Pierson-Moskowitz Spectrum Research Articles

Study on TMD Control on Stability Improvement of Barge-Supported Floating Offshore Wind Turbine Based on the Multi-Island Genetic Algorithm

A floating offshore wind turbine (FOWT) has a great potential in producing renewable energy as offshore wind resource is rich in deep sea area (water deeper than 60 m) where fixed foundations are cost-effective or deployable. However, compared with a fixed-bottom installation, FOWT has to suffer more extreme loads due to its extra degrees of freedom. Therefore, the stability of an FOWT is a key challenge in exploiting offshore deep-water wind. Focusing on the stability of barge-type FOWT, this paper is to investigate the effect of passive structural control by equipping a tuned mass damper (TMD) on the nacelle. The turbulent wind with sharp fluctuations is established both in velocity and inflow direction based on standard Kaimal turbulence spectrum as suggested in the standard IEC61400-2. The irregular wave is generated according to the Pierson-Moskowitz spectrum. The dynamic structural characteristics of FOWT are calculated based on the fully coupled aero-hydro-servo-elastic solver FAST. Evidence has shown that the proposed method of the nacelle-based TMD is effective in controlling stability of an FOWT, as the sway and roll motions of barge and the side-side displacement of tower top decreased significantly. With the increase of mass, the side-side displacement of tower-top and the amplitude of roll motion of barge reveal a trend of increasing first and then decreasing. The stiffness and damping have little effect. Furthermore, the multi-island genetic optimization algorithm (MIGA) is employed to find globally optimum structural parameters (mass, stiffness and damping) of the TMD. The optimum structure parameters of TMD are achieved when the mass is 21393 kg, damping is 13635 N/(m/s) and stiffness is 6828 N/m. By adopting the optimized TMD, stability of roll motion of barge and side-side displacement of tower-top increase up to 53% and 50% respectively when compared with the normal TMD. The simulation results verify the validity and reliability of the proposed TMD control and the optimization methods.

Generation of two correlated stationary Gaussian processes and application to ship rolling motion

A model of series expansion with random amplitudes is proposed to generate two correlated Gaussian stochastic processes. The spectral density of each process can be adjusted to fit that of any practical process, and the correlation coefficient of the two processes can be specified arbitrarily. The model is then used to simulate two correlated random ocean wave forces of Pierson–Moskowitz spectrum. Based on the model, ship rolling motion under two correlated wave forces is investigated. The focus is on the effect of the correlation of the two random wave forces. Two different situations are considered. When the wave forces are weak, capsizing will not occur, and the motion will reach the stationary state after a period of time. The stationary probability densities of the rolling angle are calculated by simulation. It is found that the effect of the correlation on the long-term behavior of the ship rolling motion may be neglected. Another situation is with strong wave forces, in which case capsizing may happen as a random event. The ship rolling motion is non-stationary transient process before capsizing occurs. The average capsizing time is calculated also by simulation. It is shown that the correlation may have significant effect on the capsizing process, depending on system and excitation parameters.

Experimental hydrodynamic investigation of a fixed offshore Oscillating Water Column device

Oscillating Water Column (OWC) is one of the pioneer devices in harnessing wave energy; however, it is not fully commercialized perhaps due to the complicated hydrodynamic behavior. Previous studies are significantly devoted to OWC devices located in nearshore and coastal regions where incident wave energy would experience dissipation more than offshore. In this paper, a 1:15 scaled fixed offshore OWC model is tested in a large towing tank of National Iranian Marine Laboratory. Wave spectrum shape effect on the efficiency of the OWC model is addressed. Moreover, the paper investigates the effects of the geometric and hydrodynamic factors on OWC device efficiency and uncovers new points in nonlinear interaction occurring inside the chamber; i.e. sloshing. The results indicate that shape of the spectrum inside the chamber is affected by the type of incident wave spectrum, especially for long waves. Pierson–Moskowitz spectrum leaded to higher efficiency rather than JONSWAP spectrum at longer incident wave periods. According to efficiency analysis, increasing wave height may lead to air leakage from the chamber followed by vortex generation, which is a reason for decreasing the efficiency of the OWC device. Furthermore, no shift in the resonant period of the OWC model, due to wave height increase, was observed at the opening ratios equal or smaller than 1.28%. Spectral analysis of water fluctuation inside the OWC chamber illustrates two modes of sloshing. The first mode can be seen at short period waves while the second mode is visible at long period waves. The sloshing modes approximately vanish by increasing draft value.

Study on Energy Spectrum Instability in the Processes of Propagating and Breaking of Focusing Waves

Based on phase focusing theory, focusing waves with different spectral types and breaking severities were generated in a wave flume. The time series of surface elevation fluctuation along the flume were obtained by utilizing 22 wave probes mounted along the mid-stream of the flume. Based on the wave spectrum obtained using fast Fourier transform (FFT), the instability characteristics of the energy spectrum were reported in this paper. By analyzing the variation of total spectral energy, the total spectral energy after wave breaking was found to clearly decrease, and the loss value and ratio gradually increased and tended to stabilize with the enhancement of breaking severity for different spectral types. When wave breaking occurred, the energy loss was primarily in a high-frequency range of f/fp>1.0, and energy gain was primarily in a low-frequency range of f/fp<1.0. As the breaking severity increased, the energy gain-loss ratio decreased gradually, which demonstrates that the energy was mostly dissipated. For plunging waves, the energy gain-loss ratio reached 24% for the constant wave steepness (CWS) spectrum, and was slightly larger at approximately 30% for the constant wave amplitude (CWA) spectrum, and was the largest at approximately 42% for the Pierson-Moskowitz (PM) spectrum.

Linear Quadratic Optimal Control of a Spar-Type Floating Offshore Wind Turbine in the Presence of Turbulent Wind and Different Sea States

This paper presents the design of a linear quadratic (LQ) optimal controller for a spar-type floating offshore wind turbine (FOWT). The FOWT is exposed to different sea states and constant wind turbulence intensity above rated wind speed. A new LQ control objective is specified for the floater-turbine coupled control, in accordance with standard requirements, to reduce both rotor speed fluctuations and floater pitch motion in each relevant sea state compared with a baseline proportional-integral (PI) controller. The LQ weighting matrices are selected using time series of the wind/wave disturbances generated for the relevant sea states. A linearized state-space model is developed, including the floater surge/pitch motions, rotor speed, collective blade pitch actuation, and unmeasured environmental disturbances. The wind disturbance modeling is based on the Kaimal spectrum and aerodynamic thrust/torque coefficients. The wave disturbance modeling is based on the Pierson–Moskowitz spectrum and linearized Morison equation. A high-fidelity FOWT simulator is used to verify the control-oriented model. The simulation results for the OC3-Hywind FOWT subjected to turbulent wind show that a single LQ controller can yield both rotor speed fluctuation reduction of 32–72% and floater pitch motion reduction of 22–44% in moderate to very rough sea states compared with the baseline PI controller.

Nonlinear dispersion for ocean surface waves

Two expressions for the nonlinear dispersion relation for gravity waves on water of constant depth are derived, one for wave fields with discrete amplitude spectra, the other for wave fields with continuous wavenumber energy spectra. Numerical examples for wave quartets and for two-dimensional Pierson–Moskowitz spectra are given, and an important possible application is discussed.

Stochastic Dynamic Analysis of an Offshore Wind Turbine Considering Frequency-Dependent Soil–Structure Interaction Parameters

This study investigates the dynamic response of a 5[Formula: see text]MW offshore wind turbine with monopile foundation subjected to wind and wave actions under parked condition. It includes dynamic interaction between the monopile and the underlying soil subjected to stochastic wind and wave loading. The offshore wind turbine tower has been modeled using the finite element software ANSYS 14 as a line structure and it comprises a rotor blade system, a nacelle, and a flexible tower under parked condition. The mass of the rotor, blade, and nacelle are lumped at the top of the tower for simplicity. Stochastic wind and wave loadings are simulated using the Kaimal spectrum and the Pierson–Moskowitz spectrum correlating wind and wave forces, respectively. The soil–structure interaction (SSI) effect at the foundation level is taken into consideration by including rotational as well as lateral spring constants derived from Wolf’s double cone model for embedded foundations. The results are studied in the frequency domain for both wind and wave loadings in the form of power spectral density functions, which show that the response of the structure depends not only on the external forces but also on the soil–structure interaction effect. Under very soft soil conditions, the displacement response is amplified to a very high value under wind loading when compared with that under wave loading at lower frequencies. Incorporation of soil–structure interaction also modified the peak value of displacement and its subsequent frequency when compared with that for the fixed base structure which does not consider soil–structure interaction.

Modeling and Sizing of an Undersea Energy Storage System

This paper presents modeling and sizing of an undersea energy storage system (USS). The USS, which is placed at the seabed, consists of a concrete sphere, a reversible pump-turbine unit, a permanent magnet synchronous machine, and a steel pipe through which water flows into/out of the sphere from/to the deep ocean. A novel mathematical model is derived to describe the governing equations of the USS operation. It is assumed that the USS operates in parallel with a direct-drive wave energy converter (DDWEC) and a novel sizing algorithm is developed to obtain the required USS capacity/size for regulating the DDWEC output power fluctuations during a designated period of time. In order to determine the DDWEC output power profile, wave spectrums are required. Real significant wave height and dominant wave period data, which are measured and collected by a buoy station in the Gulf of Maine, are used to generate wave profiles based on the modified two-parameter Pierson-Moskowitz spectrum. Eventually, Monte Carlo simulations with a thousand generated wave profiles are carried out to determine the optimal size of the USS for all possible scenarios. Furthermore, through promising results obtained from hardware-in-the-loop studies, the effective role of the USS in regulating the wave energy converter output power fluctuations is demonstrated.

Comparison of Various Spectral Models for the Prediction of the 100-Year Design Wave Height

Offshore structures are exposed to random wave loading in the ocean environment, and hence the probability distribution of the extreme values of their response to wave loading is required for their safe and economical design. In most cases, the dominant load on offshore structures is due to wind-generated random waves where the ocean surface elevation is defined using appropriate ocean wave energy spectra. Several spectral models have been proposed to describe a particular sea state that is used in the design of offshore structures. These models are derived from analysis of observed ocean waves and are thus empirical in nature. The spectral models popular in the offshore industry include Pierson-Moskowitz spectrum and JONSWAP spectrum. While the offshore industry recognizes that different methods of simulating ocean surface elevation lead to different estimation of design wave height, no systematic investigation has been conducted. Hence, the aim of this study is to investigate the effects of predicting the 100-year responses from various wave spectrum models. In this paper, the Monte Carlo time simulation (MCTS) procedure has been used to compare the magnitude of the 100-year extreme responses derived from different spectral models. Additionally, the linear random wave theory (LRWT) was implemented to simulate the offshore structural responses due to random wave loading. The models have been tested for three different environmental conditions represented by Hs = 15m, 10m and 5m respectively. The accuracy of the predictions of the 100-year responses from Pierson-Moskowitz and JONSWAP spectrums will then be investigated.

Atmospheric channel model of maritime visible light communication based on repeated coding

Visible light communication (VLC) is a new type of wireless communication technology, and its applications in offshore ships and ship-shore lamp signal systems are drawing increasing attention as a supplement of communication net. In maritime environment, VLC system is affected by many factors, of which the wave fluctuation and atmospheric turbulence are the most noticeable. The turbulence will make signal intensity fluctuate randomly, and thus reducing the performance of VLC system operating in the atmosphere. To establish an effective VLC network in the actual marine environment, an effective channel transmission model needs to be established and used to study the performance of the maritime VLC link. Considering large aperture diameter receiver with the aperture averaging effect, log-normal distribution model is employed to deduce the mathematical expression of average bit error rate of maritime VLC system in atmospheric turbulence. By using time-diversity to transmit interleaved symbols with repeated coding in a maritime VLC system, it is possible to ensure that the code-word passes through multiple channels to resist the deep fade performance, and to reduce the bit error rate due to the occurrence of deep fading in a single channel. In the actual application process, in order to improve the system performance, the average signal-to-noise ratio usually increases with the transmission power increasing, but for a VLC system, there are some difficulties in making the high-power high-rate visible light transmitters. And the power will produce light pollution and even damage the naked eye. The implementation of the repetitive coding principle is simple, and in some special cases it is even better than the complex orthogonal space-time coding and other schemes, so studying the system performance of the repetitive coding scheme is of considerable value for practical application. Based on the modified Pierson-Moskowitz spectrum, the effect of wave height, transmission distance, atmospheric turbulence intensity, receiver aperture size and visibility on the average bit error rate of VLC system are analyzed. The performance of the VLC system between lighthouse and ship is affected by the fluctuations of the sea waves, and the average bit error rate changes with randomness and complexity like the sea waves in a short distance. As the wind speed increases, the marine environment becomes worse and the average bit error rate is undulate. The average bit error rate of maritime VLC increases with the increasing of transmission distance and atmospheric turbulence intensity, and with the decreasing of receiver aperture size, wavelength and average signal-to-noise ratio. Atmospheric turbulence intensity and visibility have a significant effect on the system performance, and it should be emphatically considered to take measures to reduce the influence. Increasing receiver aperture and repetitive coding are effective to a certain extent. In the present work a new model is proposed for evaluating the performance of a maritime VLC system and providing reference for practical application.

Modeling scattering effects from time-varying sea surface based on acoustic reciprocity

In marine seismic processing, the sea surface is often considered a flat mirror; hence, the effects of different weather conditions during the acquisition are largely ignored. However, studies have shown that rough sea-surface ghosts can severely damage the 4D signal, if not handled properly in data processing. To account for realistic sea-surface effects in processing, the impact of time-varying rough sea surfaces needs to be studied. We derive a method for modeling source and receiver ghosts from the time-varying rough sea surface and their interaction with subsurface reflections. This method is based on acoustic reciprocity and leads to integral equations of nonstationary wavefields. These modeling equations can also serve as a basis for investigating source and receiver deghosting methods for time-varying rough sea surfaces. Our developed modeling algorithm is validated against a frequency-domain approach for a “frozen” rough sea surface. For a moving simple sea surface, the Doppler shift produced by our method is in very good agreement with the analytical solution. Using a Pierson-Moskowitz spectrum, we derive a time-varying rough sea surface and model the receiver ghost, the source ghost, and the source-receiver ghost for the subsurface primary reflections of a heterogeneous geologic model. The results highlight that the source and receiver ghost interactions with a time-varying sea surface differently affect the subsurface reflections, and these effects can significantly impact the seismic repeatability of 4D studies.

Analysis of axial compressive loaded beam under random support excitations

An analytical procedure to investigate the response spectrum of a uniform Bernoulli-Euler beam with axial compressive load subjected to random support excitations is implemented based on the Mindlin-Goodman method and the mode superposition method in the frequency domain. The random response spectrum of the simply supported beam subjected to white noise excitation and to Pierson-Moskowitz spectrum excitation is investigated, and the characteristics of the response spectrum are further explored. Moreover, the effect of axial compressive load is studied and a method to determine the axial load is proposed. The research results show that the response spectrum mainly consists of the beam's additional displacement response spectrum when the excitation is white noise; however, the quasi-static displacement response spectrum is the main component when the excitation is the Pierson-Moskowitz spectrum. Under white noise excitation, the amplitude of the power spectral density function decreased as the axial compressive load increased, while the frequency band of the vibration response spectrum increased with the increase of axial compressive load.

Nonlinear short term extreme response of spar type floating offshore wind turbines

In this work, the dynamic characteristics of spar based floating 5MW offshore wind turbine (OWT) are studied under operational and survival conditions for offshore Indian conditions. The spar with OWT is installed in 320m water depth. The OWT is subjected to combined wind and wave loads according to irregular Pierson-Moskowitz spectrum. After obtaining nonlinear spar platform responses, 3-h short term extreme motions are calculated which are useful for the design of an OWT. Extreme values provide realistic estimates to the associated risk of an event since the tail region governs the largest/smallest events or even the events over a threshold in a sample. In this paper, the extreme values are obtained by fitting the peaks in the tail regime using a Weibull-distribution. While one obtains Gaussian responses under the survival loads, the operational conditions have non-Gaussian responses and also larger than survival ones. The results show one should exercise caution for generating the extreme values for non-Gaussian responses and do a proper sensitivity analysis with respect to samples used for generation of extremes.

Stochastic Response of Intact and a Removed Tendon Tension Leg Platform to Random Wave and Current Forces

This work analysed the dynamic response of intact and a removed tendon tension leg platform (TLP) in simultaneous action of random wave and current loads in normal and less severe environments. Two different sea states of extreme severe sea state but less probable and less severe state and most probable are considered in this study. The artificial random wave is simulated by Monte Carlo simulation using Pierson–Moskowitz spectrum. Wave diffraction effect is not considered since the ratio of characteristic dimension to wavelength is less than limit. The coupling in all degree of freedoms and various degrees of nonlinear effects is considered. The inertia, mass, damping matrices of equation of motion and hydrodynamic force vector are formulated and solved numerically using Newmark integration scheme. The statistical results show that removal of one tendon increases surge and tendon tension values, while heave and pitch are not adversely affected in both environments. The response values of the TLP in extreme severe sea state are quite higher compared to less severe sea state. The percentage increase in all degree of freedoms and tendon tension is <5 % when one tendon is removed as compared to intact tendon TLP. It could be concluded that maximum and minimum tension in tendon constraints is equally passed by the TLP tendons.

Rotating orbits of pendulum in stochastic excitation

A method to extract energy from an excitation which is stochastic in nature is presented. The experimental rig comprises a pendulum, and a vertical excitation is provided by a solenoid. The control input assumed in the form of a direct current motor, and another motor, used in reverse, acts as a generator. The stochastic excitation has been achieved by varying the time interval between switching the RLC circuit on and off according to a random distribution. Such non-linear vertical excitations act on an oscillatory system from which a pendulum is pivoted. The Pierson-Moskowitz spectrum has been chosen as the random distribution while an inverse transform technique has been used for generation of the random excitation signal in LabVIEW environment. Moreover, a bang-bang control algorithm has been implemented to facilitate rotational motion of the pendulum. Experimental observations have been made for various noise levels of vertical excitations, and their implication on energy generation has been discussed. A positive amount of energy has been extracted for a minimal amount of the control input.

A Detailed Comparison Between a Small-Slope Model of Acoustical Scattering From a Rough Sea Surface and Stochastic Modeling of the Coherent Surface Loss

The accurate modeling of underwater acoustical reflection from a wind-roughened ocean surface is a challenging problem. Some complicating factors are the presence of near-surface bubbles and the potential for shadowing of acoustical energy by parts of the surface itself. One essential factor, which is the subject of this paper, is the specular reflection of coherent plane waves at an ocean-like rough surface. We tested the accuracy of one rough surface reflection model, the small-slope approximation (SSA) approach as used by Williams (J. Acoust. Soc. Amer., vol. 116, no. 4, pp. 1975-1984, Oct. 2004), for scenarios for which scattering was entirely in the vertical plane. The SSA model was used to compute values of the coherent plane wave reflection loss per bounce for wind speeds between 5 and 12.5 m/s, frequencies between 1.5 and 9 kHz, and grazing angles between about 1 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> . These values were compared to those obtained from a Monte Carlo approach based on the parabolic equation (PE) method, where realistic ocean surfaces were generated based on the Pierson-Moskowitz spectrum for ocean surface heights. The SSA model compared favorably with the more rigorous PE method for most of the range of parameters considered. An approximation to the SSA model was derived for application to grazing angles less than particular values, and this approximation was shown to compare well with results from PE modeling.

In Memoriam: Sergei A. Kitaigorodskii, 1934—2014

Professor Sergei A. Kitaigorodskii, one of the pioneers of air-sea interaction studies, passed away on 4 December 2014 in Helsinki, at the age of 80. He had a clear vision of the fundamental problems of marine boundary layer physics. His bold ideas opened new paths more than a half century ago, and they continue to stimulate active research. In particular, he made significant progress on the difficult problem of estimating gas transfer rates through gas-liquid interfaces and contributed to many of the Gas Transfer at Water Surfaces conferences.Kitaigorodskii was born in Moscow on 13 September 1934. His father, Professor Alexander Kitaigorodskii, was a well-known physicist. Sergei Kitaigorodskii completed his undergraduate studies at Moscow State University in 1956, and he continued his postgraduate studies on the theory of turbulent mixing at the Academy of Sciences, Moscow, where he was awarded his Ph.D. from the Institute of Physics of the Atmosphere in 1960.As a post-doctoral fellow at the Institute of Oceanology of the Academy of Sciences, he formulated the Kitaigorodskii similarity hypothesis for the wind-generated wave spectrum in 1961. He later recounted how, as a young student, he had been impressed by the simple and at the same time fundamental results of O. M. Phillips, and he had wanted to follow that path by combining it with the Russian tradition, the Kolmogorov similarity hypothesis of turbulence. The Kitaigorodskii hypothesis immediately showed its value when W. J. Pierson Jr. and L. Moskowitz were able in 1964 to collapse their empirical data into a single dimensionless curve, and to formulate the Pierson-Moskowitz spectrum for fully developed seas. At the same time Kitaigorodskii continued his studies of the atmospheric side and showed how swell could influence the marine boundary layer, a topic upon which his ideas remain important.In 1962 Kitaigorodskii became a Senior Scientist at the institute and turned his main interest back to turbulence of the oceanic mixed layer. Here he again made fundamental contributions. These, combined with his earlier studies of waves and the air-sea boundary layer, were summarized in his thesis for the Russian degree of Doctor of Science in 1968. This work was the basis of a monograph, Physics of Air- Sea Interaction, published in 1970 and translated into English in 1973; it was used widely as a textbook. In the same year, 1973, he received the Rosenstiel gold medal, and in 1978 the Liege University Award.Between 1968 and 1977 Kitaigorodskii was head of the Laboratory of the Physics of Ocean- Atmosphere Interaction in the Institute of Oceanology.During the 1970s Kitaigorodskii established contacts with the Finnish geophysics community. His Finnish wife had completed her term as a press correspondent in Moscow, and she returned to Helsinki with their twin daughters. The Soviet system allowed Kitaigorodskii to visit his family every other year, and he used this opportunity to lecture and conduct research at several Finnish institutes.

Scattering and Doppler Spectral Analysis for a Fast-Moving Target above Time-Varying Lossy Dielectric Sea Surface

A numerical electromagnetic method based on the physical optics with physical optics method (PO-PO) is employed to calculate backscattered returns from a missile-like target above sea surface. Surfaces are time-varying Monte Carlo simulations initialized as realizations of a Pierson–Moskowitz spectrum. The monostatic normalized radar cross section of composite model by the hybrid PO-PO method is calculated and compared with those by the conventional method of moments, as well as the runtime and memory requirements. The results are found to be in good agreement. The runtime shows that the hybrid PO-PO method enables large-scale time-varying Monte Carlo simulations. The numerical simulations of the Doppler spectrum from the fast-moving target above time-varying lossy dielectric sea surface are obtained, and the Doppler spectra of backscattered signals from this model are discussed for different incident angles, speed of flying target, wind speeds, incident frequencies, and target altitudes in detail. Finally, the coupling effects on Doppler spectra are analyzed. All the results are obtained at the incidence of horizontal polarization wave in this study.

Comparisons between energy-flux propagation models utilizing single- and multiple-scattering treatments of sea-surface roughness in a range-independent waveguide

Acoustic scattering from a rough sea surface in propagation modeling is generally treated using a single-scatter approximation in which only specular components are considered and the incoherent field is neglected. This work presents energy-flux calculations of transmission loss for a shallow water waveguide with a rough sea-surface following a Pierson-Moskowitz spectrum. Multiple scattering is evoked by considering conservation of energy from scattering events. Comparisons are made between the model with multiple scattering and that using the single-scatter approximation. Models are compared to the direct solution using the boundary element method. [This work was partially funded by the Department of Defense (DoD) through the National Defense Science &amp; Engineering Graduate Fellowship (NDSEG) Program, the Eric Walker Fellowship from the Applied Research Laboratory at the Pennsylvania State University, and the Achievement Rewards for College Scientists (ARCS) Foundation Pittsburgh Chapter.]

Spatial structure of directional wave spectra in hurricanes

The spatial structure of the wave field during hurricane conditions is studied using the National Data Buoy Center directional wave buoy data set from the Caribbean Sea and the Gulf of Mexico. The buoy information, comprising the directional wave spectra during the passage of several hurricanes, was referenced to the center of the hurricane using the path of the hurricane, the propagation velocity, and the radius of the maximum winds. The directional wave spectra were partitioned into their main components to quantify the energy corresponding to the observed wave systems and to distinguish between wind-sea and swell. The findings are consistent with those found using remote sensing data (e.g., Scanning Radar Altimeter data). Based on the previous work, the highest waves are found in the right forward quadrant of the hurricane, where the spectral shape tends to become uni-modal, in the vicinity of the region of maximum winds. More complex spectral shapes are observed in distant regions at the front of and in the rear quadrants of the hurricane, where there is a tendency of the spectra to become bi- and tri-modal. The dominant waves generally propagate at significant angles to the wind direction, except in the regions next to the maximum winds of the right quadrants. Evidence of waves generated by concentric eyewalls associated with secondary maximum winds was also found. The frequency spectra display some of the characteristics of the JONSWAP spectrum adjusted by Young (J Geophys Res 111:8020, 2006); however, at the spectral peak, the similarity with the Pierson–Moskowitz spectrum is clear. These results establish the basis for the use in assessing the ability of numerical models to simulate the wave field in hurricanes.