta /dt} as an integrating factor , which would eventually yield which is an implicit solution involving an elliptic integral . This "solution" generally does not have many uses because most of the nature of the solution is hidden in the nonelementary integral (nonelementary unless C 0 = 2 {\displaystyle C_{0}=2} ). Another way to approach

3102-511: Is very general in that x {\displaystyle x} can be any sensible mathematical object (number, vector, function, etc.), and the function f ( x ) {\displaystyle f(x)} can literally be any mapping , including integration or differentiation with associated constraints (such as boundary values ). If f ( x ) {\displaystyle f(x)} contains differentiation with respect to x {\displaystyle x} ,

3196-590: The Ocean Ranger , a semisubmersible mobile offshore drilling unit that sank in Canadian waters on 15 February 1982. In 2007, the United States' National Oceanic and Atmospheric Administration (NOAA) compiled a catalogue of more than 50 historical incidents probably associated with rogue waves. In 1826, French scientist and naval officer Jules Dumont d'Urville reported waves as high as 33 m (108 ft) in

3290-546: The Australian National University has stated that 10 rogue waves exist in the world's oceans at any moment. Some researchers have speculated that roughly three of every 10,000 waves on the oceans achieve rogue status, yet in certain spots ‍ — such as coastal inlets and river mouths ‍ — these extreme waves can make up three of every 1,000 waves, because wave energy can be focused. Rogue waves may also occur in lakes . A phenomenon known as

3384-739: The Black Sea in 2004 which was far more extreme than the Ucluelet wave, where the Datawell Waverider buoy reported a wave whose height was 10.32 metres (33.86 ft) higher and 3.91 times the significant wave height, as detailed in the paper. Thorough inspection of the buoy after the recording revealed no malfunction. The authors of the paper that reported the Black Sea event assessed the wave as "anomalous" and suggested several theories on how such an extreme wave may have arisen. The Black Sea event differs in

3478-474: The Derbyshire with the emerging science on freak waves, concluding that the Derbyshire was almost certainly destroyed by a rogue wave. Work by sailor and author Craig B. Smith in 2007 confirmed prior forensic work by Faulkner in 1998 and determined that the Derbyshire was exposed to a hydrostatic pressure of a "static head" of water of about 20 m (66 ft) with a resultant static pressure of 201 kilopascals (2.01 bar; 29.2 psi). This

3572-529: The Navier–Stokes equations in fluid dynamics and the Lotka–Volterra equations in biology. One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to construct general solutions through the superposition principle . A good example of this

3666-402: The significant wave height (SWH). In a storm sea with an SWH of 12 m (39 ft), the model suggests hardly ever would a wave higher than 15 m (49 ft) occur. It suggests one of 30 m (98 ft) could indeed happen, but only once in 10,000 years. This basic assumption was well accepted, though acknowledged to be an approximation. Using a Gaussian form to model waves has been

3760-579: The "Three Sisters" is said to occur in Lake Superior when a series of three large waves forms. The second wave hits the ship's deck before the first wave clears. The third incoming wave adds to the two accumulated backwashes and suddenly overloads the ship deck with large amounts of water. The phenomenon is one of various theorized causes of the sinking of the SS ; Edmund Fitzgerald on Lake Superior in November 1975. A 2012 study reported that in addition to

3854-400: The 20th century, "people who encountered 100-foot [30 m] rogue waves generally weren't coming back to tell people about it." Unusual waves have been studied scientifically for many years (for example, John Scott Russell 's wave of translation , an 1834 study of a soliton wave). Still, these were not linked conceptually to sailors' stories of encounters with giant rogue ocean waves, as

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3948-419: The Draupner wave was more than twice as tall and steep as its neighbors, with characteristics that fell outside any known wave model. The wave caused enormous interest in the scientific community. Following the evidence of the Draupner wave, research in the area became widespread. The first scientific study to comprehensively prove that freak waves exist, which are clearly outside the range of Gaussian waves,

4042-711: The Draupner wave was recorded in 1995, early research had already made significant strides in understanding extreme wave interactions. In 1979, Dik Ludikhuize and Henk Jan Verhagen at TU Delft successfully generated cross-swell waves in a wave basin. Although only monochromatic waves could be produced at the time, their findings, reported in 1981, showed that individual wave heights could be added together even when exceeding breaker criteria. This phenomenon provided early evidence that waves could grow significantly larger than anticipated by conventional theories of wave breaking. This work highlighted that in cases of crossing waves, wave steepness could increase beyond usual limits. Although

4136-455: The Indian Ocean with three colleagues as witnesses, yet he was publicly ridiculed by fellow scientist François Arago . In that era, the thought was widely held that no wave could exceed 9 m (30 ft). Author Susan Casey wrote that much of that disbelief came because there were very few people who had seen a rogue wave and survived ; until the advent of steel double-hulled ships of

4230-638: The Maritime Court investigation concluded that the severe weather had somehow created an "unusual event" that had led to the sinking of the München . In 1980, the MV Derbyshire was lost during Typhoon Orchid south of Japan, along with all of her crew. The Derbyshire was an ore-bulk oil combination carrier built in 1976. At 91,655 gross register tons, she was ‍ — and remains to be ‍ — the largest British ship ever lost at sea. The wreck

4324-481: The Peregrine soliton reaching up to about 3 times the height of the surrounding sea, a hierarchy of higher order wave solutions could also exist having progressively larger sizes, and demonstrated the creation of a "super rogue wave"—  a breather around 5 times higher than surrounding waves ‍ — in a water tank . Also in 2012, researchers at the Australian National University proved

4418-554: The Ucluelet wave. In the paper, they infer that in 2006 a 21-metre (69 ft) wave appeared in the Pacific Ocean off the Port of Coos Bay , Oregon, with a significant wave height of 3.9 metres (13 ft). The ratio is 5.38, almost twice that of the Ucluelet wave. The paper also reveals the MV ; Pont-Aven incident as marginally more extreme than the Ucluelet event. The paper also assesses

4512-424: The behavior of a nonlinear system is described in mathematics by a nonlinear system of equations , which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations ) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations,

4606-404: The case of transient, laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is laminar and one dimensional and also yields the simplified equation. Other methods include examining the characteristics and using the methods outlined above for ordinary differential equations. A classic, extensively studied nonlinear problem is the dynamics of

4700-432: The creation of a "super rogue wave" (a breather around five times higher than surrounding waves) in a water-wave tank . A 2012 study supported the existence of oceanic rogue holes, the inverse of rogue waves, where the depth of the hole can reach more than twice the significant wave height. Although it is often claimed that rogue holes have never been observed in nature despite replication in wave tank experiments, there

4794-580: The equation is not a linear function of u {\displaystyle u} and its derivatives. Note that if the u 2 {\displaystyle u^{2}} term were replaced with u {\displaystyle u} , the problem would be linear (the exponential decay problem). Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield closed-form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered. Common methods for

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4888-627: The equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it. As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ( linearization ). This works well up to some accuracy and some range for

4982-590: The existence of "rogue wave holes", an inverted profile of a rogue wave. Their research created rogue wave holes on the water surface in a water-wave tank. In maritime folklore , stories of rogue holes are as common as stories of rogue waves. They had followed from theoretical analysis but had never been proven experimentally. "Rogue wave" has become a near-universal term used by scientists to describe isolated, large-amplitude waves that occur more frequently than expected for normal, Gaussian-distributed, statistical events. Rogue waves appear ubiquitous and are not limited to

5076-476: The existence of all waves that have been detected, several different mechanisms are likely, with localized variation. Suggested mechanisms for freak waves include: The spatiotemporal focusing seen in the NLS equation can also occur when the non-linearity is removed. In this case, focusing is primarily due to different waves coming into phase rather than any energy-transfer processes. Further analysis of rogue waves using

5170-425: The fact that it, unlike the Ucluelet wave, was recorded with a high-precision instrument. The Oceanic rogue waves paper also reports even more extreme waves from a different source, but these were possibly overestimated, as assessed by the data's own authors. The Black Sea wave occurred in relatively calm weather. Furthermore, a paper by I. Nikolkina and I. Didenkulova also reveals waves more extreme than

5264-489: The formation of near-vertical jets". On 17 November 2020, a buoy moored in 45 metres (148 ft) of water on Amphitrite Bank in the Pacific Ocean 7 kilometres (4.3 mi; 3.8 nmi) off Ucluelet , Vancouver Island , British Columbia , Canada , at 48°54′N 125°36′W  /  48.9°N 125.6°W  / 48.9; -125.6 recorded a lone 17.6-metre (58 ft) tall wave among surrounding waves about 6 metres (20 ft) in height. The wave exceeded

5358-458: The input values, but some interesting phenomena such as solitons , chaos , and singularities are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of

5452-1766: The largest wave likely to be encountered was 21.4 m (70 ft). Smith observed in 2007 that the navy now believes that larger waves can occur and the possibility of extreme waves that are steeper (i.e. do not have longer wavelengths) is now recognized. The navy has not had to make any fundamental changes in ship design due to new knowledge of waves greater than 21.4 m because the ships are built to higher standards than required. Non-linearity Collective intelligence Collective action Self-organized criticality Herd mentality Phase transition Agent-based modelling Synchronization Ant colony optimization Particle swarm optimization Swarm behaviour Social network analysis Small-world networks Centrality Motifs Graph theory Scaling Robustness Systems biology Dynamic networks Evolutionary computation Genetic algorithms Genetic programming Artificial life Machine learning Evolutionary developmental biology Artificial intelligence Evolutionary robotics Reaction–diffusion systems Partial differential equations Dissipative structures Percolation Cellular automata Spatial ecology Self-replication Conversation theory Entropy Feedback Goal-oriented Homeostasis Information theory Operationalization Second-order cybernetics Self-reference System dynamics Systems science Systems thinking Sensemaking Variety Ordinary differential equations Phase space Attractors Population dynamics Chaos Multistability Bifurcation Rational choice theory Bounded rationality In mathematics and science ,

5546-555: The largest waves recorded by scientific instruments up to that time. The authors noted that modern wave prediction models are known to significantly under-predict extreme sea states for waves with a significant height (H s ) above 12 m (39.4 ft). The analysis of this event took a number of years and noted that "none of the state-of-the-art weather forecasts and wave models ‍ — the information upon which all ships, oil rigs, fisheries, and passenger boats rely ‍ — had predicted these behemoths." In simple terms,

5640-466: The laser used for the Draupner wave might be somewhat confused by the spray at the top of the wave if it broke, and this could lead to uncertainties of around 1.0 to 1.5 m (3 to 5 ft) in the wave height. They concluded, "... the onset and type of wave breaking play a significant role and differ significantly for crossing and noncrossing waves. Crucially, breaking becomes less crest-amplitude limiting for sufficiently large crossing angles and involves

5734-476: The latter were believed to be scientifically implausible. Since the 19th century, oceanographers, meteorologists, engineers, and ship designers have used a statistical model known as the Gaussian function (or Gaussian Sea or standard linear model) to predict wave height, on the assumption that wave heights in any given sea are tightly grouped around a central value equal to the average of the largest third, known as

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5828-424: The media, and are not examples of open-ocean rogue waves. Often, in popular culture, an endangering huge wave is loosely denoted as a "rogue wave", while the case has not been established that the reported event is a rogue wave in the scientific sense – i.e. of a very different nature in characteristics as the surrounding waves in that sea state] and with a very low probability of occurrence. This section lists

5922-429: The most common causes are or whether they vary from place to place is premature. The areas of highest predictable risk appear to be where a strong current runs counter to the primary direction of travel of the waves; the area near Cape Agulhas off the southern tip of Africa is one such area. The warm Agulhas Current runs to the southwest, while the dominant winds are westerlies , but since this thesis does not explain

6016-408: The nonlinear equation has u = 1 x + C {\displaystyle u={\frac {1}{x+C}}} as a general solution (and also the special solution u = 0 , {\displaystyle u=0,} corresponding to the limit of the general solution when C tends to infinity). The equation is nonlinear because it may be written as and the left-hand side of

6110-511: The number of solutions. A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter sequences . Nonlinear discrete models that represent a wide class of nonlinear recurrence relationships include the NARMAX (Nonlinear Autoregressive Moving Average with eXogenous inputs) model and

6204-491: The oceans. They appear in other contexts and have recently been reported in liquid helium, nonlinear optics, and microwave cavities. Marine researchers universally now accept that these waves belong to a specific kind of sea wave, not considered by conventional models for sea wind waves. A 2015 paper studied the wave behavior around a rogue wave, including optical and the Draupner wave, and concluded, "rogue events do not necessarily appear without warning but are often preceded by

6298-553: The past 50 years at least. We don’t even have the start of a theory." In 2006, Smith proposed that the IACS recommendation 34 pertaining to standard wave data be modified so that the minimum design wave height be increased to 19.8 m (65 ft). He presented analysis that sufficient evidence exists to conclude that 20.1 m (66 ft) high waves can be experienced in the 25-year lifetime of oceangoing vessels, and that 29.9 m (98 ft) high waves are less likely, but not out of

6392-535: The problem is to linearize any nonlinearity (the sine function term in this case) at the various points of interest through Taylor expansions . For example, the linearization at θ = 0 {\displaystyle \theta =0} , called the small angle approximation, is since sin ⁡ ( θ ) ≈ θ {\displaystyle \sin(\theta )\approx \theta } for θ ≈ 0 {\displaystyle \theta \approx 0} . This

6486-429: The qualitative analysis of nonlinear ordinary differential equations include: The most common basic approach to studying nonlinear partial differential equations is to change the variables (or otherwise transform the problem) so that the resulting problem is simpler (possibly linear). Sometimes, the equation may be transformed into one or more ordinary differential equations , as seen in separation of variables , which

6580-653: The question. Therefore, a design criterion based on 11.0 m (36 ft) high waves seems inadequate when the risk of losing crew and cargo is considered. Smith also proposed that the dynamic force of wave impacts should be included in the structural analysis. The Norwegian offshore standards now consider extreme severe wave conditions and require that a 10,000-year wave does not endanger the ships' integrity. W. Rosenthal noted that as of 2005, rogue waves were not explicitly accounted for in Classification Society's rules for ships' design. As an example, DNV GL , one of

6674-497: The recording of a rogue wave by the Gorm platform in the central North Sea in 1984. A stand-out wave was detected with a wave height of 11 m (36 ft) in a relatively low sea state. However, what caught the attention of the scientific community was the digital measurement of a rogue wave at the Draupner platform in the North Sea on January 1, 1995; called the "Draupner wave", it had

6768-499: The related nonlinear system identification and analysis procedures. These approaches can be used to study a wide class of complex nonlinear behaviors in the time, frequency, and spatio-temporal domains. A system of differential equations is said to be nonlinear if it is not a system of linear equations . Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are

6862-420: The result will be a differential equation . A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a linear equation . For a single equation of the form f ( x ) = 0 , {\displaystyle f(x)=0,} many methods have been designed; see Root-finding algorithm . In the case where f is a polynomial , one has

6956-468: The role played by a Peregrine soliton that may explain those waves that appear and disappear without leaving a trace. Rogue waves in other media appear to be ubiquitous and have also been reported in liquid helium , in quantum mechanics, in nonlinear optics , in microwave cavities, in Bose–Einstein condensate , in heat and diffusion, and in finance. Many of these encounters are reported only in

7050-652: The sea-floor, and oceanographic research vessels. In February 2000, a British oceanographic research vessel, the RRS Discovery , sailing in the Rockall Trough west of Scotland, encountered the largest waves ever recorded by any scientific instruments in the open ocean, with a SWH of 18.5 metres (61 ft) and individual waves up to 29.1 metres (95 ft). In 2004, scientists using three weeks of radar images from European Space Agency satellites found ten rogue waves, each 25 metres (82 ft) or higher. A rogue wave

7144-454: The ship sank because of structural failure and absolved the crew of any responsibility. Most notably, the report determined the detailed sequence of events that led to the structural failure of the vessel. A third comprehensive analysis was subsequently done by Douglas Faulkner, professor of marine architecture and ocean engineering at the University of Glasgow . His 2001 report linked the loss of

7238-442: The ship to survive flooding in hold one unless loading is restricted. Rogue waves present considerable danger for several reasons: they are rare, unpredictable, may appear suddenly or without warning, and can impact with tremendous force. A 12 m (39 ft) wave in the usual "linear" model would have a breaking force of 6 metric tons per square metre [t/m ] (8.5 psi). Although modern ships are typically designed to tolerate

7332-445: The shore is sometimes called a sneaker wave . In oceanography , rogue waves are more precisely defined as waves whose height is more than twice the significant wave height ( H s or SWH), which is itself defined as the mean of the largest third of waves in a wave record. Rogue waves do not appear to have a single distinct cause but occur where physical factors such as high winds and strong currents cause waves to merge to create

7426-645: The small angle approximation, this approximation is unstable, meaning that | θ | {\displaystyle |\theta |} will usually grow without limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is literally an unstable state. One more interesting linearization is possible around θ = π / 2 {\displaystyle \theta =\pi /2} , around which sin ⁡ ( θ ) ≈ 1 {\displaystyle \sin(\theta )\approx 1} : This corresponds to

7520-509: The sole basis of virtually every text on that topic for the past 100 years. The first known scientific article on "freak waves" was written by Professor Laurence Draper in 1964. In that paper, he documented the efforts of the National Institute of Oceanography in the early 1960s to record wave height, and the highest wave recorded at that time, which was about 20 metres (67 ft). Draper also described freak wave holes . Before

7614-482: The superposition principle An equation written as is called linear if f ( x ) {\displaystyle f(x)} is a linear map (as defined above) and nonlinear otherwise. The equation is called homogeneous if C = 0 {\displaystyle C=0} and f ( x ) {\displaystyle f(x)} is a homogeneous function . The definition f ( x ) = C {\displaystyle f(x)=C}

7708-486: The surrounding significant wave heights by a factor of 2.93. When the wave's detection was revealed to the public in February 2022, one scientific paper and many news outlets christened the event as "the most extreme rogue wave event ever recorded" and a "once-in-a-millennium" event, claiming that at about three times the height of the waves around it, the Ucluelet wave set a record as the most extreme rogue wave ever recorded at

7802-412: The system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems. This term is disputed by others: Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. In mathematics ,

7896-494: The time in terms of its height in proportion to surrounding waves, and that a wave three times the height of those around it was estimated to occur on average only once every 1,300 years worldwide. The Ucluelet event generated controversy. Analysis of scientific papers dealing with rogue wave events since 2005 revealed the claims for the record-setting nature and rarity of the wave to be incorrect. The paper Oceanic rogue waves by Dysthe, Krogstad and Muller reports on an event in

7990-479: The typhoon load on the hatches was more than 10 times the design load. The forensic structural analysis of the wreck of the Derbyshire is now widely regarded as irrefutable. In addition, fast-moving waves are now known to also exert extremely high dynamic pressure. Plunging or breaking waves are known to cause short-lived impulse pressure spikes called Gifle peaks . These can reach pressures of 200 kPa (2.0 bar; 29 psi) (or more) for milliseconds, which

8084-445: The waterline. The pins had been bent back from forward to aft, indicating the lifeboat hanging below it had been struck by a wave that had run from fore to aft of the ship and had torn the lifeboat from the ship. To exert such force, the wave must have been considerably higher than 20 m (66 ft). At the time of the inquiry, the existence of rogue waves was considered so statistically unlikely as to be near impossible. Consequently,

8178-430: The wave known as a " hundred-year wave ", which is a purely statistical description of a particularly high wave with a 1% chance to occur in any given year in a particular body of water. Rogue waves have now been proven to cause the sudden loss of some ocean-going vessels. Well-documented instances include the freighter MS München , lost in 1978. Rogue waves have been implicated in the loss of other vessels, including

8272-449: The waves merge to create a single large wave; or when nonlinear effects cause energy to move between waves to create a single extremely large wave. Once considered mythical and lacking hard evidence, rogue waves are now proven to exist and are known to be natural ocean phenomena. Eyewitness accounts from mariners and damage inflicted on ships have long suggested they occur. Still, the first scientific evidence of their existence came with

8366-662: The waves studied were not as extreme as rogue waves, the research provided an understanding of how multidirectional wave interactions could lead to extreme wave heights - a key concept in the formation of rogue waves. The crossing wave phenomenon studied in the Delft Laboratory therefore had direct relevance to the unpredictable rogue waves encountered at sea. Research published in 2024 by TU Delft and other institutions has subsequently demonstrated that waves coming from multiple directions can grow up to four times steeper than previously imagined. The Draupner wave (or New Year's wave)

8460-469: The world's largest international certification bodies and classification society with main expertise in technical assessment, advisory, and risk management publishes their Structure Design Load Principles which remain largely based on the Significant Wave Height, and as of January 2016, still have not included any allowance for rogue waves. The U.S. Navy historically took the design position that

8554-601: Was found in June 1994. The survey team deployed a remotely operated vehicle to photograph the wreck. A private report published in 1998 prompted the British government to reopen a formal investigation into the sinking. The investigation included a comprehensive survey by the Woods Hole Oceanographic Institution , which took 135,774 pictures of the wreck during two surveys. The formal forensic investigation concluded that

8648-485: Was not necessarily as expected. If waves met at an angle less than about 60°, then the top of the wave "broke" sideways and downwards (a "plunging breaker"). Still, from about 60° and greater, the wave began to break vertically upwards, creating a peak that did not reduce the wave height as usual but instead increased it (a "vertical jet"). They also showed that the steepness of rogue waves could be reproduced in this manner. Lastly, they observed that optical instruments such as

8742-490: Was published in 1997. Some research confirms that observed wave height distribution, in general, follows well the Rayleigh distribution . Still, in shallow waters during high energy events, extremely high waves are rarer than this particular model predicts. From about 1997, most leading authors acknowledged the existence of rogue waves with the caveat that wave models could not replicate rogue waves. Statoil researchers presented

8836-442: Was the first rogue wave to be detected by a measuring instrument . The wave was recorded in 1995 at Unit E of the Draupner platform , a gas pipeline support complex located in the North Sea about 160 km (100 miles) southwest from the southern tip of Norway. The rig was built to withstand a calculated 1-in-10,000-years wave with a predicted height of 20 m (64 ft) and was fitted with state-of-the-art sensors, including

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