## Abstract

Extreme wave–structure interactions are investigated using second-order diffraction theory. The statistics of surface elevation around a multi-column structure are collected using Monte Carlo-type simulations for severe sea states. Within the footprint of a realistic four-column structure, we find that the presence of the structure can give rise to extreme crest elevations greater than twice those at the same return period in the incident wave field. Much of this extra elevation is associated with the excitation of second-order near-trapped modes. A ‘designer’ incident wave can be defined at each point around the structure for a given sea state as the average input wave to produce extreme crest elevations at a given return period, and we show that this wave can be simply vertically scaled to estimate the response at other return periods.

## 1. Introduction

The phenomenon of near-trapping is a near-resonant local response excited by waves of a certain frequency interacting with arrays of vertical surface-piercing columns and other geometries. Each near-trapping frequency is associated with a mode of strong local free-surface oscillation which decays to zero at large distance from the array and is damped owing to wave radiation. However, the excitation periods of all but the lowest one or two near-trapped modes are usually too short to be excited linearly by a typical storm wave for most semi-submersible platforms. Second-order sum excitation of the higher near-trapped modes by waves with an incident period twice that of the mode excitation period can form a large component of extreme wave–structure interactions.

In this paper, we examine peak free-surface elevation around a four-column semi-submersible geometry in random seas. The waves excite near trapped modes via both linear and second-order frequency doubling interactions. We show that, in a severe but realistic sea state, water can reach more than twice the elevation of the waves away from the structure. We also use the results of a statistical study to assemble a ‘designer wave’, a compact wave packet, which can trigger such high levels of response. This ‘designer wave’ is the average shape of a series of random waves which reach this high elevation; thus, it is a function of the incident wave field, the structure geometry and the probability of occurrence in the specified sea state. However, we will show for both extreme crests and troughs the designer wave scales as the probability of occurrence of the waves is changed.

Quadratic transfer functions (QTFs) can be found using codes such as WAMIT [1,2] or DIFFRACT (e.g. [3,4,5]) to give the total surface elevation to second order in the vicinity of a structure for a given incident wave. Most analysis of second-order effects on structures has concentrated on forces, but we look at surface elevation. Calculation of QTFs can be very computationally intensive and so Taylor *et al.* [6] introduced a near-flat sum QTF matrix approximation for surface elevation around cylinder arrays. Grice *et al.* [7] investigated this approximation further to see if it could be used to estimate extreme wave–structure interactions while minimizing the number of QTF calculations necessary. This investigation examines extreme surface elevation time histories using the full QTF matrices.

There are various possible approaches to obtaining extreme response statistics, including those proposed by Tromans, Winterstein and Naess. The Tromans method [8] incorporates a weakly nonlinear wave model into a spectral response surface method for broad-banded, unidirectional, deep-water sea states. To avoid lengthy time-domain simulations, all of the calculations are performed in the probability domain. This uses the transfer function accounting for the presence of the structure, relative to open ocean, and its effect on wave surface elevation. Then a numerical search is carried out in probability space to find the Fourier representation of the most probable time history for the specified level of system response. (This is a first-order reliability FORM method.) The Winterstein method [9] uses water-depth-dependent analytic formulae to predict the skewness and kurtosis of nonlinear random waves. A Hermite model is used to predict the wave elevation and crest heights with specified return periods. The Naess method [10] expresses extreme responses in terms of a second-order Volterra series (including a linear and a quadratic term). The mean upcrossing frequency is then found and asymptotic expressions are used to obtain closed-form solutions to the extreme-value problem.

A brute force method is also possible using the linear transfer functions (LTFs) and QTFs, as discussed previously. A preliminary attempt was made by Eatock Taylor & Kernot [11], who made some limited comparisons of Monte Carlo simulations and closed-form solutions of sum frequency responses of a taut moored platform. Until recently, however, a purely brute force method such as Monte Carlo-type simulations has been generally considered impractical for regular use in predicting second-order interactions [12,13], but definitely of use in validating any other method used. However, in this work, we show that given the nature of the statistics being obtained the majority of calculations can take place in the frequency domain with just a single inverse fast Fourier transform (FFT) to obtain the associated time history. This means that, once the LTFs and QTFs have been calculated for a structure, the computation time needed for a brute force method might not be much slower than for some of the other methods suggested earlier. On a PC with a 2.2 GHz processor and 6 Gb RAM, it takes approximately 10 min to generate 10^{7} s of random response time history. This brute force statistical method is therefore used here.

The relative size of the first- and second-order components of surface elevation is investigated using the collected statistics. A ‘designer’ wave is discussed as a useful tool when studying extreme wave–structure interactions. It is shown in §2 how statistics can be collected on a large number of wave–structure interactions in a random realization of a target sea state. Section 3 describes how the largest crests or troughs can then be identified and the average surface elevation taken for several periods either side of the maximum amplitudes. The average incident surface elevation time history leading to these largest crests or troughs is investigated in §4 in the context of defining a ‘designer’ wave for targeted further analysis in wave tank model testing or computationally intensive computational fluid dynamics (CFD) simulations using programs such as OpenFOAM. The implications of varying the return period are discussed in §5, followed by some conclusions about the applicability of the approach.

## 2. Collection of random statistics

A series of long random realizations of the surface elevation is required, with each one based on the same user-defined sea state. In the results presented here, JONSWAP spectra are used to model the sea state, with *ω*_{p} the frequency of the peak of the spectrum, an upper cut-off at *ω*_{Max}=2.5*ω*_{p}, and the JONSWAP peak enhancement factor *γ* taken to be 3.3. The simulations reported here are for waves in a unidirectional sea state with an approach direction diagonal across the four columns. Other approach directions produce qualitatively similar results, but because of the symmetry of the structure different near-trapped modes are excited for the broadside wave approach. And intermediate directions produce excitation of a mixture of modes. Each approach direction requires a new set of QTFs to be computed. Although the second-order methodology could be extended for multi-directional seas, this would require a quadruple sum (a double sum arising from the frequencies of the pair of linear components, and a second double sum over the approach directions of the pair of these linear components). We have not attempted this.

The linear part of the sea surface elevation is assumed to be a stationary Gaussian random process with zero mean. The amplitudes of each of the Fourier components are taken to be random variables with Gaussian distributions and zero means. Two sets of coefficients are used, *a*_{n} and *b*_{n}, where *n*=1:*N*/2 and *N* is the length of the signal being generated. The surface elevation is then given by the equation
2.1
where *ω*_{n} is the angular frequency of the *n*th component. The variance of the Fourier coefficients, *a*_{n} and *b*_{n}, is based on the one-sided power spectrum, *S*(*ω*_{n}), for the desired sea state and *dω* is the frequency discretization
2.2

A random number generator is used to produce a set of Gaussian distributed random numbers, *z*_{n}, with zero mean and unit variance. These are then scaled up using the standard deviation given by the target sea state,
2.3
The square root arises because the JONSWAP spectrum, *S*, represents the power spectrum. The amplitude components, *a*_{n} and *b*_{n}, are used in the surface elevation spectrum (where *b*_{n} is also generated by equation (2.3), but with a different random number *z*_{n}). Once this surface elevation amplitude spectrum has been generated, an inverse FFT is then used to find the associated time history. As all the manipulation occurs in the frequency spectrum, the method requires modest computation time to generate the spectra. Rather than generating one long-time history from which to collect statistics, it is more convenient to generate a large number of short-time histories that can then be combined. Sets of 10^{6} waves were produced using 10^{4} short-time histories of length 1033 s with a sampling frequency of 1.98 Hz. An FFT calculation in Matlab is more efficient for signals where the number of components is a power of 2, so 1024 frequency components were used.

After generating the 10^{4} random spectra and their associated time histories, a check was made to confirm that the correct spectral information was conserved. The autocorrelation was found for each of the time histories by computing 〈*η*(*t*)*η*(*t*+*τ*)〉, with *τ* the time delay within the autocorrelation function, averaging across the ensemble. This ensemble averaging meant that any sampling noise in the data was minimized without reducing the temporal resolution. The average autocorrelation so obtained can then be compared with a NewWave focused wave group produced from the same parent JONSWAP spectrum as used in the random spectra generation (NewWave is the (scaled) Fourier transform of the power spectrum). The results are shown in figure 1*a*, scaled to unit peak amplitude, and there is a very good match between the two.

A further check to confirm whether the results are reliable is to compare the largest events in a random sea with a NewWave focused wave group. NewWave has been shown to be an accurate model for the average extreme crest surface elevation and so by identifying the largest crests in a large set of random time histories and averaging over these crests at each point in time, the resulting mean surface elevation should be very close to a NewWave based on the same sea state spectrum [14,15]. A set of 10^{4} random sea states were generated based on a JONSWAP spectrum with a peak period of *T*_{p}=17 *s* and a significant wave height of 10.8 m.

Figure 1*b* shows a comparison between the average linear surface elevation for the largest 500 crests and an ideal NewWave in the same sea state with the amplitudes matched at time *t*=0 s. The results are very close to the ideal NewWave with only small deviations several periods away from the focus point, which are to be expected owing to the uncertainty introduced by the random spectra used.

Having established that large sets of linear random surface elevation time histories can be generated in open ocean for a given sea state one can then modify these incident spectra with LTFs and QTFs to find the total response to second order in the presence of a large offshore structure. To allow comparison of these second-order results in the presence of a structure with the open-ocean results, QTFs for wave–wave interactions without a structure present were found using the method of Forristall [16]. By calculating the response both in open ocean and in the vicinity of a structure one can find the change in surface elevation caused by the presence of the structure. The structure under study here is a simplified version of a typical large offshore platform with four vertical columns with centres located at (±41.42 m,±41.42 m) in deep water (taken to be 350 m), connected by pontoons of height 11.52 m with a draft of 30 m. The cross section of the columns is 23 m square with bevelled corners, radius of curvature 7.68 m. Figure 2 shows an example mesh for the structure with two planes of symmetry used to minimize computation time, hence only one quadrant is shown. The wave approach direction used in these results was taken to be diagonal, moving from bottom left to top right in the axes of figure 2*b*. In this analysis, the structure is assumed to be rigidly fixed, so there is no motion. Thus, the analysis is (approximately) representing a tension leg platform (TLP), where the stiff vertical tendons suppress motion in heave and pitch. The long period surge and sway modes of a TLP are also ignored; the structure is assumed to be rigid. We believe that this is reasonable as the motions in such modes are so slow that linear and second-order sum near-trapping is unlikely to be significantly affected. Hence, although there is no structure below pontoon level, the methodology and results described in this paper could be expected to carry over directly to a four-column bottom-founded gravity base structure comparable in form to several platforms installed in the North Sea.

Near-trapped frequencies were predicted for the structure using the method of Linton & Evans [17] to enable selection of a sea state peak period to maximize the response within the structure. Near-trapping may be thought of as quasi-resonant modes between columns at critical geometrical spacings relevant to the incident wavenumber and is associated with particularly violent expected free-surface responses. The near-trapped modes found for this large offshore platform were used to select the random sea state peak period so that it would be in the range of typical storm peak periods and the second-order sum responses would excite one or more of the near-trapped modes. Detailed discussion on the calculation of LTFs and QTFs for this structure using DIFFRACT and the excitation to second order of individual near-trapped modes for the same structure is given in Grice *et al.* [7,18].

To find the average surface elevation for the largest crests, the zero upcrossing locations were found for each of the random time histories generated. The crest and trough between each pair of these zero upcrossings were then identified and the largest 500 crests were selected. For each of the 500 largest crests, the surface elevation time histories 50 s either side of the largest crest were isolated and then shifted in time so that the largest crest was at time *t*=0 s. The time history ±50 *s* was isolated because, when averaged, a NewWave-like focused wave group is a short event lasting only a few periods.

For the random sea states used in this section, averaging over the top 500 waves represents a one-in-1000 to one-in-1500 wave crest. A typical storm lasts around 1000 waves so the average extreme crests shown in these results are approximately the largest events one would expect to see in a typical storm of given severity.

The wave–structure interactions presented in this paper take no account of the effects of a possible surface current in the open sea, arising from tides, wind shear or set-up effects. The mathematical nature of the ‘forward speed’ problem for wave structure interactions is such that a different formulation and solution technique from that used here is required (e.g. [19,20] for a study of linear and second-order diffraction with a current). The influence of a current on wave near-trapping and violent wave diffraction around realistic multi-column structures is left as an unresolved issue.

## 3. Extreme crests and troughs

The previous section confirmed that the generation of random surface elevation time histories works in open ocean for linear sea states. This section begins by looking at the effect of second-order sum interactions. Second-order difference interactions for waves in deep water are much smaller than the linear and second-order sum components and are therefore not considered here. Figure 3 plots the average components of the total surface elevation for the largest 500 crests and 500 troughs in open ocean with the peak period *T*_{p}=15.16 s and significant wave height *H*_{s}=12 m. This gives a *k*_{p}*H*_{s} value of 0.21, which combined with the peak period of 15.16 s gives a severe but not unrealistic storm sea state. The second-order interactions provide a significant contribution to the total surface elevation, although still small relative to the linear component. The second-order component is almost identical in both the crest and trough figures. This is to be expected as there is no structure within the QTFs to introduce a phase change. The linear components of the trough focused signals are 180° out of phase with the crest focused signals but the quadratic nature of the second-order terms leads to this phase difference cancelling out. The overall effect of the second-order component is for the peaks in the total surface elevation to be slightly raised and the troughs slightly reduced relative to a linear NewWave. This averaged phase alignment has been used previously to identify second-order corrections in the analysis of field data (e.g. [14,15]). In terms of the variation across the 500 time histories, the 5–95% estimate of the mean elevation at the peak is ±0.71 m about the average elevation of 12.56 m, with this increasing to the RMS surface elevation in a random signal far away from this point as the effect of the conditioning weakens.

Of more interest is the interaction of extreme waves with a structure. Figure 4*a* shows the components of the average of the top 500 crests in the total surface elevation with the large semi-submersible structure present, calculated at (12 m,12 m) near the geometric centre of the structure (0 m,0 m). The components shown are the total average surface elevation (bold solid line), second-order sum (thin dashed line) and linear (thin solid line). These were calculated from random time histories based on a JONSWAP spectrum with *T*_{p}=15.16 s and *H*_{s}=10.8 m. The linear component is fairly symmetrical and NewWave-like in shape. The second-order component is also a localized packet, but with double the frequency and an amplitude equal to 40% of the linear amplitude. These add to give a maximum crest amplitude of 18.88 m. Figure 4*b* shows the average of the incident waves from figure 4*a* at the location of the structure but with the structure absent. Forristall coefficients were used instead of the DIFFRACT QTFs to find the open-ocean surface elevation from the same incident linear spectra. This results in a 48% reduction in the maximum amplitude and a delay of 8.6 s in the focus time. The ramp up in the open-ocean case is slower than the ramp down.

Figure 4*c* shows the components of the average of the largest 500 troughs in the total surface elevation with the structure present. The total average surface elevation is again a NewWave-like packet with a maximum trough depth of −12.68 m. By selection of the largest responses we are effectively maximizing the total trough depth but there are two competing processes. The linear component needs to be large but the second-order component, which is bound to the wave group, must lead to minimal reduction of the trough depth. Two possible situations that allow this are by minimizing the amplitude of the second-order envelope at the focus point or ensuring the phase is optimal. The second-order elevation in figure 4*c* is an example of an occurrence of the first with an obvious dip in the second-order envelope at the focus point. Figure 4*d* plots the open-ocean surface elevation for the same incident wave as figure 4*c*, but without the structure present. The peak amplitude is −8.63 m, which shows that the presence of the structure increases the peak amplitude by 47% (noting the previous value of −12.68 m). It is interesting to note that the deepest trough in figure 4*b* has a larger amplitude than the deepest trough in figure 4*d* even though in figure 4*b* the incident wave leads to the largest crest in the presence of the structure, rather than the deepest trough.

## 4. ‘Designer’ waves

In order to gain a better understanding of the physics of extreme waves one can ask what kind of incident wave leads to an extreme event, rather than simply extract examples of major events from a long dataset. For each of the top 500 crests and troughs, the incident wave field that led to these major events was identified. This was done using the same incident spectra but instead of DIFFRACT QTFs to find the response in the presence of the structure, the method of Forristall [16] was used to find the surface elevation at the location of the structure but with the structure absent. The surface elevation for ±50 s was then isolated from this open-ocean time history about the same focus time as the largest event from the response with the structure present. These incident surface elevation time histories were collected together for two sets: one that led to the largest crests and one that led to the largest troughs. An average was taken of each set of incident waves to give two ‘designer’ waves.

These ‘designer’ waves should lead to an extreme event if used as an incident field, with the total response field equal to the average total surface elevation found by taking the average of the largest collected responses. Figure 5*a* plots this ‘designer’ incident wave, and figure 5*b* plots the total response surface elevation when this wave is incident on the structure. It is reassuringly similar to the average collected extreme response, and this is confirmed by plotting both together in figure 6. There are slight differences between the two which are down to the variation introduced by averaging over a finite set of random signals. A ‘designer’ wave such as this could be used instead of more lengthy random time history generation for future analysis of extreme wave behaviour. In particular, one could use this for very careful physical experiments using a transient ‘designer’ wave to simulate an extreme event without worrying about the reflections associated with random wave tests in wave tanks. One could also use it for detailed CFD fluid loading calculations to look for new physics that second-order calculations do not capture such as third-order and higher harmonics, ringing, vertical water jetting, etc.

## 5. Varying return period

So far the analysis presented has looked at the top 500 events in each case in the range one in 1000 to one in 1500 waves. The following results look at what happens when the probability of occurrence is varied. A wave probability of occurrence of one in 1000 would be approximately the largest event expected in a typical storm duration and so if this is coupled with a sea state with a long return period it represents an extreme wave. This section looks at the effect of varying the wave probability of occurrence in a given storm while maintaining the same sea state. As well as the short-term intra-storm wave variability considered here, in a full air-gap reliability analysis, it would be necessary to take into account the number and severity of storms over the lifetime of the installation, and also the likelihood of sea-level rise over decades at least for a bottom-founded structure. This would not affect a freely floating semi-submersible platform but might be of relevance for a TLP, as would seabed subsidence owing to reservoir compaction such as occurred for the Ekofisk field in the central North Sea.

A sum of 10^{4} random time histories were generated using the same method discussed earlier. The crests and troughs were identified and then combined into one large dataset. The sea state used to generate the random time histories was based on a JONSWAP spectrum with *H*_{s}=12 *m* and *T*_{p}=15.16 s. The peak period of 15.16 s is within the range of typical values for a severe extra-tropical mid-latitude winter storm. For the Gulf of Mexico, the appropriate period is likely to be approximately 12–15 s. This sea state was chosen both because it is within the typical range for storms and because it should excite near-trapped modes at second order giving a large response.

Figure 7*a* shows four sets of results for the surface elevation crests near the geometric centre of the structure at (12 m,12 m) against their probability of occurrence. This location was chosen as it is known to give large responses at certain near-trapped modes that will be excited by the chosen sea state. The values at a probability of 10^{−4} (marked as point A), therefore, show the crest elevation for a one-in-10 000 wave. The four sets of results show the distribution of crest elevation to first and second order in open ocean (dashed and solid thin lines, respectively), and with the structure present (dashed and solid bold lines, respectively). The dotted vertical lines marked A–D show the four probabilities of occurrence for the average surface elevations which will be plotted in figure 8.

Figure 7*a* shows that second-order effects contribute far more to the total elevation when the structure is present than in open ocean. In open ocean, the largest waves have a linear component of 15 m elevation and a second-order component of less than 2 m. However, with the structure present the linear component is increased to a maximum of 21 m and the second-order component to 9 m. This means that for the most extreme waves almost one-third of the total surface elevation is due to second-order interactions. The highest elevations shown have a probability of occurrence of just over 10^{−6} per wave. As a given storm lasts around 1000 waves on average it is very unlikely that the largest shown wave would occur in a typical storm, but still possible. The range of crest elevations within the centre of the figure is of the most interest as the maximum expected elevation in a given storm is likely to be in this range. It is interesting to see how the addition of second-order results has a large effect on the amplitude of this maximum expected crest elevation.

The maximum surface elevation of 29.7 m in figure 7*a* is huge given that the sea state has an *H*_{s} value of 12 m. Even the one-in-1000 wave elevation with the structure present is almost 20 m. To put this into context, the large offshore platform under study could have a steady-state airgap of 17.5 m, shown in figure 7*a* as a black dashed line. If predictions are based on first-order results then the probability of water–deck impact in this sea state is approximately 2.5×10^{−5} per wave. If the results are extended to second order then the probability of water–deck impact in this sea state is approximately 2.5×10^{−3} per wave. This suggests that if this sea state occurred for more than 400 waves (around 1.7 h), one would expect there to be an impact on the deck, whereas linear predictions suggest it would take around 40 000 waves for impact to occur. It should be noted that these results are not necessarily representative of actual offshore platforms as the analysis presented here is for a structure fixed both vertically and horizontally.

Figure 7*b* plots the distribution of trough depth with probability of occurrence for the same dataset as the previous figure. The pattern is similar to the crest distribution with the four plots diverging at lower probabilities of occurrence. The top of the pontoons is 18.48 m below the surface, shown in figure 7*b* as a black dashed line, so for this sea state there is a very low probability of the pontoons being exposed.

Returning to wave crests, figure 8 shows how varying return period changes the average extreme surface elevation response with the structure present and the ‘designer’ wave leading to each average extreme elevation response. The surface elevation is again evaluated at (12 m,12 m). Each pair of subplots represents a different probability of occurrence per wave with figure 8*a*,*b* representing the ‘designer’ wave and average response elevation at point A in figure 7*a* with a probability of 10^{−4} or one in 10 000 waves. The other pairs have probabilities of 10^{−3}, 10^{−2} and 10^{−1}, respectively (as shown in figure 7*a* as points B–D), with the last pair almost certain to occur in a typical storm of 1000 waves.

At each probability of occurrence, the crest surface elevations are still averaged over 500 waves using the method discussed earlier except previously the average was taken of the surface elevation ±50 s either side of the largest 500 crests. Instead of averaging over the largest crests, a sliding window of 500 events beginning at the stated probability of occurrence is used. For example, if a set contained 200 000 waves then the ‘designer’ wave at a probability of occurrence of 0.01 would average over the 2000–2499 largest waves.

Figure 8 shows that the general shape of the signals is similar at each probability of occurrence but the level of nonlinearity greatly increases as probability of occurrence decreases. The second-order component plays a much larger part at more extreme responses, as one would expect. The maximum elevation reached at a probability of occurrence of 10^{−4} per wave is 20.66 m and this is reduced to less than half at 9.79 m for a probability of 10^{−1} per wave. The ratio of second order to linear components is 0.415 at the more extreme case of 10^{−4} per wave and this is reduced to 0.216 for the probability of 10^{−1} per wave case.

Figure 9 again looks at how varying return period changes the average extreme surface elevation response with the structure present and the ‘designer’ wave leading to each average extreme elevation response. However, the surface elevation is now taken just upstream of the downstream column at (32 m,32 m) in the same target sea state as the previous example. The same probabilities as the previous case (A–D) were used for each pair of figures: 10^{−4}, 10^{−3}, 10^{−2} and 10^{−1}. These results show that even after changing position the general shape of the signals is very similar at each probability of occurrence and the level of nonlinearity greatly increases as probability of occurrence decreases.

To emphasize how, when scaled appropriately, the shapes of the incident linear component, the response linear component and the response second-order component are extremely close at any given probability of occurrence, figure 10 shows the scaled results from the four probabilities at (12 m,12 m) and plots them on top of each other. The left half of the figure shows the crest results: figure 10*a* shows the linear component of the incident waves, figure 10*c* shows the linear component of the response with the structure present and figure 10*e* shows the second-order sum component of the response with the structure present. The scaling factor for the linear components is taken from the maximum amplitude of the incident waves. The 10^{−4} probability plots are unscaled, with the other three cases scaled up so that they have an equal maximum amplitude in the incident case. This incident linear scaling factor is also used for the linear response component and then squared for scaling the second-order sum components. The scaling factor between the 10^{−4} and 10^{−3} linear wave components is *C*_{3}=1.088; for 10^{−2}, *C*_{2}=1.303; for 10^{−1}, *C*_{1}=1.869; and the second-order scaling is simply the square of the linear value for each case. The linear scaling ratios are close to what would be expected for extremes in Rayleigh distributed waves (factors , , 2), providing a check on the statistical results. The right half of figure 10 (subplots *b*, *d* and *f*) shows the equivalent results for the troughs.

The figures show that when each component is scaled appropriately the shapes of the elements of the incident and response wave packets are almost identical for each probability of occurrence, for both crests and troughs. This is true for the three longest return period profiles shown and is a potentially important result. It demonstrates that, once a crest or trough ‘designer’ wave has been found at a given probability of occurrence, simple vertical scaling is all that is necessary to reach other long return periods at any point close to the structure. It should be noted though that the designer wave changes as the response location is moved. The second-order near-trapped modes are spatially complex so this is to be expected.

The forms of the various wave components show some interesting crest/trough differences. Both the linear and second-order sum *crest* response components have their maximum at *t*=0 s, and are both compact wave groups. Clearly, the selection of the crest extremes in the total response results in both the linear and second-order terms being aligned in time. The behaviour is quite different for the ‘designer’ *trough* components. In order to have an extreme trough in response, it is necessary to have a deep linear trough with its maximum depth very close to *t*=0 s. If the local structure of the ‘designer’ trough was similar to that of the crest (i.e. simply inverted), there would be a considerable second-order positive crest to be added to the negative trough, considerably reducing the size of the total trough response. Instead, the specification of the average shape of the deep trough actually leads to the second-order response passing through zero at *t*=0 s at least approximately, thus the total linear and second-order trough value at *t*=0 s is the same as the linear trough alone. The shape of the linear trough component to the total trough is different from that for the crest as a different arrangement of Fourier phases is required to achieve a zero-crossing condition when combined in the second-order signal. Thus, the inclusion of the second-order components into the total surface responses changes the linear contribution into the ‘designer’ waves away from what would have been obtained by random searching of linear simulations, where the ‘designer’ crests and troughs would have been simply inverted but otherwise identical in form (and would be NewWaves in the linear response field). For the full problem, where the surface elevation is a combination of linear and second-order components, the average linear component, the ‘designer wave’, is not simply a response NewWave because of the conditioning achieved by the combination of nonlinear time histories.

In figure 7*a*,*b*, the curves plotting crest and trough height against probability lose their smoothness at high amplitudes, with jumps in the data appearing owing to sample variability. This is due to the randomness of the dataset and the rarity of waves at such high elevations. To confirm this, the method used to produce figure 7*a* was repeated to give 10 datasets, each containing 10^{4} random spectra and an associated time history for each of length 1033 s. Crest elevation was then plotted against probability of occurrence for each dataset, as shown in figure 11. This confirms that the steps in each dataset at high elevations are due to the randomness of the data.

## 6. Conclusion

Large numbers of random surface elevation time histories can be generated for a sea state efficiently by performing diffraction transfer function calculations in the frequency domain. Only a single inverse FFT is then needed to find the surface elevation time history associated with the response spectrum to second order. Using QTFs to modify the incident spectra in the frequency domain allows a fast, efficient method of calculating surface elevation time histories to second order without needing to implement the more complex methods of Tromans *et al.* [8] or Jha & Winterstein [9].

The average surface elevation for a one-in-*N* wave can be found along with the average incident wave that would cause it. It has been shown that second-order components could make a large contribution to the total surface elevation, particularly in steep sea states. The comparison of predictions of occurrence of first- and second-order water–deck impact showed that first-order predictions greatly underestimated the likelihood of violent responses. The average incident waves or ‘designer’ waves could be used in targeted wave-tank model testing rather than general random wave simulations. They require far less time and have the added benefit of minimal interference from side and end wall reflections in a real wave tank. The results presented for ‘designer’ wave interactions at varying return periods suggest that, once a ‘designer’ wave has been found for a given sea state, simple vertical scaling can be used to find the expected surface elevation at other return periods.

The second-order methodology presented here can be applied to a freely floating structure where the structure could be expected to ‘ride’ the wave crests. A further paper addressing this issue is in preparation.

## Funding statement

The Engineering and Physical Sciences Research Council (EPSRC) and BP plc are thanked for their financial support.

## Footnotes

One contribution of 12 to a Theme Issue ‘Advances in fluid mechanics for offshore engineering: a modelling perspective’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.