Experimental Study on Sloshing Characteristics in the Elastic Tank Based on Morlet Wavelet Transform

Hydroelastic effect of sloshing is studied through an experimental investigation. Different excitation frequencies are considered with low-fill-depth and large amplitude. Morlet wavelet transform is introduced to analyze the free surface elevations and sloshing pressures. It focuses on variations and distributions of the wavelet energy in elastic tanks. The evolutions of theoretical and experimental wavelet spectra are discussed and the corresponding Fourier spectrums are compared. Afterwards, average values of the wavelet spectra are extracted to do a quantitative study at various points. From the wavelet analysis, sloshing energies are mainly distributed around the external excitation frequency and expanded to high frequencies under violent condition. In resonance, experimental wavelet energy of the elevation in elastic tanks is obviously less than that in the rigid one; for sloshing pressures, the elastic wavelet energy close to the rigid one and conspicuous impulse is observed. It recommends engineers to concern the primary natural frequency and impulsive peak pressures.


Introduction
Oil and gas are produced geographically in limited areas and transported to major consuming regions mostly by ship in cargo tanks. When the ship motion contains peaked energy close to the lowest natural tank mode, violent sloshing may appear inside the tank. Large fluid movements will create highly localized impact pressure on the bulkheads which may cause the structural damage (Hirdaris et al., 2010). In engineering applications, the tank is usually elastic and will respond to sloshing loads (Faltinsen and Timokha, 2009). As in LNG carriers, large-sized bulkheads of the membranetype tanks are typical elastic structures (Paik et al., 2011). The fluid-structure interaction plays an important role for the impact due to the elasticity of the containment system. Meanwhile, safety of the structure is strongly influenced by the dynamical response, such as the structural strain or movement. With the hydroelastic development, research on sloshing in elastic tanks has been started from the 1990s. Nash et al. (1989), Zhou and Liu (2007) and Faltinsen and Timokha (2009) did some preliminary researches on sloshing mechanisms in theory. Hwang and Ting (1989), Koh et al. (1998), Lee and Choi (1999), Cho et al. (2008) and Hirdaris et al. (2010) examined the hydroelastic effect in tanks numerically.
However, in theoretical and numerical investigations, many assumptions have to be made in the calculation, which may cause the solution to deviate from the actual value (Faltinsen and Timokha, 2009). There is a large consensus in the community that experiments are considered to be the most reliable method. For instance, Xue et al. (2017) developed a precisely liquid sloshing experimental setup and studied the effects of the vertical baffles with different con-figurations in reducing the sloshing impact pressure on rigid tank walls. But to the authors' knowledge, there are only a few literatures addressing the experimental study in elastic tanks. In an LNG-tank model test (Jung et al., 2008), the elasticity of the tank structure was found to have a significant influence on the height and shape of the impact pressure peak. Hydroelastic effects were modeled in an experiment by comparing the peak pressures between stiff and flexible panels (Bunnik and Huijsmans, 2007). To explore the hydroelastic effect in elastic tanks, a systematic experiment was conducted to analyze the wave elevation and pressures with the statistical approach (Jiang et al., 2014). Meanwhile, the resonant variation characteristics of the wave elevations, pressures and structural strains were analyzed in the time and frequency domain with the Fourier transform (Jiang et al., 2015a).
Fourier transform is the global transformation to analyze data either completely in the time or frequency domain (Frandsen, 2004;Jiang et al., 2015a); however, it cannot analyze data in the two domains simultaneously. For the interaction process of sloshing on the tank structure, it can be generally divided into two categories; the impact instant and the action process of the non-impulse pressure. The most significant characteristics of the impact instant is the short duration, very large magnitude and typical high-frequency behavior, whereas the latter one presents a relatively long duration corresponding to a low-frequency process with the excitation frequency (Hamlin et al., 1986). Therefore, the sloshing-structure interaction is a non-stationary variation process. For analyses of the impact instant, a narrow timewindow is needed to accurately determine high-frequency behavior of the impulse signal; however, for the non-impulsive pressure, a wider time-window is required to fully analyze the low-frequency features. Owing to the limitation, Fourier transform cannot describe the localized time-frequency behavior of the non-stationary signal, which is the most essential and critical property of this kind of signal.
To deal with this dynamic signal, the wavelet transform is proposed, which is the natural extension of the Fourier transform. The general ideas of the two are the same, to decompose and approximate the signal series by basis functions (Daubechies, 1992). While the difference is that for the wavelet transform there are infinite basis functions, but only one fixed for the Fourier transform. In the wavelet transform process, the constraint relation between the frequency and time resolution can be solved very well by changing the shape of the frequency-time window. And this transform has good localization property both in the frequency and time domain (Dong et al., 2008). For the highfrequency components of the analyzed data, a lower frequency resolution is obtained upon a narrower time window. In contrast, for the low-frequency components, a wider time window is chosen to obtain a higher frequency resolution. This adaptation feature makes it widely to be used in engineering technology and signal process (Mallat, 1999;Liu, 2000;Dong et al., 2008). More recently, this method has been brought to the sloshing field. Colagrossi et al. (2004) reported the evolution of the wave-elevation spectrum in the experiment to show the nonlinear sub-harmonic. This transform was also introduced to analyze the baffle effect (Xue et al., 2013), and when the baffles were installed, the spectral energy of the wave-elevation decreased along all the frequencies.
In the previous works, only the Fourier transform was brought to study the sloshing pressure, which cannot display the behaviors in the time and frequency domain at the same time. While the wavelet transform just has the timefrequency characteristics, which can simultaneously present time-frequency behaviors of the impulsive pressure at different stages through the change of the scale. Meanwhile, only preliminary analysis on sloshing is conducted by the wavelet transform, which is concentrated on the wave elevation in rigid tanks. Here for the wave elevation in the elastic tank, a further investigation will be conducted together with the descriptions of the sloshing waves. Furthermore, for nonlinear impulse pressures in the non-stationary state, an intensive mechanical study is very necessary at present and the research with wavelet transform has not been found yet. In addition, for the wave elevation and impact pressure in actual elastic tanks, it is also an urgent engineering need to get more understanding of their features.
Therefore, as a supplement and improvement of the previous works, the wavelet transform is adopted to conduct a research on the time-frequency characteristics for the wave elevation and impulsive pressure. Based on the Fourier spectral analysis (Jiang et al., 2015a), an experimental investigation with the wavelet transform is performed to explore the hydroelastic effect of sloshing. The elastic tank is excited by the harmonic excitation and different frequencies. Morlet wavelet transform is adopted to study the characteristics of the free surface elevations and sloshing pressures. The evolution of the theoretical and experimental wavelet spectra at different locations are discussed qualitatively and compared with the corresponding Fourier spectrum. Subsequently, the average values of the wavelet energy spectra are employed to present the preliminary quantitative evaluation at various points.

Model tank
The experiment under horizontal excitation is conducted at State Key Laboratory of Coastal and Offshore Engineering. The structural physical model in the experiment is a simplified model of the LNG tank. According to the practical situation and performance of the vibration table, including the vibration direction, excitation frequency and working area, the experimental sloshing tank is determined to be a rectangular tank. Two types of rectangular tanks, rigid and elastic, are modeled in the experiment. The rigid model is made of the plexiglass sheet with the thickness of 0.012 m, while for the elastic model, the top, left and right bulkheads are made of the 0.002 m thick plexiglass sheet, and the thickness of others are 0.012 m. The horizontal simple harmonic excitation is set as the vibration pattern and the length of the tank is 0.500 m along the direction of the vibration. Sloshing in the rectangular tank is often treated as a two-dimensional flow. While the external excitation frequency is close to the lowest order natural frequency or the excitation amplitude is large enough, the liquid in the tank will slosh violently and result in turbulence, wave breaking and three-dimensional effect. In order to reduce the occurrence of three-dimensional effects, the ratio of the length to the width of the tank should be as large as possible. In this case, the ratio of 5 is selected to obtain the relevant characteristics of two-dimensional sloshing. For the convenience of comparing with the length of the tank and making the model, the height of the tank is also chosen as 0.500 m. Therefore, the internal dimensions of the tank are L×H×B = 0.500 m×0.500 m×0.100 m. The sketch of the tank and the coordinate system xoy are plotted in Fig. 1.

ρ ν
Clean tap water at the room temperature is used as the liquid in the tank. Its density and viscosity are = 1.0×10 3 kg/m 3 and = 1.005×10 -3 N·s/m 2 , respectively. In order to catch the free surface, the red fluorescent dye rhodamine is added to the water, which does not affect the density and viscosity of the water.

Instrumentation
Sloshing test facility is made by ETS Solutions Ltd., which is capable of performing harmonic motions with three degrees of freedom. Here, the horizontal harmonic excitation is applied along the length direction of the tank (x direction). Displacements of the motion are , where A and f are the excitation amplitude and frequency, respectively. In the experiment, the shaking table provides an initial momentum for the movement of the tank, and the momentum is continuously provided by the table during the experiment. After the initial and sustained momentum is applied, the tank begins to move and the fluid in the tank also starts to move along with the tank. In the process of the movement, the sloshing fluid interacts with the elastic tank to have an exchange of the momentum. On the one hand, the tank transfers the momentum from the external excitation to the liquid in the tank, which makes the liquid obtain the initial momentum. On the other hand, after obtaining the momentum, the liquid presents the violent movement in the tank, which will generate the traveling wave and hydraulic jump. The sloshing wave impacts the wing bulkhead and the tank roof and applies the obtained momentum to the elastic structure in turn.
An image acquisition and data analysis system is designed to obtain the wave shape and elevation. The sampling frequency is 30 frames per second with a precision of ± 0.0005 m. Two measuring points, E01 and E02, are shown in Fig. 2 with the x coordinate of E01 being 0.125 m and E02 being 0.240 m. Capillary effect has an influence on the acquisition of the wave elevation, which is the result of adhesion and surface tension. Adhesion of water to the walls of the tank will cause an upward force on the liquid at the edges and result in a meniscus which turns upward. The surface tension acts to hold the surface intact, so instead of just the edges moving upward, the whole liquid surface is dragged upward. The capillary effect caused by surface tension is considered in the experimental study. The wave elevation data in the experiment is extracted from the acquisition images. In order to make the data more representative and avoid overlarge elevation due to the capillary effect near the boundaries, the selected elevation points in the image are on the middle line of the free surface which is along the length of the tank. In the meanwhile, the distortion correction has been done to all the images.
The sloshing pressures are measured using DS30-MD14 multi-point measuring system, made by Tianjin Research Institute of Water Transportation Engineering. All the pressure signals are sampled at 1000 Hz with a precision of ± 0.001 kPa. Pressure transducers are fixed on the bulkhead at different positions in the tank, as shown in Fig. 2, P01 and  P02. More detailed information can be found in Jiang et al. (2013).

Case studies ω n,T
For a tank with the fixed geometry, the natural period of the sloshing depends on the liquid depth, shown in Fig. 3. At present, the estimation of the theoretical natural period is usually calculated by traditional linear formula (Abramson, 1966) where L is the width of the tank, h is the liquid depth, the modal number n = 0, 1, 2, and the natural frequency f 0, T can be obtained by . Owing to the high nonlinearity or the randomness, in some cases the available analytical solution cannot provide a satisfactory estimate of the natural frequency, as shown in Fig. 3. Therefore, according to the theoretical natural frequency f 0, T , the actual lowest-order natural frequencies f 0, e in the present experiment are obtained through the sweep test (Jiang et al., 2014). A frequency segment of the width 0.2 Hz with the theoretical lowest-order natural frequency f 0 as the center is chosen for performing the sweep test. The frequency interval is 0.001 Hz. Then, the acquired time-domain sloshing pressure is analyzed through the fast Fourier transform, and the spectral peak frequency corresponding to the maximum pressure spectrum is the actual lowest-order natural frequency of the liquid in tanks. Around the actual lowest-order natural frequency f 0, e , a relatively wide range of the excitation frequencies is selected, which occur between 0.5 Hz and 1.7 Hz.
A low-fill-depth h = 0.080 m is chosen in the present study, and its relative depth h/L is 0.16. The excitation amplitude A is set to be a relatively large value of 0.010 m. In order to obtain periodic steady-state sloshing waves, most of the tests run last at least 100 s, where the running time is determined by the excitation period and specific circumstances. Each test is repeated at least three times, and the mean value of the wave elevation and sloshing pressure is taken over the three experiments.
In the low-fill-depth case, under different external excit-ation periods, five various types of sloshing phenomena might be generated inside the tank; standing wave, traveling wave, hydraulic jump, solitary wave and the combination form among the former four. The sloshing liquid acts on the bulkhead and could, in general, create two types of dynamic pressure, that is, impulsive and non-impulsive pressures. The free surface deformation is the main source of the energy dissipation, which has significant influence on the wave elevation and sloshing pressure. For excitation periods around resonance, the low-fill-depth case is characterized by the formation of hydraulic jumps or traveling waves (Abramson et al., 1974;Jiang et al., 2014). When hydraulic jumps or traveling waves are present, extremely high impact pressures can occur on the tank walls. Figs. 4a and 5a show typical wave elevation and impulsive pressure traces recorded under this condition from the present experi- In order to verify the correctness of the present test from one side, the pattern and shape of the free surface elevation and impulsive pressure are compared between the present experiment and the available experiments. Faltinsen et al. (2005) analyzed a nearly steady resonant regime in a rigid square tank. The tank had a square base with the breadth/width 59 cm and the height of 80 cm. The non-dimensional water depths were 0.508, 0.34 and 0.27. The forced tank velocity is expressed as for the longitudinal excitations. Colagrossi et al. (2010) studied the sloshing flow inside a rigid prismatic tank. The tank of the   The chosen h implies a nearly shallow-water regime. However due to the limitations of the test conditions, the elevation and pressure data in the same amplitude and time range are lacked between the present test and the available experimental work. So only the data in the same order of the magnitude are used to do the comparison. Upon the excitation parameter with the same order of the magnitude, the present results are very similar to the available experimental data (Figs. 4b and 5b) (Faltinsen et al., 2005;Colagrossi et al., 2010). = A/L is the non-dimensional forcing amplitude, which is the ratio between the forcing amplitude A and the horizontal length dimension L of the tank along the sway direction. = f/f 0 is the non-dimensional forcing frequency, which is the ratio between the forcing frequency f and the analytical natural frequency f 0 of the tank. Together with the former comparison between the analytical and experimental natural frequency (Fig. 3), it can be proved that the experimental elevation and pressure data presented here are reliable.

Continuous wavelet transforms
Owing to the insufficient Fourier transform, the wavelet transform is proposed. It has good resolution both in the time and frequency domain and can do multi-scale analysis for the signal transform and data processing. Therefore, it is known as the mathematical microscope, which is widely applied in the field of the signal processing (Mallat, 1999;Dong et al., 2008).
The function and , that is . And the function is defined as the basis or the mother wavelet function. The wavelet function is constructed by translating or dilation of the mother wavelet function: 1/ √ a where a and b are the scale and translational parameter, respectively. For all a and b, the normalization constant follows the conservation of energy: For a known continuous signal , the continuous wavelet transform is defined as: where , b and t are continuous variables and the asterisk denotes the complex conjugate. q n (t) However, in the practical application, the discrete signals are always sampled and acquired for the convenience of computer's processing. The continuous transform of a discrete time series is defined as: n ′ ∆t where, denotes the translation quantity of time; is the sampling interval of the discrete time series. To make the wavelet function in each scale have the specific energy, a dimensionless method is adopted to compare the result after the transform: To reduce the amount of calculation, upon the convolution theory, Eq. (5) can be converted to the inverse Fourier transform to calculate and : q nψ * (aω n ) q n (t) where the series and are the Fourier transforms of the signal and the mother wavelet, respectively: To achieve continuous wavelet transform of the discrete data, appropriate scale must be selected firstly, usually with the binary method. When the scale is set as 2, the fractional power can be expressed as: δ where a 0 is the minimum scale; and decides the value of the scale, the maximum of which does not exceed 0.5.

Wavelet energy spectrum q(t)
In the above discussion, the mathematical expression of the wavelet transform coefficient for the signal is defined in Eq. (4). Wavelet transform is governed by the conservation of energy: C ψ where is the admissible condition of the wavelet function, 11) Based on Eq. (7), the wavelet energy spectrum is defined as: q(t) For the signal , Eq. (10) shows that the wavelet energy spectrum in the time and frequency domain is presented in a two-dimensional area, which is constructed by the scale parameter a (frequency) and the translational parameter b (time).
A three-dimensional spectrum is composed of the frequency a, time b and the wavelet energy spectrum P W . Here for comparison convenience, a two-dimensional color spectrum is introduced with the time and the frequency as the x and y axis respectively, as shown in Fig. 6. Wavelet energy spectra appeared and changed in the two-dimensional area with the color, which represents the spectral energy value. At the distributed location, the redder the color is, the greater the wavelet energy is.

Selection of the wavelet function
The basis wavelet function has a great influence on the signal analysis, and its choice becomes the key point of the wavelet transform. Morlet wavelet is the most commonly used complex wavelet function (Mallat, 1999;Liu, 2000). The mathematical expression is: /2 (13) and its Fourier transform can be written as: ω 0 ⩾ 5.34, e −ω 2 0 /2 ≈ 0. When the variable Morlet wavelet and its Fourier transform can be simplified as follows: Since Morlet wavelet function decays with the squared exponential, its waveform is very similar to the features of the impulse signal that there is the impact pressure in the model test. For a more comprehensive understanding of the Morlet wavelet, let . For the scale factor a, 2, 1 and 0.5 are chosen, Morlet wavelet in the time domain and the corresponding Fourier spectra in the frequency domain are presented in Fig. 7 to further illustrate the characteristic of this function.
From the comparison, a significant variation is observed when changing the scale parameter a. As the variable a becomes smaller, the time window of the Morlet wavelet function increases; on the contrast, the frequency window decreases and the center frequency also decreases gradually, from 2 Hz, 1 Hz to 0.5 Hz. This shows that the Morlet wavelet has the feature of "zoom". Whereas the variation of b only has the influence on the time domain, which is performed on the translation along the time axis and the center of the time axis is t = b in the time history.
The above analysis shows that the sampling step of the wavelet transform is regulatory in the time domain versus different frequencies. In another word, for the wavelet transform in the low frequency, poorer time resolution and higher frequency resolution would be obtained; while in the high frequency, this transform has the characteristic of a higher time resolution and poorer frequency resolution. It keeps with the slow transformation of the low-frequency signal and the quick change of the high-frequency signal.

Result and discussion
Violent sloshing could be created in the tank and then would have an impact on tank structures. In the impact instant, the high-frequency behavior could be presented, and after the impact or during the action stages of the smallamplitude sloshing, the low-frequency one would be shown. Just through the change of the scale, the time-frequency characteristics of these two phases can be captured and distinguished by the wavelet transform. In the light of previ- ous statistical (Jiang et al., 2014) and Fourier spectral analysis (Jiang et al., 2015a), here Morlet wavelet transform is employed to do a comprehensive study on the behavior of the wavelet spectra for the experimental elevations and pressures.

Wavelet analysis for the free surface elevations
For the free surface elevation in the elastic tank, the time-frequency behavior and statistical distribution are studied based on Morlet wavelet transform. The resonant condition is selected. Figs. 8 and 9 present the time history, Fourier and wavelet energy spectra of the free surface elevation under resonance, where the excitation frequency f is 0.890 Hz.
Fourier spectrum is unitary in Figs. 8b and 9b. The theoretical spectra only have the primary energy in the excitation frequency f. While in the experimental spectra, except for taking the excitation frequency f as the primary peak frequency, other peak frequency components exist and are multiples of the excitation frequency kf (k = 2, 3, 4), which are very small compared with the primary one (Jiang et al., 2015). The rigid and elastic spectra are close to each other very much, both of which are less at E01 but greater at E02 than the analytical one.
A nonlinear analytical solution for a rigid tank is intro-duced (Faltinsen, 1978) and compared with the experiment result. This solution is widely used by many researchers, such as Wu et al. (1998), Liu and Lin (2008) and Jiang et al. (2015b), to verify the experiments and numerical simulations. The analytical solution is derived from the theory of nonlinear potential flow, which assumes that the fluid is incompressible and the flow is irrotational. The velocity potential function satisfies the Laplace equation in the fluid domain. In the actual sloshing, there is damping of the liquid motion due to viscous effects. From the experimental wave shape, elevation and sloshing pressure, liquid sloshing in most test runs reach a steady state or quasi steady state in 2-10 sloshing periods. This shows that the liquid motion has a relatively large damping in the low-fill-depth and it dissipates a large amount of energy from the external excitation, which is also summarized by Faltinsen and Rognebakke (1999). The damping of the liquid is mainly caused by the viscous effects, or by the viscous resistance of liquids, which is related to the free surface deformation and the wall shear. In the ideal state, there is no liquid damping in the theoretical solution; however, there is damping of liquid motion in practice. Therefore, to approximate the reality, an artificial damping term is introduced in the analytical solution, which is chosen based on trial-and-error from a variety of parameters in the tank. Different damping coeffi- cients are calculated, and in the case of the damping coefficient of 5%, the theoretical solution and the experimental data are most close to each other. Thus, 5% of the artificial damping coefficient is adopted in our calculation. Here, this theoretical solution is compared with the experimental data to highlight nonlinearities in the physical test, such as nonlinear phenomena of hydraulic jump, double-peak and traveling waves. Besides the features displayed by the Fourier spectra, there still have much more information that can only be presented in the wavelet spectra due to its time-frequency behavior. In the wavelet energy spectra of Figs. 8 and 9 during the evolution process, the primary energy is directly observed to distribute around the external excitation frequency (f = 0.890 Hz) in the whole time. In this frequency domain, the experimental wavelet energy has a wider extension than that of the theoretical energy spectrum. It is not only mainly distributed around the excitation frequency, but also has a certain energy distribution near and above the two-times frequency of the excitation frequency (2f = 1.780 Hz), that does agree well with Xue et al.'s (2013) analysis result for the wavelet energy in rigid tanks. Meanwhile, this part of energy presents a discontinuous spread over time and concentrated on the wave crest. The variation and distribution is in accordance with the rigid result of Colagrossi et al. (2004).
Moreover, due to the elastic effect of the bulkhead, the wavelet elevation energy at each frequency region in the elastic tank is less than the corresponding one in the rigid tank obviously (Figs. 8d, 8e, 9d and 9e). These are not very clear in the Fourier spectra due to the global transformation effect, but are very obvious in the wavelet spectra as it shows the evolution of the energy variation at each frequency region over time.
Above analysis further demonstrates that the nonlinear analytical solution (Faltinsen, 1978) is not suitable to predict the resonant free surface elevation. In addition, since the nonlinear effect of hydraulic jump near the bulkhead, the energy spectrum of the free surface elevation near the bulkhead (E02) is larger than that far away from the bulkhead (E01), and is stronger in the high frequency region where the continuity is also better (Figs. 9d and 9e).
Upon the qualitative analysis for the wavelet energy distribution, the average values P W, Avg of the wavelet energy spectrum for the analyzed area are extracted to do a quantitative analysis. Table 1 shows the average value for the wavelet energy spectrum of the free surface elevation, where Δ TE is the deviation between the theoretical and experimental value in the elastic tank; Δ RE is the deviation between the experimental values in the rigid and elastic tank. As shown in the table, at the position far away from the bulkhead (E01), the experimental value in the elastic tank is less than the theoretical value by about 25.7%, and is also smaller than the rigid one by about 16.5%. However, at the position near the bulkhead (E02), due to the influence of the strongly nonlinearity near the bulkhead, the wave elevation energy is larger than the theoretical one by 221.0%, but is still smaller than the rigid one by about 24.4%.

Wavelet analysis for the sloshing pressure
μ This section shows the time-frequency characteristics and statistical distributions for the sloshing pressure in the elastic tank based on Morlet wavelet transform. The resonant condition is selected. The pressure analytical solution used here is also from Faltinsen (1978) and based on the theory of the nonlinear potential flow, in which the viscosity term is introduced and the boundary conditions are linearized. The pressure analytical solution equation can be ob-tained by solving the modified style of Bernoulli equation: where, δ where is the damping coefficient.
At this moment, the static liquid-depth is at the location P01 (y = 0.000 m). When the water falls below this level, the recorded value of the pressure transducer at this static level is zero. Therefore, for the measuring point above the level, only the positive pressures are analyzed when the analytical solution (Faltinsen, 1978) is compared with the experimental result. Besides, due to the limitations of the theoretical solution, there are no theoretical values at the measuring positions above the free surface.
Figs. 11 and 12 show the time history, Fourier and wavelet energy spectra of the sloshing pressure under the resonant condition, where the excitation frequency f is 0.890 Hz. As seen from Figs. 11a and 12a, the impulsive pressures occur on the bulkhead in the model test, and the double-peak phenomenon is observed on the crest of the ex-perimental pressure, which is caused by the impacting of the traveling wave together with the climbing and falling of hydraulic jump (Fig. 10) (Zhao, 2013;Jiang et al., 2014). The shape of the experimental pressure on the elastic bulkhead is almost the same as that on the rigid bulkhead, but its magnitude is slightly smaller than the rigid result. The analytical solution is a regular cosine curve, the magnitude of which is smaller than that obtained from the test.
For the Fourier pressure spectra, the primary peak frequency is the excitation frequency f, the same as the elevation spectra; and in the model test there still have some other sub-peak frequencies which are multiples of the excitation frequency kf.
Owing to the obvious impact feature of the pressure occurred at the locations above the static surface, the nonlinear theoretical solution is not suitable to predict the pressure at the corresponding locations. Therefore, the deviation between the wavelet energy spectra of the experimental pressure and the analytical one is very great. From the wavelet energy spectra, the range of the experimental energy is very wide in the frequency domain; the primary energy is concentrated around the excitation frequency (f = 0.890 Hz) and a part of energy has a continuous distribution in the double frequency (2f = 1.780 Hz). Moreover, an obvious energy dispersion is near the quadruple frequency (4f = 3.560 Hz) concentrating on the wave crest (Figs. 11d, 11e, 12c and 12d). Lastly but most significantly, the conspicuous impulse behavior is observed in the wavelet energy distribution above the quadruple frequency 4f. This behavior cannot be described in the Fourier spectra since it cannot explore sloshing waves in the time and frequency domain simultaneously. In addition, compared with the rigid energy spectra, the elastic energy distribution in each frequency range is slightly smaller than the corresponding rigid one at the two locations.
Accordingly, this impulsive behavior is corresponding to the double peak of the impact pressure as mentioned before (Jiang et al., 2014). In membrane tanks, the primary problem associated with sloshing is the potential damage to the tank walls from this peak impact sloshing pressures (Abramson et al., 1974). In essence, although the duration of Fig. 10. Snapshots of the wave elevation for the traveling wave and water jump in the rigid test. this impulsive peak is short, the peak is extremely great and the effect area is small, so it is prone to cause larger local failure to the structure. Slosh-related loads causing tank damages have occurred on two ships with membrane tanks, the "Polar Alaska" and the "Artic Tokyo" (Gavory and de Seze, 2009).
Furthermore, it can be generally observed that the magnitude of the peak pressure at the static level (P01) is less than that at the location P02, but the spectral energy at P01 is greater than that at P02 apparently. This may be attributed to the pressure integrated over the duration of the impact (Hamlin et al., 1986), which can represent the distributed energy to some extent (Bunnik and Huijsmans, 2007). From the comparison of a typical pressure sample in Fig. 13, it is found that the integrated area in a period at P01 (0.229 kPa·s) is obviously larger than that at P02 (0.094 kPa·s), which can verify the former inference.
The average value P W, Avg of the wavelet energy spec-trum for the sloshing pressure is presented in Table 2. From the table, at the location of the static surface (P01), the energy value in the elastic tank is less than that in the rigid tank, while the theoretical value is larger than the elastic one by 25%. Because of the larger integrated pressure pulse at P01, the energy value at P01 is larger than that at the location P02 obviously.

Conclusions
Morlet wavelet transform is employed to study the characteristics of the free surface elevations and sloshing pressures in the elastic tank. Upon the time-frequency behavior, variations and distributions are analyzed for the wavelet energy spectra.
From the wavelet analysis, sloshing energies are mainly concentrated around the external excitation frequency and expanded to high frequencies under violent condition. Besides, for the free surface elevations, the experimental wavelet spectra have a certain energy spread near and above the Table 2 Average value of wavelet energy spectra for sloshing pressure under the resonant condition (h = 0.080 m, A = 0.010 m, f = 0.890 Hz).

Location
Model Elastic (kPa 2 ) Theoretical (kPa 2 ) Deviation Δ TE (%) Rigid (kPa 2 ) Deviation Δ RE (%) P01 (y = +0.000 m) 3.6 4.8 25.0 3.9 7.7 P02 (y = +0.045 m) 0.9 --0.9 0.0 double frequency 2f, which presents a discontinuous or continuous distribution over time and is concentrated on the wave crest; as to sloshing pressures under resonance, a part of experimental energy has a continuous distribution in the double frequency 2f, and an obvious energy dispersion is near the quadruple frequency 4f, which is concentrated on the wave crest. Meanwhile, the conspicuous impulse behavior is observed in the pressure wavelet energy distribution above the quadruple frequency. Moreover, experimental wavelet energy of the elevation in the elastic tank is less than the rigid one in resonance; otherwise, there is no significant difference between the two ones. The energy of the pressure spectra in the elastic tank is close to the rigid one.
The results can recommend engineers to concern the primary sloshing natural frequency and the impulsive peak pressure, which are of great energy and may cause structural damage. It can also help designers to ensure the effective implementation of technological innovations for the engineering measures to suppress sloshing.   JIANG Mei-rong et al. China Ocean Eng., 2018, Vol. 32, No. 4, P. 400-412 411