1 Introduction

The frequency domain second-order wave exciting forces are usually decomposed into three different terms. One component is called the steady drift force, which is basically a mean value that is frequency dependent, but time independent. The other components are a consequence of the interactions between pairs of harmonics composing the sea state. They are called the sum frequency and difference frequency components and oscillate in time with a frequency resulting from the summation or subtraction of the harmonics composing the pair.

In the case of the difference frequency second-order forces, they result on the slowly varying wave drift excitation in irregular seas, which are important for floating moored structures. Usually, the natural period of the floater plus mooring is large compared to the wave period. However, the slowly varying drift forces have longer periods, and therefore they may excite the floater and mooring system at their natural frequency. This will induce large horizontal motions of the floater and tensions on the mooring lines.

Slowly varying drift motions calculation requires the solution of the boundary value problem. The problem needs to be formulated up to second order, meaning that the boundary conditions must include all quadratic and second-order terms and the second-order velocity potential must be calculated. The complete formulation for the second-order wave exciting forces and motions has been thoroughly presented by, for example, Ogilvie [1]. The problem solution is not trivial and the main difficulty is related to the integration of the quadratic forcing function in the free surface boundary condition. Kim and Yue [2] have suggested a procedure in which the free surface around the floating body is divided into two annular regions. The integration in the inner region (close to the body) is performed numerically and in the outer region it is based in Fourier–Bessel series.

The numerical solution complexity has motivated the research of simplifications to allow the slow drift force calculation. The most widely used in the offshore industry is probably the well-known Newman’s approximation, which was introduced before the complete calculation had ever been done. It was presented by Newman [3] and uses the steady drift force transfer function to calculate the whole quadratic transfer function. Later, Pinkster and Huijsmans [4] proposed an approximation to be used in shallow waters which neglected the second-order interaction between the linear incident, diffracted and radiated waves, but took into account the loads associated to the setdown phenomenon. Recently Stansberg and Kristiansen [5] have used a similar approximated method for shallow waters which considers a linear excitation from the setdown low-frequency wave, showing that in shallow waters the second-order loads are highly correlated with this excitation component. Another approximation proposed by Chen [6] neglects the free surface integral calculation, maintaining all remaining second-order terms of the boundary value problem. These yield approximate results have been shown to work well in deep waters. In shallow waters, it has been suggested that they may not work as well and particularly Newman’s approximation has been shown to provide faulty results [6, 14].

Some Scientists have studied this problem experimentally. Hsu and Blenkarn studied the slow drift loads in a moored vessel in 1970 [7], and in 1990, Krafft and Kim tested a rectangular barge in bichromatic waves. They analyzed the data using the Volterra model approach. More recently, in 2004, motivated by the LNG transfer problem, Naciri et al. [8] measured the slow drift motions of a tanker model in shallow waters, subjected to an irregular sea state.

The Volterra model is usually the procedure used for analyzing the slow drift motions in irregular sea states. This method has some limitations, though, and special care should be taken when preparing the experiments to make sure the slow drift motions are not on the wave spectrum’s more energetic frequency range, as suggested by Pinkster [9]. A way for better measuring the slow drift motions is to use bichromatic wave trains, and measure the difference frequency component of the motions. This was first done by Pinkster in 1980 [9] for a semisubmersible and later by Krafft and Kim [10] on a rectangular barge. In 2011, Fonseca et al. [11] presented extensive experimental results on the slow drift force acting on a restrained cylinder subjected to incident bichromatic waves in shallow, intermediate and deep waters. Pessoa et al. [12] presented partial results for the slow drift motions on the same geometry and waves. Results for the steady drift force in the presence of current in the same geometry are presented by Mazarakos and Mavrakos [13].

The present study follows from the authors’ previous works [12, 14] where the slowly varying drift forces and motions on a vertical cylinder were investigated both numerically and experimentally for three different water depths representing shallow, intermediate and deep waters. In [14], the authors presented a study on the second-order slow drift forces on the body restrained from moving and considering three different water depths. Three different levels of approximation to the quadratic transfer function were tested: the Newman’s approximation [3], the approximation proposed by Chen [6], and the exact calculation as formulated by Ogilvie [1] and solved by Kim and Yue [2]. Two additional methods combine Newman’s approximation with the second-order incident wave potential effects by assuming that the difference frequency wave length is much larger than the floater dimensions. It was concluded that Newman’s approximation is a valid method for calculating the surge second-order force in deep and intermediate waters, but not for shallow waters. Nor is it valid for the heave second-order force, unless in deep waters. Chen’s approximation proved to be a good approximation in any of the tested cases, although it tends to predict higher energetic second-order low-frequency force spectra on the higher frequency range than the exact second-order calculation.

In the present study, the investigation presented in [14] is generalized for the second-order slowly varying wave-induced motions. Even though this paper is focusing on the second-order low-frequency motions, it should be perceived as a follow-up from [14].The same approximation methods are applied to calculate the slowly varying drift forces, however, now accounting for the unrestrained body motions. The slow drift motion results are compared with experimental data in bichromatic waves and irregular sea states. The purpose is to understand how the depth effects influence the quality of the approximations to the second-order forces when the body is free to float and how this affects the second-order motions. It is shown that while the Hooft based method was good for calculating the second-order loads in a fixed body, it tends to underestimate the motions when the body is free to oscillate. Likewise, Newman’s approximation proved to underestimate the low-frequency motions even in deep waters. This happens due to the highly resonant behavior of the floating body. This means that the mentioned simplified methods are not appropriate for calculating slow drift second-order loads and motions of floating structures that exhibit strong resonant motions, such as, for example, wave energy converters.

2 Theory

The theory for the numerical calculation of linear and second-order wave exciting forces and motions on floating bodies has been thoroughly presented and discussed by several authors and is widely available in the literature. The reader may look into the work of Ogilvie [1], Kim and Yue [2], Pinkster [9] or Lee [15] for details on second-order theory.

In this paper, only a brief explanation of the theory used in the calculations of the second-order motions in irregular seas will be presented to explain the different levels of approximation for the second-order forces.

2.1 Second-order force approximations

Five approximations are used in the present work to calculate the second-order difference frequency wave exciting forces [represented in the frequency domain by the force quadratic transfer functions (QTFs)].

2.1.1 Complete second-order solution

In the complete second-order solution, the potential flow boundary value problem is expanded into power series, the expansions truncated above second order and the first- and second-order terms grouped separately resulting in the first-order and the second-order problems. The related second-order hydrodynamic forces include all of the second-order contributions. This methodology is explained in Lee [15].

2.1.2 No free surface forcing solution

The second approximation is similar to the complete solution. The difference lies in how the free surface boundary condition is evaluated. The boundary value problem of the complete solution includes a boundary condition in the free surface given by:

$$ \frac{{\partial^{2} \varphi^{(2)} }}{{\partial t^{2} }} + g\frac{{\partial \varphi^{(2)} }}{\partial z} = Q_{F} ,\,\quad {\text{on}}\,\,\,z = 0 $$
(1)

where Q F is the quadratic free surface forcing function, t is the time in seconds, g is the gravitational acceleration, z is the vertical coordinate pointing upwards of an orthogonal coordinate system placed on the average free surface position. The superscript (2) refers to second-order quantities.

Equation (1) combines the kinematic and the dynamic boundary conditions. This second-order boundary conditions must consider second-order terms of both Bernoulli’s equation and velocity terms of the free surface elevation \( \eta \) expanded about its mean position. The resulting quadratic forcing function is:

$$ Q_{F} = \frac{1}{g}\frac{{\partial \varphi^{(1)} }}{\partial t}\frac{\partial }{\partial z}\left( {\frac{{\partial^{2} \varphi^{(1)} }}{{\partial t^{2} }} + g\frac{{\partial \varphi^{(1)} }}{\partial z}} \right) - \frac{\partial }{\partial t}\left( {\nabla \varphi^{(1)} \cdot \nabla \varphi^{(1)} } \right) $$
(2)

In this approximated method proposed by Chen [6], the second-order boundary value problem is formulated without considering the free surface forcing Q F in Eq. 1.

2.1.3 Newman’s approximation

This approximation consists of using the zero difference frequency force components, which represent the steady drift loads in monochromatic waves, to calculate the whole force QTF. It was proposed by Newman [3] and it is based on the assumption that the force QTF surface is smooth and there is little variation of magnitude around the steady drift diagonal.

The force QTF off diagonal term T jk corresponding to a pair of interacting frequencies ω j and ω k is represented in terms of the steady drift force coefficients corresponding to the frequencies j and k, respectively, T jj and T kk:

$$ T_{\text{jk}} = \frac{1}{2}\left( {T_{\text{jj}} + T_{\text{kk}} } \right) $$
(3)

Since the steady drift terms do not depend on the second-order potential, the problem is significantly simplified.

2.1.4 Hooft’s approximation

The fourth approximated method consists of adding a contribution from the difference frequency second-order incident wave potential to Newman’s approximation.

The contribution to the second-order force due to the incident wave potential is calculated with approximated methods suggested by Hooft [16] to calculate linear forces on bodies with small dimensions relative to the incident wave length. Under this assumption, scattering effects can be simplified. In Hooft’s work, the methods were used to calculate linear forces. Here, the method is applied to calculate the exciting force contribution due to the difference frequency second-order incident wave potential. Neglecting the remaining second-order velocity potential components requires assuming that the body diameter is small compared to the second-order incident wavelength (\( L_{w}^{ - } = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {\left( {k_{j} - k_{k} } \right)}}} \right. \kern-0pt} {\left( {k_{j} - k_{k} } \right)}} \), where \( k_{j} \) and \( k_{k} \) are, respectively, the wave numbers for the jth and the kth harmonic components).

Following Liu et al. [17], the second-order difference frequency surge force can be divided into real and imaginary parts:

$$ T_{1jk} \left( {\omega_{j} ,\omega_{k} } \right) = P_{1} \left( {\omega_{j} ,\omega_{k} } \right) + {\text{i}}Q_{1} \left( {\omega_{j} ,\omega_{k} } \right) $$
(4)

In the present method, \( P_{1} \left( {\omega_{j} ,\omega_{k} } \right) \) is given by Newman’s approximation, and it accounts only for the contributions related to quadratic interaction of linear quantities; therefore, it does not consider contributions from the second-order potential. \( Q_{1} \left( {\omega_{j} ,\omega_{k} } \right) \) is the contribution from the second-order incident wave potential to the total second-order force. Since the body is assumed to be small compared to the incident second-order wavelength, the scattered potential will be accounted for by an added mass coefficient, and thus the wave-body second-order boundary value problem does not need to be solved. A solution for the second-order incident bichromatic wave potential is given by (see for example Lee [15]):

$$ \begin{gathered} \varphi_{w}^{ - } = \frac{{q^{ - } }}{{ - \left( {\omega_{j} - \omega_{k} } \right)^{2} + g\left( {k_{j} - k_{k} } \right)\tanh \left[ {\left( {k_{j} - k_{k} } \right)h} \right]}} \hfill \\ \quad \quad \cdot \frac{{\cosh \left[ {\left( {k_{j} - k_{k} } \right)\left( {z + h} \right)} \right]}}{{\cosh \left[ {\left( {k_{j} - k_{k} } \right)h} \right]}} \cdot \sin \left[ {\left( {k_{j} - k_{k} } \right)x - \left( {\omega_{j} - \omega_{k} } \right)t} \right] \hfill \\ \end{gathered} $$
(5)

where x is the longitudinal coordinate of the position vector \( {\mathbf{x}} \), h is the water depth and:

$$ q^{ - } = - \frac{1}{2}A_{j} A_{k} \left( {\frac{{\omega_{j}^{3} }}{{\sinh^{2} \left( {k_{j} h} \right)}} - \frac{{\omega_{k}^{3} }}{{\sinh^{2} \left( {k_{k} h} \right)}}} \right) - A_{j} A_{k} \omega_{j} \omega_{k} \left( {\omega_{j} - \omega_{k} } \right)\left( {\frac{1}{{\tanh \left( {k_{j} h} \right)\tanh \left( {k_{k} h} \right)}} + 1} \right) $$
(6)

According to Hooft [16], the surge force can be related to the horizontal acceleration of the fluid. The same approach is used here, however, applying the second-order incident potential (as done by Liu et al. [17]):

$$ Q_{1} = \frac{{\partial^{2} \varphi_{w}^{ - } }}{\partial x\partial t}\left( {1 + C_{a} } \right)\rho \forall {\mathbf{e}}_{1} = {\text{i}}\left( {\omega_{j} - \omega_{k} } \right)\left( {k_{j} - k_{k} } \right)\left( {1 + C_{a} } \right)\rho \forall \varphi_{w}^{ - } {\mathbf{e}}_{1} $$
(7)

where \( C_{a} \) is the surge added mass coefficient for infinite period, \( \forall \) is the immersed volume of the body, e 1 is a unit vector with the positive x-axis direction and \( \varphi_{w}^{ - } \) is evaluated at the body center of buoyancy.

In the heave force case, the additional contribution due to the incident difference frequency second-order wave potential is real, and so Eq. (4) does not apply. In this case the additional second-order incident wave potential contribution shall be referred to as \( P_{3}^{ - } \); therefore, the difference frequency heave force is represented by:

$$ T_{3jk} \left( {\omega_{j} ,\omega_{k} } \right) = P_{3} \left( {\omega_{j} ,\omega_{k} } \right) + P_{3}^{ - } \left( {\omega_{j} ,\omega_{k} } \right) $$
(8)

\( P_{3} \left( {\omega_{j} ,\omega_{k} } \right) \) is given by Newman’s approximation and, as for \( P_{1} \left( {\omega_{j} ,\omega_{k} } \right) \), it considers only contributions from the quadratic interaction of linear quantities. The second-order incident potential related heave force is calculated in the present work by integrating the second-order incident wave pressure (\( p_{w} \)) at the base of the cylinder (\( S_{0} \)).The cylinder bilge is disregarded for this calculation. The base is assumed to be a flat disk with the cylinder diameter. Since the assumption is, again, that the wave length is much larger than the body dimensions, this is referred in the text also as Hooft’s approximation. The incident difference frequency second-order pressure is given by:

$$ p_{w} = - \rho \frac{{\partial \varphi_{w}^{ - } }}{\partial t} = - {\text{i}}\left( {\omega_{j} - \omega_{k} } \right)\rho \varphi_{w}^{ - } $$
(9)

The force is thus given by:

$$ P_{3}^{ - } = \iint\limits_{So} {p_{w} .{\mathbf{n}}}dS = - {\text{i}}\left( {\omega_{j} - \omega_{k} } \right)\rho \iint\limits_{So} {\varphi_{w}^{ - } }dS( - {\mathbf{e}}_{3} ) $$
(10)

and assuming that the body is small relative to \( {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {\left( {k_{j} - k_{k} } \right)}}} \right. \kern-0pt} {\left( {k_{j} - k_{k} } \right)}} \):

$$ P_{3}^{ - } = {\text{i}}\left( {\omega_{j} - \omega_{k} } \right)\rho \varphi_{w}^{ - } \iint\limits_{So} {dS}{\mathbf{e}}_{3} = {\text{i}}\frac{{\left( {\omega_{j} { - }\omega_{k} } \right)\rho \pi D^{2} }}{4}\varphi_{w}^{ - } {\mathbf{e}}_{3} $$
(11)

where D is the cylinder diameter, e 3 is a unit vector with the positive z-axis direction and \( \varphi_{w}^{ - } \) is evaluated at the cylinder base.

The off diagonal vertical force QTF terms calculated by Newman’s approximation are assumed to have real parts only (usual assumption). Regarding the second-order incident wave potential, it induces a setdown on the free surface, which appears as a mean water level decrease associated to groups of large amplitude waves and a mean water level increase for groups of small amplitude waves. When the related pressures are integrated over the body wetted surface, the associated vertical forces are in phase with the setdown. Newman’s approximation vertical forces and Hooft’s contribution for the vertical forces are both considered real and directly sum up.

2.1.5 Newman’s approximation with setdown correction

This approximation applies only to the vertical difference frequency second-order forces. It consists on applying Newman’s approximation to the vertical steady drift forces in monochromatic waves accounting for additional bichromatic waves setdown effects.

It was pointed out by Eatock Taylor in 1989 [18] that there is a discontinuity in the vertical second-order loads caused by an inconsistency in the calculation of the wave setdown in monochromatic and bichromatic waves in finite water depths. It is still not clear if the discontinuity is in fact inconsistent. Nevertheless in 2006 Chen [19] suggested that the problem could overcome with a correction to the setdown already present in second-order regular Stokes waves. The new contribution is derived from the second-order incident bichromatic wave potential as the difference frequency tends to zero and the result matched to the classical form of second-order Stokes wave elevation. The result is:

$$ C_{j} = - \frac{{k_{j} A_{j}^{2} }}{4}\left[ {\frac{{4S_{j} + 1 - \tanh^{2} (k_{j} h)}}{{4S_{j}^{2} k_{j} h - \tanh (k_{j} h)}}} \right] $$
(12)

where A j and k j are, respectively, the wave amplitude and wave number, h is the water depth and S j is given by:

$$ S_{j} = \frac{{\sinh (2k_{j} h)}}{{2k_{j} h + \sinh (2k_{j} h)}} $$
(13)

A related steady vertical second-order force component can be derived, which is simply given by the wave setdown multiplied by the body water plane area S wp:

$$ R_{3} \left( {\omega_{j} } \right) = \rho gS_{wp} C_{j} $$
(14)

The vertical mean drift force in monochromatic waves accounting for bichromatic wave setdown effects is thus given by:

$$ T_{3jj} \left( {\omega_{j} } \right) = P_{3} \left( {\omega_{j} } \right) + \rho gS_{wp} C_{j} $$
(15)

The quadratic transfer function is then calculated applying Newman’s approximation (Eq. 3).

3 Experimental model

An experimental program was performed to test the validity of the described methods in restricted water depths. The studied geometry is a cylinder with a rounded bilge. It was designed this way so that viscous non-linear effects such as vortex shedding would be minimized in the experiments. An advantage of the simple cylindrical shape is that it can be easily reproduced numerically or analytically by any researcher or scientist. The resulting experimental data can thus be used as validation tool for a large variety of theoretical models.

Figure 1 shows the profile of the axis-symmetrical body. The main dimensions of the body are presented in Table 1.

Fig. 1
figure 1

Curve whose 360º revolution originates the wetted surface

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Table 1 Main dimensions of the axis-symmetrical body
Full size table

The tests were carried out at Danish Hydraulics Institute, namely at the shallow water basin for the water depths of 40 and 55 cm, and at the offshore basin for the water depth of 300 cm. The origin of the orthogonal coordinate system used in the experiments and the numerical calculations is located in the center of the model at the calm free surface level. X is positive in the direction contrary to the propagation of the waves and Z is positive upwards.

The body was placed in the middle of the wave tank, so that wall effects could be minimized. In addition to that, only long crested waves were produced, with a direction of propagation which is parallel to the side walls, so no reflection occurred on the side walls. At the other end of the wave tank, dissipation beaches were placed so that only a very small part of the energy was reflected back into the domain once the model reached a steady state of motion. The wave paddles are equipped with a reflection absorption system that prevents any energy that could come back to the wave maker to be reflected back into the domain.

The model was kept in place with a very soft mooring system. Four lines were attached to each quadrant of the model in such way that they would not touch the water, and with an inclination angle such that the mooring forces would pass approximately through the model’s center of gravity. This way the mooring systems effect on the dynamics of the model was minimized.

Center of gravity tests and decay tests were carried out to assess the physical properties of the model, as well as to get an estimation of the real damping for the relevant modes of motion. Due to the restricted depth effects, they differ from one water depth to the other. The physical and hydrodynamic properties of the moored body system are presented in Table 2.

Table 2 Physical characteristics of the model
Full size table

The damping factors in surge and pitch were obtained from analysis of decay tests. The initial displacements were chosen such that the decay test amplitudes would represent typical motion amplitudes during the free motion tests. The forced motion amplitudes varied over a wide range and in fact the experimental damping coefficients are not linear over these ranges. The method consisted on selecting the (linear) damping coefficients that best fit the decay tests over the following ranges: between 140 and 20 cm in surge and between 10 and 4° in pitch.

The model was subjected to a long crested irregular sea state, following a JONSWAP target spectrum with 1.55 s peak period (T P) and peak enhancement factor of 1. The significant wave height (Hs) was chosen so that the model would not touch the basin bottom while heaving and pitching. According to this, a significant wave height of 3.6 and 9.4 cm was chosen, respectively, for the 55 cm and for the 300 cm water depth. The duration of the irregular sea state runs was 15 min.

Three sets of bichromatic waves with difference frequency of 0.5, 1.5 and 4 rad/s were also tested. These tests were performed in both deep waters and in shallow waters. The duration of the bichromatic wave runs was 5 min. The wave characteristics are presented in Table 3.

Table 3 Characteristics of the tested bichromatic waves
Full size table

In the bichromatic wave tests, the data are analyzed with a least squares regression-based method which allowed separating the linear and second-order response components. This method is described in detail in [11]. A qualitative discussion on the uncertainties and error of the experimental data was presented in [12].

4 Slow drift motions in bichromatic waves

The slowly varying drift motion results of the cylinder in bichromatic waves were presented by Pessoa et al. [12]. In that paper the authors compare the experimental data with numerical calculations obtained with the complete second-order solution. The agreement between the numerical and experimental results was better for the smaller difference of frequencies, but satisfactory overall. The depth effect was identified both numerically and experimentally.

In this section, the experimental results will be compared with motions resulting from the five second-order force approximations described in Sect. 2.1. The second-order motion amplitudes are normalized by the incident wave amplitudes A j and A k , and the cylinder diameter D such that:

$$ \frac{{\xi^{(2)} }}{{A_{j} A_{k} /D^{n} }} $$
(16)

where n is equal to 1 for surge and heave and equal to 2 for pitch. The results are plotted as function of the non-dimensional water depth (\( h/\lambda_{j} \)), normalized by the wave length of the harmonic with the lowest frequency. This new scale permits an interesting depth effects analysis.

Figure 2 shows the surge non-dimensional slowly varying motion amplitudes in bichromatic waves as a function of \( h/\lambda_{j} \), with a difference of frequencies of 0.5, 1.5 and 4 rad/s, and the three tested water depths. Equivalent results for the heave and pitch low-frequency motions are shown, respectively, in Figs. 3 and 4. The plots include both the experimental results and the numerical calculations according to the different approximation methods.

Fig. 2
figure 2

Numerical and experimental surge motion QTF in bichromatic waves with a difference frequency of 0.5 rad/s (left), 1.5 rad/s (center) and 4 rad/s (right), in water depths of 300 cm (top), 55 cm (center) and 40 cm (bottom)

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Fig. 3
figure 3

Numerical and experimental heave motion QTF in bichromatic waves with a difference frequency of 0.5 rad/s (left), 1.5 rad/s (center) and 4 rad/s (right), in water depths of 300 cm (top), 55 cm (center) and 40 cm (bottom)

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Fig. 4
figure 4

Numerical and experimental pitch motion QTF in bichromatic waves with a difference frequency of 0.5 rad/s (left), 1.5 rad/s (center) and 4 rad/s (right), in water depths of 300 cm (top), 55 cm (center) and 40 cm (bottom)

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In the case of bichromatic waves, the period of the slow drift motions is the one related to the difference of frequencies, and it is constant in the plots. This means that 0.5, 1.5 and 4.0 rad/s differences of frequency bichromatic waves oscillate, respectively, in the periods of 12.57, 4.19 and 1.57 s. This way, the 0.5 rad/s bichromatic waves induce forces oscillating close to the surge natural period and the difference frequency of 4.0 rad/s is close to the pitch natural period.

The resulting resonant behavior of the second-order responses is clear in the plots of Figs. 2 and 4. In fact, the surge non-dimensional amplitudes for \( \Delta \omega \) = 0.5 rad/s are around 100, which means a bichromatic wave composed by two harmonic waves of amplitude 1 induce a second-order motion of amplitude 100. The very large dynamic amplification is observed for both the experimental and numerical results. The same type of behavior can be observed for the pitch second-order motions in the 4.0 rad/s case. The large dynamic amplifications occur, in spite of the second-order wave exciting forces small magnitudes, because resonant conditions and very light damping occur simultaneously.

The effects of first-order motions on the second-order motions are clearly visible in the surge and pitch graphs. This is because the second-order forces depend on quadratic interactions between first-order motions. For example, the slow drift surge at the smaller frequency difference (\( \Delta \omega \) = 0.5 rad/s) shows three peaks. Two of the peaks are related to the pitch natural period, one when the first incident harmonic has the natural pitch period and the other when the same occurs for the second incident wave harmonic. The third peak is associated to the heave natural period. The pitch slow drift motion of Fig. 4 also shows dynamic amplifications associated to the first-order resonant motions.

The second-order heave motion is very small, however, it is interesting to see that it was identified in the experiments and that it increases significantly for shallow water compared to deep water. The depth effect is very clear in all results since the second-order motions are always larger as the water depth decreases. A dramatic increase of the second-order motions is also observed when the water depth ratio is smaller than around 0.1–0.2.

Regarding the comparison between experimental data and the complete second-order solution results, one can say that the agreement is quite reasonable. The numerical model is able to identify all aspects detected in the experimental namely the order of magnitude of the slow drift amplitudes, the tendencies with \( h/\lambda_{j} \) and the dynamic amplifications. The agreement is not as good as the one obtained for the first-order responses (see [12]); however, this is in great part related to the uncertainty of the second-order experimental data itself. The experimental surge motions corresponding to the larger \( h/\lambda_{j} \) (high-frequency incoming waves) are considerably smaller than the numerical predictions. The error is likely related to limitations of the experimental setup. During the experiments, high-frequency waves had very small amplitudes. Consequently, the related mean second-order forces are also very small. In this case, the friction in the pulley system (not accounted for in the numerical model) may become important and reduce the experimental amplitudes compared to numerical predictions.

The approximations for the second-order surge forces induce similar surge motions in most of the non-dimensional water depth range, although while the no free surface forcing approximation and the complete calculation resulting motions are virtually the same; Newman’s approximation and the Hooft based method results deviate from the former. There is a sharp increase on the surge motions on shallow and intermediate water depths when (\( h/\lambda_{j} \)) < 0.1–0.2, which is predicted by the higher-order methods, but not by Newman’s approximation. This increase in the motion is corroborated by the experiments. The former result shows that the second-order incident wave potential cannot be neglected in restricted water depths.

The two peaks that can be seen in the surge motion are related to a strong increase of the second-order force when any of the composing frequencies coincides with the pitch natural frequency. The peaks are mostly caused by the quadratic interaction of the linear quantities involving the pitch motion and are thus predicted by all of the numerical test cases. Newman’s and Hooft based methods underestimate these peaks, as compared to the complete solution results.

As the difference frequency increases, the second-order potential contribution to the slow drift force increases as well. For this reason, larger discrepancies are observed between the motions calculated by different approximations.

Regarding the second-order heave motion results (Fig. 3), although the experimental data show some spreading, clear tendencies have been identified. The second-order amplitudes increase very much as the water depth decreases, especially for (\( h/\lambda_{j} \)) < 0.2. The agreement between the predictions and experiments is good, except for Newman’s approximation in restricted water depths, demonstrating, again, that the second-order incident wave potential effects cannot be neglected. The complete and the no free surface solutions give very similar results for the two smaller difference frequencies; however, significant differences are observed for \( \Delta \omega \) = 4.0 rad/s showing that second-order free surface effects become important for larger difference frequencies.

The pitch second-order amplitudes numerical results in Fig. 4 include also the Hooft’s based method results, even though the pitch moment is not corrected by the second-order incident potential effects. These results are included because pitch is coupled with sway.

There is no contribution from the free surface forcing to the second-order potential moment in pitch, since both the higher-order approximations yield the same results in any difference frequency and water depth. The agreement between the experiments and the predictions by the complete and the no free surface approximations is quite good, while the Newman’s and Hooft’s approximations tend to overestimate the experiments by a large margin in some frequency ranges. The sharp peaks observed in the graphs are related to the pitch first-order resonant motions, since these first-order motions contribute to the second-order wave exciting pitch moments.

As general conclusions, one can say the experimental program successfully identified the second-order difference frequency surge, heave and pitch motions. The surge and heave second-order amplitudes increase as the water depth decreases, especially for small depth ratios (\( h/\lambda_{j} \)) where a steep increase is identified. Results from the complete second order and the no free surface solutions compare quite well with the experiments. Newman’s approximation compares well with the experiments in deep water; however, it underestimates the experiments in shallow water. The approximations which consider the second-order incident wave potential effects are adequate to calculate the second-order motions, except for large difference frequencies where the second-order scattering potentials cannot be neglected.

5 Slow drift motions in irregular waves

The experimental wave elevation and the surge, heave and pitch motions were measured for JONSWAP sea states (with T P = 1.55 s and peak enhancement parameter γ of 1). This Section presents the analysis of results in terms of linear auto spectra, low-frequency spectra and coherence spectra. Two conditions were selected for presentation and discussion: (a) water depth of 300 cm and Hs = 9.4 cm, (b) water depth of 55 cm and Hs = 3.6 cm. The spectra of experimental data, obtained by Fourier analysis of the measured time records, are compared with the spectra of predicted motions. The latter, named here as “numerical spectra”, includes the linear and low-frequency spectra.

The linear amplitude spectra \( S_{\xi }^{(1)} \) are calculated by:

$$ S_{\xi }^{(1)} = S_{w}^{{}} \cdot \xi^{(1)} $$
(17)

where S w is the complex incident amplitude wave spectrum, and \( \xi^{(1)} \) is the complex motion linear transfer function.

The second-order low-frequency motion spectrum \( S_{\xi }^{(2)} \) is calculated by equating the contributions from each combination of wave frequencies present in an irregular sea state in a geometric manner. The method has been described in Duncan and Drake [20] for the sum frequency component of the second-order wave elevation and in Agarwal and Manuel [21] for the complete second-order wave model of both the wave elevation and kinematics, including the sum and difference components. These references present the method details. Although the method was developed to calculate the second-order wave elevation, the same procedure can be applied to obtain the second-order motion spectrum. In [21], the method is applied to a complex bi-frequency wave amplitude matrix \( \left\{ A \right\} \cdot \,\left\{ A \right\}^{T} \) and a wave elevation QTF; while in the present study, it is applied to the slow drift motion complex matrix given by \( \left\{ A \right\} \cdot \,\left\{ A \right\}^{T}\, \cdot \,\left[ {\xi^{ - } } \right] \) (where \( \xi^{ - } \) is the difference frequency motion QTF). The result of the mentioned procedure is the complex second-order motion spectrum \( S_{\xi }^{(2)} \). In Figs. 5 and 8, Agarwal’s and Manuel’s procedure was used for calculating the second-order wave elevation numerical spectrum. It should be noted that it includes only the incident wave component of the second-order wave elevation, so it does not include the diffracted and radiated second-order waves.

Fig. 5
figure 5

Wave elevation, surge, heave and pitch experimental and numerical linear and second-order response auto spectra. Water depth of 300 cm. Sea state with Hs = 9.4 cm, T P = 1.55 s, g = 1 and g = 9.806 ms−2

Full size image

The results are presented as a real auto-spectrum, which is calculated by:

$$ S_{\xi \xi } = S_{\xi }^{*} \cdot S_{\xi }^{{}} $$
(18)

where the superscript * stands for the complex conjugate.

Five numerical low-frequency spectra are compared, which correspond to the low-frequency motion QTFs calculated by the five approximations described in the previous sections.

Figure 5 presents the wave, surge, heave and pitch auto spectra in deep water (300 cm at model scale) for the seastate with Hs = 9.4 cm. The results are plotted in the figure as a black solid line for the complete second-order solution, a green dotted dash line for the no free surface forcing approximation, a red dotted line for Newman’s approximation, red dotted lines with cross markers for the Newman + Setdown approximation and blue line with cross markers for the Hooft’s based method. In addition, the circles represent the experimental results and the large dashed light brown lines stand for linear predictions.

The experimental surge motion spectrum shows much larger variance at the low-frequency range (below 2 rad/s) than around the wave frequency range (between 3 and 6 s). In fact, the surge motion is excited at the natural frequency (0.5 rad/s) by the slowly varying second-order drift forces, which, although of very small amplitude, induce large motions because the damping is small. The results from the surge motion induced by any of the second-order approximations is in good agreement with the experiments in the low-frequency range, clearly showing that the motion in this range is caused by the second-order slow drift force. This also seems to indicate that any of the approximations can be used to predict the surge slow drift motion in deep water. A closer look at the surge spectrum low-frequency zoomed graph (right side graph) shows the complete second order and no free surface results compare slightly better with the experiments than the other approximations.

Figure 6 presents the difference frequency exciting force QTFs (left graphs) calculated by four approximated methods and the corresponding surge motion QTFs (right graphs). It is interesting to observe that while the force QFTs by different approximations are quite different, especially for large difference frequencies, the resulting surge motion QTFs are qualitatively similar. In fact, the slow drift motion QTFs are dominated by the resonant dynamic amplifications around the surge natural frequency (0.5 rad/s): all of the surge motion QTFs exhibit the two peaks along the 0.5 rad/s and −0.5 rad/s diagonals. Forces at difference frequency away for 0.5 rad/s are filtered by the body inertia and the motion responses are negligible.

Fig. 6
figure 6

Surge force (left) and motion (right) quadratic transfer function in deep water (300 cm) calculated with the four different methods

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Regarding the heave motion spectrum in Fig. 5, the motions are very well reproduced in the linear range (above 2 rad/s). However, for frequencies smaller than 2 rad/s, the correlation between the measured and the numerical looks poor. These spectral values are very small. It is very difficult to measure accurately the large linear motions and the small second-order motions with the same measuring equipment calibration. Since the heave motions tend to follow the second-order wave elevation in the low-frequency range, and the experimental second-order wave elevation agrees with the theory, it is possible that the disagreement is caused by measuring errors. Nevertheless, the results show that the heave motions can be completely characterized by the linear model.

The pitch motion experimental spectrum (Fig. 5) is dominated by the linear response, since the pitch natural frequency coincides with the sea state peak frequency. In that frequency range, the linear forces are around two orders of magnitude larger than the second-order ones, but it is possible to see that the numerical motions resulting from the second-order approximations exhibit a dynamic amplification around the pitch natural frequency, which was shown to be accurate in bichromatic waves. Low-frequency pitch variances are very small, but still a good agreement is observed between the experimental data and the numerical predictions.

The analysis of the plots in Fig. 5 led to the conclusion that the low-frequency spectral variances are related to second-order low-frequency motions, while the variances for frequencies between 2 and 6 rad/s are related to first-order motions. The coherence spectrum provides a way to evaluate this aspect. In its original form, the coherence spectrum is calculated by means of a balance between the cross spectrum of two data series and the product of the auto-spectrum of each series:

$$ Coh{\kern 1pt} \,1 = \frac{{\left| {S_{xy} } \right|^{2} }}{{S_{xx} \cdot S_{yy} }} $$
(19)

It is a measure of how well a linear transformation will turn one series into the other. There is no similar function to evaluate the relation of the second-order type, unless one looks at the problem in the bi-frequency space. We can, however, relate the numerical auto-spectrum (\( S_{{\xi \xi_{num} }} \)) of the second-order motions with the experimental auto-spectrum of the motions (\( S_{{\xi \xi_{exp} }} \)) to see if they correlate well. It is not a traditional coherence spectrum, but it has been used by scientists before (e.g., Kim and Kim [22]), and it allows for some conclusions to be drawn. This modified version of the coherence spectrum can be calculated as:

$$ Coh{\kern 1pt} \,2 = \frac{{S_{{\xi \xi_{num} }} }}{{S_{{\xi \xi_{exp} }} }} $$
(20)

In both of the coherence spectra versions, the closer the result is to 1, the better the correlation between the two time series.

Figure 7 presents the coherence spectra for the surge, heave and pitch motions. Coh1 (black dots) stands for the experimental data; therefore, the two time series consist on the experimental wave elevation and wave exciting force. Coh2 results are calculated from the experimental force time series and numerical time series, where different Coh2 results stand for the: complete second-order solution (black line), no free surface forcing approximation (green dashed line), Newman’s approximation (red dotted line), Hooft based method (blue solid line with cross markers), Newman + setdown (red dotted line with cross markers) and linear exciting forces (brown dashed line).

Fig. 7
figure 7

Surge, heave and pitch experimental and numerical linear and second-order force coherence spectra. Water depth of 300 cm. Sea state with H S = 9.4 cm, T P = 1.55 s, g = 1 and g = 9.806 ms−2

Full size image

Analyzing the coherence spectra, it becomes clear that in the surge case the second-order motion is dominant in the frequency range below 2 rad/s. The coherence is better for the no free surface forcing and the complete calculation than for Newman’s approximation in that range, indicating better quality of these methods even at deep water. It was shown by Pessoa and Fonseca [14] that with the body restrained from moving, Newman’s approximation predicted similar slow drift force as the other approximations in deep waters. The present results indicate that with the body free to move this does not happen, and the results tend to underestimate the experiments. This is most likely due to the highly resonant pitch motion on this particular geometry and sea state which causes the assumption of a fairly constant force QTF not to be valid in this case.

In the surge mode, for frequencies higher than 2 rad/s, the motions are dominated by the linear forces, but since these motions are quite small, the quality of measurements is not as good. As a consequence, the coherence curves are not exactly close to 1.

The pitch coherence plot also clearly shows the dominance of the second-order forces in the low-frequency range.

Finally, Fig. 8 presents the response spectra for the 55 cm water depth. Although the wave peak period as the same as the one corresponding to the deep water results (Fig. 5), the significant wave height is around 2.6 time smaller; therefore, direct comparison of results is not possible. Anyway, the results are qualitatively similar for the two water depths, as described in the following sentences. Experimental surge spectral values are very large at the low-frequency range, since difference frequency wave forces excite the surge natural period. Experimental difference frequency slow drift motions are identified also for heave and pitch, although with very small magnitudes. The agreement between the numerical results and the experimental data is good for the low-frequency range, with slightly better agreement for the complete second-order solution and the no free surface solution, compared to the other approximations.

Fig. 8
figure 8

Wave elevation, surge and heave experimental and numerical linear and second-order force auto spectra. Water depth of 55 cm. Sea state with Hs = 3.6 cm, Tp = 1.55 s, g = 1 and g = 9.806 ms−2

Full size image

The heave motions in the low-frequency range are in much better agreement that in the case presented in Fig. 5. A possible explanation for this is that it is very hard to be able to measure accurately the large linear motions and the small second-order motions with the same measuring equipment. That is apparently the reason why the low-frequency motions are not correctly predicted in the low-frequency heave motions in Fig. 5. However, in this case the incident waves are much less energetic than the ones tested in Fig. 5. This allows calibrating the measuring equipment for acquiring smaller motions. In addition, the heave motions QTFs at this water depth are much larger due to the shallow water effects (see Fig. 3). This makes it easier to accurately measure the second-order motions in the set up used for the experimental data shown in Fig. 8, than in the one used in Fig. 5.

6 Conclusions

A numerical and experimental investigation of the slowly varying motions of a floating body subjected to bichromatic waves and long crested irregular waves is presented. Five approximations for the second-order slowly varying wave exciting forces were tested, namely: the complete second-order solution, the complete solution with no free surface forcing, Newman’s approximation plus second-order incident wave field effects (Hooft based method), Newman’s + wave setdown correction approximation, and Newman’s approximation.

Regarding the bichromatic waves conditions, the experimental program successfully identified the second-order difference frequency surge, heave and pitch motions. Experimental slow drift surge motions are very much amplified when the difference frequency is close to the surge natural period. This behavior is well represented by the several numerical methods. Surge slow drift motion amplifications are also identified when the bichromatic waves induce pitch resonant motions. In this case, Newman’s and Hooft based approximations underestimate the surge dynamic amplifications.

The surge and heave second-order amplitudes increase as the water depth decreases, especially for small depth ratios (\( h/\lambda_{j} \)) where a steep increase is identified. Results from the complete second order and the no free surface solutions compare well with the experiments. Newman’s approximation compares well with the experiments in deep water; however, it underestimates the experiments in shallow water. The approximations which consider the second-order incident wave potential effects are adequate to calculate the second-order motions, except for large difference frequencies where the second-order scattering potentials cannot be neglected.

Regarding the irregular wave results, the experimental surge spectral values are much larger at the low-frequency range than at the wave frequency range, which is correctly captured by all second-order numerical approximations, with a very slight advantage for the complete second order and no free surface forcing solutions. It was concluded that while the force QFTs by different approximations are quite different, especially for large difference frequencies, the resulting surge motion QTFs are qualitatively similar. In fact, the slow drift motion QTFs are dominated by the resonant dynamic amplifications around the surge natural frequency (0.5 rad/s) and forces at difference frequency away for 0.5 rad/s are filtered by the body inertia and the motion responses are negligible. Heave and pitch low-frequency responses are very small compared to the first-order ones. The experimental conditions tested were not appropriate to take conclusions regarding the depth effects on the slow drift motions in irregular waves.