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FOURIER TRANSFORM PROFILOMETRY FOR PURE WATER WAVES
This chapter presents a brief description of Fourier Transform Profilometry (FTP) method for water waves and its influence on water properties, manifested by an enhanced energy dissi-pation. The physical basis of this effect are discussed. It is followed with the article published in Experiments in Fluids [68] describing the improvement to the FTP technique, which over-comes the enhanced damping effect and enables a proper investigations of the wave phe-nomena.
Fourier Transform Profilometry – method description
Experimental investigations of the physical phenomena demands the measurement tech-nique and appropriate equipment. Constant progress in water waves experiments spurs re-searchers to invent new methods, which can make use of new technologies. The most valu-able quantity in the study of surface waves is the free-surface deformation (FSD). The devel-opment of a water wave field is revealed through its FSD. The technique for studying the FSD underlies the fundamental issue in the experimental investigations of phenomena occurring on the free surface and its vicinity. In the last decades various techniques have been invented, such as refractive and reflective techniques [15], gradient detector techniques [83, 84], diffus-ing light photography [76], and others. Unfortunately, all these methods present limitations (only one or few-points measurements, intrusive methods) and cannot be easily applied in a wide range of experiments on water waves (for the extensive monograph of different fringe projection techniques and its recent developments see Gorthi and Rastogi [35]). Advanced research of the FSD in the laboratory scale requires a technique that allows to reconstruct, non-intrusively, the 3D surface deformation with high space and time resolution.
A Fourier Transform Profilometry method (FTP) seemed to be a good answer to our needs. Originally developed for solid bodies [72, 73] it has been used to measure water waves firstly in 2002 [82]. Later, it has been adapted by our group for fully resolved space-time measurements of FSD. The scheme of our setup configuration is presented in the Fig. 2.1. The basic idea is as follows: the fringe pattern of known characteristic is projected onto the flat surface and recorded by a camera (reference image). When surface deforms the fringe pattern acquired by the camera is also distorted. The deformation depends on the object’s profile and the per-spective from which it is seen by the camera. This deformed fringe pattern is then compared to the reference one, revealing the phase difference between them (see Fig. 2.2). The informa-tion about FSD is directly encoded in this phase shift map. The appropriate detection of the underlying phase shift map 4’ of the captured patterns and conversion from 4’ to absolute height • are the key tasks in the FTP method for water waves.
where D is a distance between camera and videoprojector, L is a distance between surface at rest and the camera/videoprojector (see Fig. 2.1) and p the periodicity of the fringes. These values are known a priori, thus the phase shift 4’ has to be found to reconstruct FSD.
Omitting the background variations, the intensity of light recorded by the camera for refer-ence image I0 and deformed image I , while projecting sinusoidal wave can be mathematically described as:
Figure 2.2: A reference fringe pattern of known characteristic is projected onto a surface at rest (on the left) and onto a deformed surface (on the right). The distortion of the fringe pat-tern on the deformed image with respect to the undeformed one is clearly visible. Determi-nation of the phase shift 4’ between both patterns is fundamental to reconstruct the free surface deformation (FSD) by means of the FTP method.
A stands for the amplitude intensity of the captured pattern. This light intensity amplitude A depends not only on the projection power, but also on a surface light diffusivity and a camera sensitivity. Note also that due to the inhomogeneous projection by the videoprojector (more intense at the center than close to the borders) its value is not constant on the recording area. Now, an analytical form to obtain phase shift 4’ ˘ ’(x)¡’0(x) has to be found. Let’s perform a Hilbert transform of both quantities:
where i stands for the imaginary unit. Complex conjugation of the expression (2.4) multiplied by the expression (2.5) gives:
To separate the value of phase shift 4’ from the above formula, it is enough to take the imag-inary part of the natural logarithm:
This procedure allows one to obtain ¢’ completely isolated from the light intensity varia-tion A.
This method has been successfully employed in our laboratory and used by our group to study water-wave trapped modes [11, 13] and drop impact on thin liquid film [45]. It has also been used by other groups of scientists, e.g. in the case of wave turbulence on the surface of fluid [40].
Since the FTP method uses images reflected from the surface, this surface has to be light diffusive. The most natural and cheapest liquid for studying wave phenomena is obviously water, for which the sufficient surface diffusivity cannot be obtained. Our previous experi-mental investigations (quoted above) used white paint dissolved in water (at 1 : 200 ratio by volume). This solution does not change significantly the density and gives the sufficient light scattering to employ the FTP method.
While performing experiments it has been found that the attenuation of waves is strongly enhanced compared to the one of pure water. At the beginning a reason for such behavior was unknown. The solution to that problem was a first task of my thesis and gave unexpected results.
Surface wave attenuation – resonance phenomenon
Water waves suffer attenuation due to the viscosity „ (not considered in the linear wave the-ory developed in section 1.2) through two main processes of energy dissipation. The first one is the bottom friction. Its presence is significant, whenever the wavelength ‚ is large com-pared to the depth of the liquid H . In that case, the wave motion is far from being cylindrical and takes an elliptical form. Horizontal motion near the bottom induces the friction and in consequence the attenuation.
Second type of attenuation connected with the water viscosity „ is known as internal (bulk) dissipation. If we take into account water viscosity, then the stress tensor is altered and governing equation changes, namely the linearized momentum equation (1.2) takes a form:
and this in consequence changes boundary condition on the free surface (it is no more only surface tension and gravity that are balanced on the surface). The new set of boundary con-ditions, which incorporate the viscous losses, has to be satisfied now, resulting in the wave energy dissipation. Based on the conservation of energy, one can derive the relationship be-tween the spatial decay of the wave amplitude and the fluid viscosity [2] (presented in eq. (5) in the following article). The resulting internal attenuation by viscous shearing is substantial only for high frequency waves (low wavelengths).
None of that effects could be responsible for the largely enhanced dissipation of the in-termediate waves. It has been finally discovered that the application of paint has a big disad-vantage not considered before. Paint requires substances stabilizing emulsion. These surface active substances, so important in normal use of paint, can have a dramatic impact directly on the water surface and indirectly on water wave character (amplitude and energy attenua-tion). Due to the surfactant’s structure, containing the hydrophobic and hydrophilic groups, their molecules move towards liquid’s interface (water-air interface in this case) spreading on the water surface until creating a monolayer. Hence, relatively small amount of surfactant can cause an extensive area of fluid to be covered. It may seem questionable whether the surfac-tant film of the thickness of few 10¡9 m can have a considerable influence on wave attenu-ation, however, the investigations of the surface film rheology, carried out in the end of the XXth century, distinctly proved the enhance of the wave energy dissipation due to the phys-ical properties of the monomolecular film. Passing wave locally expands (at the wave back) and compresses (at the wave front) the surfactant monolayer causing locally higher and lower surface tension areas (due to the local differences in surfactant density) and in consequence of surface gradients – tangential stresses.
Figure 2.3: Due to the horizontal velocity on the surface of the traveling wave, the surfactant density on the wave front is increased and reduced on its wave back. Resulting surface tension gradients induce longitudinal waves (Marangoni waves). The scheme proposed by Behrozzi et al. [3]
In the consequence of elastic properties of the surface, which causes resistance to stretch-ing and compressing, the underlying fluid changes its motion pattern from cylindrical to el-liptical. For this reason, the energy is drawn away stronger by viscous friction. The result-ing damping coefficient is significantly enhanced. The experiments performed in 1940s [47], in 1950s [19] and in the next decade [52, 60] showed that surface elasticity has strong influ-ence on capillary wave damping. They revealed that damping is the strongest for low surface elasticity values, however, the precise explanation for this effect remained unanswered. Fi-nally, in 1968, Lucassen [50] made a discovery. He proved an existance of longitudinal waves (called Marangoni waves) accompanying the capillary ones. He stated that in contrast to cap-illary waves, the Marangoni waves are governed by the surface elastic modulus rather than by surface tension. Due to the tangential surface stresses, the kinematic boundary condi-tion on the surface is changed, thus the linearized Navier-Stokes equation has two different solutions: one describing well-known transversal waves and another describing Marangoni waves. These waves occur independently of each other. He also reported [50] that surface elasticity has practically no influence on surface transversal waves, while it has strong influ-ence on Marangoni waves, which reaches maximum damping for a low surface elasticity.
Further investigations revealed that not only capillary waves, but also short-gravity as well as intermediate ocean waves, can be significantly damped while Marangoni waves exist (see [8, 42, 51]). An in-depth explanation to these observations has been given by Alpers and Hühnerfuss [1]. They stated that increased energy attenuation is caused by the resonance-type wave damping between Marangoni and short-gravity waves.
In the second section of this chapter my article published in Experiments in Fluids [68] is presented. The article describes in details the theory of surface contamination and proposes the use of new pigment to obtain water-air interface light scattering with clean surface. The experimental measurements of water colored by standard paint and non-surface active pig-ment are given and compared. The resonance-type damping arising from the surface tensions gradients is confirmed experimentally. The mathematical formulas and physical interpreta-tion of the calming effect due to the presence of surface film in the capillary-gravity regime is also discussed.
This study was crucial for subsequent experimental studies conducted during my thesis. It opened a possibility of proper studying of wave turbulence by means of FTP method as well as increased a quality of water wave refocalisation in time-reversal experiments.
Fourier, transform profilometry for water waves: how to achieve clean water attenuation with diffusive reflection at the water surface?
Abstract We present a study of the damping of capillary-gravity waves in water containing pigments. The practical interest comes from a recent profilometry technique (FTP for Fourier Transform Profilometry) using fringe projection onto the liquid-free surface. This experimental technique requires diffusive reflection of light on the liquid surface, which is usually achieved by adding white pigments. It is shown that the use of most paint pigments causes a large enhancement of the damping of the waves. Indeed, these paints contain surfactants which are easily adsorbed at the air–water interface. The resulting surface film changes the attenuation properties because of the resonance-type damping between capillary-gravity waves and Marangoni waves. We study the physicochemical properties of col-oring pigments, showing that particles of the anatase (TiO2) pigment make the water surface light diffusive while avoiding any surface film effects. The use of the chosen particles allows to perform space-time resolved FTP measurements on capillary-gravity waves, in a liquid with the damping properties of pure water.
Przadka B. Cabane P. Petitjeans (&)
Free surface waves are an important subject in fluid dynamics due to their practical applications in the industry (such as naval architecture, coastal and ocean engineering) and also due to the wealth of physical phenomena that they display. Most experimental studies on liquid surface deformation have used qualitative direct visualizations of the 2D surface or quantitative one-point temporal mea-surements. Recently, our group has proposed a full space-time resolved measurement of the surface elevation, using a technique called Fourier transform profilometry (FTP). This technique was first developed by Takeda et al. (1982), Takeda (1983) for solid surfaces. We have improved and implemented it for liquid surfaces (for details and biblio-graphy see Maurel et al. 2009; Cobelli et al. 2009; Gorthi 2010). Typical examples of measured wave fields are shown in the Fig. 1. In the first example, a plane wave propagates with defined frequency x over a nonuniform bottom producing wave scattering. Owing to the temporal resolution, the total measured displacement field P h(x, y, t) can be easily expanded in h(x, y, t) = n Re[hn(x, y)einxt] to extract the complex field h1 that is later analyzed. The second example is of particular interest with respect to the purpose of this paper.1 In the case of wave turbulence experiments, the water wave field results from the nonlinear interactions of random waves. The theoretical predictions in weak turbulence theory are done in the Fourier (k, x)-space, and this space becomes accessible a real part of the wave field h1(x,y) in experiments of wave propagating over a nonuniform bottom. The complex field h1 is obtained by extracting the Fourier component of the total measured field. Colorbar is in mm. b Instantaneously measured velocity field in experiments of wave turbulence. The measure of h(x, y, t) gives access to the Fourier (k, x)-space for comparison with theoretical predictions. Colorbar is in m/s experimentally thanks to our full space-time resolved method.
Since the FTP method uses the deformation of fringes projected onto the free surface, this surface must scatter light. It is not possible with pure water, which is transparent and has a very low surface reflectance. Previously, light diffu-sivity was achieved by diluting white paint in water. This produced a substantial increase in the scattering from water (in the reflectance of the water surface). However, it was also found that the attenuation of surface waves was dramatically enhanced compared to that of pure water. This spurious attenuation is particularly harmfull when studying wave phenomena. It was soon found that it is caused by the pres-ence of a film at the surface of water. Indeed, water is a liquid that has an unusually high cohesion and, therefore, a very high surface tension. Consequently, many species adsorb to the water–air interface, either from the water side (dissolved species) or from the atmosphere (airborne molecules or particles). These species quickly create a film at the water surface and this film may strongly change the attenuation properties of surface waves: this phenomenon is well known since the observations of the ‘‘calming effect of oil on water’’ by Benjamin Franklin in the eighteenth century (Franklin 1774; Behroozi et al. 2007). Since then, the attenuation due to the excitation of surface film vibrations by water waves has been studied in details by several authors (Alpers 1989; Levich 1962; Miles 1967; Lucassen 1968, 1982; Lange 1984).
We have found a way to make the water light diffusive while keeping its low attenuation property. The solution to this problem required investigation into the surface chem-istry of the pigments. It also involved the use of rigorous methods to ensure good pigments dispersion without release of any surface active molecules in water (not even traces!).
This paper is organized as follows: Section 2 is dedi-cated to the choice and characterization of nonsurface active aqueous dispersions that can make water light dif-fusive while avoiding the formation of a surface film. To verify the absence of a surface film, the measurement of the surface tension of the dilute aqueous dispersions was per-formed. In Sect. 3, we present FTP measurements of the water wave attenuation, comparing water colored by plain paint to the dispersion of nonsurface active particles. It is also confirmed that the chosen particles allow us to restore the attenuation of pure water. Incidentally, it is shown that the enhancement of attenuation by plain paint is associated with a resonance of the film surface, in agreement with the theory (Alpers 1989; Lucassen 1968, 1982).
The desired characteristics of the pigment particles are threefold: (a) they must provide a high reflectance of the water surface; (b) they must be well dispersed in water;
(c) they must not release any surface active molecules or ions that would form a film at the surface of water. To a large extent, these constraints are in conflict with each other and this is what makes the choice of the pigments difficult.
In order to meet condition (a), the classical choice is to use titania (TiO2) particles. Indeed, titania has a very high refractive index (n = 2.7), nearly the highest among min-erals. With this high refractive index, the optimum size of the titania particles is about 300 nm, which is the average size of commercially available TiO2 pigments. Particles of this size are strongly agitated by Brownian motions and sediment quite slowly, unless they aggregate.
Condition (b) is not easily met for pure titania particles, for two reasons. Firstly, titania surfaces are ionized in water according to the classical reaction schemes:
If one of these reactions prevails, the surface acquires a net electrical charge and the corresponding electrical potential attracts a double layer of counterions in the vicinity of the surface. When particles approach each other, the overlap of the diffuse layers of counterions gives rise to the classical DLVO repulsions, which keep the surfaces apart and prevent particle aggregation (Evans 1994; Israelachvili 1991; Hunter 1981). However, for titania, their reactions are balanced at pH = 6, which means that the surface has an isoelectric point at this pH. Consequently, the titania surfaces do not retain any electrical double layers of counterions, and when particles approach each other, the classical DLVO repulsive forces are absent. This is true for both common crystalline forms of titania, that is, anatase and rutile.
In order to prevent the aggregation of titania particles in water, the particles’ surfaces are usually covered either with a layer of another oxide (e.g., silica or alumina) or with adsorbed polyelectrolytes, such as polyacrylates. These changes shift the zeta potential curve and the loca-tion of the isoelectric point on the pH scale. Figure 2 shows the results of measurements of the electrophoretic mobility of various titania dispersions, here expressed in terms of the zeta potential, which is the electrical potential at the shear plane near the particle surface. The titania pigments provided by Kronos International, Inc. have been applied in all our dispersions.
These values of the zeta potential make it possible to predict the aggregation behavior of the various titania dispersions. Indeed, it is known from DLVO theory that zeta potentials above 40 mV provide strong electrical double layer repulsions and, therefore, predict adequate colloidal stability. Conversely, zeta potentials below 30 mV do not provide sufficiently strong electrical double layer repulsions, especially for the case of particles that have strong Van der Waals attractions as is the case for titania particles.
According to the results shown in Fig. 2, the rutile TiO2 coated with alumina would only be acceptable at pH B 3, which is too acidic to be handled in a large water tank with delicate electrical and optical equipment. Similarly, the rutile TiO2 coated with alumina and silica would not aggregate in a solution with pH C 9, but this is not acceptable in a laboratory with standard instrumentation.
Nowadays, many titania dispersions have gained a sur-face charge and a high zeta potential through the adsorption of polycarboxylates, e.g., sodium polyacrylate or sodium citrate. However, the molecules are only physically bound to the titania surfaces through ionic interactions with the surface sites. A fraction of these molecules is always released in water, either because the adsorption forces are not infinitely strong or because the manufacturer has added an amount of the molecules that exceeds the saturation level of the surface (most likely). It is then in conflict with condition (c), because these excess molecules are surface active and, therefore, spontaneously adsorb and form a film at the free water surface. Indeed, we have found that all water-dispersible pigments that are made of particles with ‘‘dispersants’’ adsorbed on their surfaces fail to pass con-dition (c), as defined by the criteria presented below.
Figure 2 shows that the anatase pigment is for us the best choice since it has a zeta potential Zp \ -40 mV in all aqueous solutions with pH[4. It is then possible to use pure water to disperse these particles. The water used in our experiment is purified. Its resistivity is greater than 16 MX cm, and its surface tension is 71 mN/m.
It is then necessary to verify that condition (c) is met with the chosen pigment. For this purpose, some experi-mental criteria and procedures must be defined. A criterion for the presence or absence of a film at the water surface is the value of the surface tension. If pfilm is the film pressure, c the measured surface tension and c0 the surface tension of pure water, then Typical values of pfilm are 20 mN/m for water surfaces exposed to open air and 40 mN/m for water containing dissolved surface active molecules. Thus, a practical cri-terion for assessing the absence of adsorbed films at the water surface is that the measured surface tension should be within 5 mN/m of that of pure water. As noted previ-ously, the absence of any surface film is not easily achieved with water containing dispersed pigments, as most com-mercially available pigments release surface active species in water. In contrast, aqueous dispersions made with the anatase pigment had surface tensions above 70 mN/m. This surface tension decreased very slowly with time because of adsorption of surface active species that have migrated from the air or other parts of the equipment (Prisle et al. 2008). A brief aspiration of the surface layer brought it back to its initial value.
For FTP experiments, it is useful to characterize the reflectance of the water surface (condition (a)) in terms of the contrast. The contrast is defined as the number of intensity gray levels between the white and black fringes.
The better resolved is the fringe signal recorded by the camera, the better resolution on the measured surface shape. This is because the resolution in the FTP measure-ments depends on this contrast (for more technical details see Sect,. 3.1). This contrast is a combination of the abso-lute reflectance of the particles, of the characteristics of the projecting device and of the characteristics of the camera. In our studies, we have obtained accurate measurements for at least 50 gray levels between white and black fringes, while the contrast below 20 gray levels was found to be critical.
Figure 3 illustrates the contrast as a function of the concentration of anatase (rutile pigments are given for comparison). For the chosen anatase particles, the contrast of 50 is obtained for a concentration of 2 g/l.
Finally, the particle sedimentation has been analyzed for the chosen anatase pigment. Settling out process causes the light to be diffused not from the surface, but from the lower layer of the dispersion and this induces an error in the measurements. The sedimentation speed v can be calcu-lated by equaling the gravity force and viscous resistance of the water where r stands for particle radius, g for the gravity accel-eration, and Dq ¼ 2:8 g/cm3 for the density difference between the particles and water, and l = 10-3 Pa.s for the dynamical viscosity. The measurements of the particles’ size for the pH in the electrically stable range showed the particle diameter to be 350 ± 50 nm.
Figure 4 illustrates the effect of the particles’ sedimen-tation on the FTP measurements. The liquid was agitated at the initial time, and then the elevation of the free surface was measured at several times in two cases: I) the liquid was not shaken anymore (it remained at rest) during the whole experiment, and II) the liquid was excited by the wavemaker for a short time between two measurements.
Fig. 3 Contrast as a function of the particles’ concentration for anatase pigment and rutile pigments. The camera worked at 8 bits (256 gray levels)
Fig. 4 The apparent surface position, as measured by FTP, as a function of time for two cases: no liquid shaking (blue curve), the liquid was excited by waves produced by a wavemaker during a short time between two measurements (red curve). Black curve shows the theoretical prediction for the sedimentation (Eq. 2, using the mean size of the particles). The dispersion used here is the anatase with a concentration of 4 g/l. The real position of the free surface is at zero (the errorbar is 0.02 mm) In the first case, the free surface position seems to decrease in time which is connected with particles’ sedimentation. Initially, sedimentation follows reasonably the theoretical prediction (black curve in the Fig. 4) using Eq. 2 with r the mean size of the particles. Then, the particles seem to sediment slower. It is connected with the existence of smaller particles which settle out slower. In the case II, the free surface position varies around the zero level within the error bar. This means that the motion induced by the wave propagation provides enough mixing to prevent the sedi-mentation effect.
FTP measurement of water wave attenuation
In 1872, Marangoni described the effect of surface tension gradients. He found that they alter the tangential stresses balance on the surface: while the wave passes, it locally expands (at the wave back) and compresses (at the wave front) the surfactant monolayer causing locally higher and lower surface tension areas due to the local differences in surfactant density. In 1968, Lucassen proved the existence of longitudinal waves on the surface film accompanying the capillary-gravity waves. These waves are governed by the surface elastic modulus of the film rather than by the surface tension and the gravity. Because of the tangential surface stresses, the kinematic boundary condition on the surface is changed, implying that the linearized Navier– Stokes equation has two different solutions: one describing well-known water waves and another one describing lon-gitudinal waves (often called Marangoni waves).
In the past, the calming effect due to the monomolecular slicks existence has been supposed to affect only the cap-illary waves. Few years later, Lucassen (1982), Cini et al.
(1987) showed that such a film can induce the resonance-type wave damping between Marangoni and capillary-gravity waves, which results in a large attenuation enhancement also in the short-gravity-wave region. At the same time, the experimental evidence of the higher energy dissipation of the short gravity waves was given by Lange (1984) in the laboratory scale and by Ermakov et al. (1986) in the open sea. The mathematical description of the problem, originally proposed by Alpers (1989), can be found in the ‘‘Appendix’’.
Experimental details
The presented experiments were performed in the rectan-gular basin of 160 9 60 cm2. The industrial titanium dioxide anatase particles (Kronos 1002) with a concentra-tion 4 g/l were used to color water of 5 cm depth. The surface tension c was measured before and regularly during experiments by Kru¨ss K100 Tensiometer using the ring method and was equal to 71 ± 1 mN/m for water with titania particles (and 32 ± 1 mN/m for water colored by plain paint). Water temperature was maintained at 17 ± 1LC.
The FTP method implemented by Maurel et al. (2009) and Cobelli et al. (2009) was applied to measure the sur-face elevation h(x, y, t). In the FTP method, a fringe pat-tern is projected onto the free surface and observed from a different position by the camera. The surface elevation information is encoded in the fringes deformation in comparison with the original (undeformed) grating image. It is, therefore, the phase shift between the reference and deformed images which contains all information about the deformed surface (see Lagubeau et al. 2010; Cobelli et al. 2009, 2011a, b, for the examples of applications).
A Phantom v9 high-speed camera was used to record the surface of 30 9 30 cm2 area with a 560 9 560 pix2 and a sampling frequency fs = 160 fps. The horizontal resolution was equal to 0.53 mm, which corresponds to the pixel size. The vertical precision was estimated to 0.1 mm with an error of 0.05 mm. It was found to be sufficient to recon-struct the waves with typical wavelength 3–15 cm and amplitude of few millimeters (to avoid the nonlinear effects).
The experiment consists in recording the transient wave produced by a broadband wavepacket signal (modulated with a blackman window). In the case of water colored by the titania particles, the central frequency of the signal was 8 Hz and the sampling was chosen such that 20 points were recored per period. A single input signal allowed us to treat broad frequency range between 4 and 10 Hz. For water colored by plain paint, the damping is enhanced, so the frequency range around the central frequency with energy above the noise level is reduced. Thus, five different input signals were used with central frequency at 4, 5, 6, 7 and 8 Hz. The images recording finished after capturing 16,000 images (100 s), which was enough for the waves to be totally attenuated.
Since the depth of the water is constant, in the harmonic regime, at pulsation x, it can be shown that the height perturbation is the solution of the Helmholtz equation:
where k is linked to the pulsation x by the linear dispersion relation. It is important to notice that k has an imaginary part due to the attenuation (k ¼ j þ ib; where j, b are real). In the above equation, k is the unknown and H stands for the time Fourier transform of the measured transient height h(x, y, t):
The FTP measurement of the transient h allowed to obtain the H fields for a broad range of frequencies with 0.01 Hz step.
An example of the measured H field is shown in the top of the Fig. 5. Owing to the spatial resolution of the FTP (the real part of the wavenumber k) and damping coefficient b (the imaginary part of the wavenumber k).
Fig. 8 Absolute damping coefficient bcalc in the function of the frequency. Comparison of the water suspensions obtained by adding anatase pigment and white paint. The black line is the theoretical attenuation b0 given by Stokes (Eq. 5)
Fig. 9 Relative damping coefficient (bcalc/b0) in the function of the frequency. Comparison of the water suspensions obtained by adding anatase pigment and white paint.
The field corresponding to -DH is shown in the Fig. 5 (bottom). Both patterns are very similar, indicating that the calculation of the wavenumber from the Helmholtz equa-tion 3 can be achieved.
Table of contents :
1 Introduction
1.1 Motivation
1.2 Linear wave theory
1.3 Organization of the manuscript
2 Fourier TransformProfilometry for pure water waves
2.1 Fourier TransformProfilometry – method description
2.2 Surface wave attenuation – resonance phenomenon
2.3 Main results
2.4 Supplementary results
3 Time-reversal of water waves
3.1 Motivation
3.2 Time-reversal – problemformulation
3.3 One-channel time-reversal in chaotic cavity
3.4 Main results
3.5 Supplementary results
4 Different regimes for water wave turbulence
4.1 Motivation
4.2 Introduction
4.3 Wave turbulence – a real experimental challenge
4.4 Main results
4.5 Supplementary results
5 Local measurement of liquid depth
5.1 Introduction
5.2 Circlemethod principles
5.3 Experimental setup
5.4 Results – circle method depth reconstruction
5.5 Results – depth reconstruction by use of Helmholtz equation
5.6 Conclusions
6 Conclusions and perspectives
6.1 Summary and conclusions
6.2 Perspectives
Bibliography
