Polarization singularities and Möbius strips in sound and water-surface waves
K. Y. Bliokh, M. A. Alonso, D. Sugic, M. Perrin, F. Nori, E. Brasselet
PPolarization singularities and M¨obius strips in sound and water-surface waves
Konstantin Y. Bliokh, Miguel A. Alonso,
2, 3
Danica Sugic, Mathias Perrin, Franco Nori,
1, 5 and Etienne Brasselet Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan CNRS, Centrale Marseille, Institut Fresnel, Aix Marseille University, UMR 7249, 13397 Marseille CEDEX 20, France The Institute of Optics, University of Rochester, Rochester, NY 14627, USA Universit´e de Bordeaux, CNRS, LOMA, UMR 5798, Talence, France Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
We show that polarization singularities, generic for any complex vector fields but so far mostlystudied for electromagnetic fields, appear naturally in inhomogeneous (yet monochromatic) soundand water-surface (e.g., gravity or capillary) wave fields in fluids or gases. The vector properties ofthese waves are described by the velocity or displacement fields characterizing the local oscillatorymotion of the medium particles. We consider a number of examples revealing C-points of purelycircular polarization and polarization M¨obius strips (formed by major axes of polarization ellipses)around the C-points in sound and gravity wave fields. Our results (i) offer a new readily accessibleplatform for studies of polarization singularities and topological features of complex vector wavefieldsand (ii) can play an important role in characterizing vector (e.g., dipole) wave-matter interactionsin acoustics and fluid mechanics.
I. INTRODUCTION
Polarization and spin are inherent properties of vec-tor waves. These are typically associated with classicalelectromagnetic/optical fields or quantum particles withspin [1–3]. Recently, it was noticed that sound waves influids or gases [4–10] as well as water-surface (e.g., grav-ity) waves [11, 12] also possess inherent vector properties,and the notions of polarization and spin are naturallyinvolved there. These properties are related to wave-induced motion of the medium particles in fluids andgases. Such motion can be characterized by the vectorvelocity field 𝒱 ( r , 𝑡 ) or the corresponding displacementfield ℛ ( r , 𝑡 ), 𝒱 = 𝜕 𝑡 ℛ , in a way entirely analogous to,e.g., the electric field ℰ ( r , 𝑡 ) or the corresponding vector-potential 𝒜 ( r , 𝑡 ), ℰ = − 𝜕 𝑡 𝒜 , in an electromagnetic wave.The main difference between electromagnetic andsound-wave polarizations is that the former are trans-verse (the fields ℰ and 𝒜 are orthogonal to the wavevec-tor k for a plane wave), while the latter are longitudinal (the fields 𝒱 and ℛ are parallel to the wavevector for aplane wave). In the case of gravity or capillary waves,which appear on surfaces of classical fuids or gases [13],a plane wave has two velocity components: longitudi-nal along the wavevector lying in the unperturbed water-surface plane, and vertical, normal to the surface. Akinto other surface or evanescent waves [4, 5, 14, 15], thesecomponents are mutualy 𝜋/ elliptically -polarized in the propaga-tion plane [16].However, when one considers structured (inhomoge-neous) wave fields, consisting of many plane waves, thesedifferences between transverse, longitudinal, and mixedplane-wave polarizations are largely eliminated. Indeed,at a given point r , a vector monochromatic field, whetherelectromagnetic, acoustic, or water-surface, traces an el-lipse which can have arbitrary orientation in 3D. Consid-ering the spatial distribution of such ellipses across the r -space, one deals with inhomogeneous polarization tex- tures. Important generic and topologically-robust char-acteristics of inhomogeneous wave fields are singularities :phase singularities in scalar fields and polarization singu-larities in vector polarization fields [17].Historically, both phase singularities and 2D polar-ization singularities were first observed in the scalarand 2D-current representations of tidal ocean waves [18–21]. However, a systematic treatment of structured wavefields has only been developed within the framework of singular optics [17, 22–24]. According to this approach,generic singularities of 2D (paraxial) and 3D (nonparax-ial) polarization fields are C-points or lines of purely cir-cular polarizations (as well as
L-lines of purely linearpolarizations) [17, 22, 24–27], and polarization M¨obiusstrips [27–36] which are formed (solely in 3D fields) bymajor axes of polarization ellipses around C-points/lines.These objects are very robust because of their topolog-ical nature; they also have important implications inthe geometric-phase and angular-momentum propertiesof the field [27].Being thoroughly described and observed for opticalfields, polarization singularities and topological polariza-tion structures have not been noticed so far in soundand gravity or capillary waves. In this work, we fill thisgap. We consider both random and regular structuredacoustic and water-surface wavefields and show that po-larization singularities and M¨obius strips are also ubiqui-tous for them. These results can have a twofold impact.First, they provide a new platform for studying polar-ization singularities and topological structures. Impor-tantly, while one cannot directly observe elliptical mo-tion of the electric field ℰ or the vector-potential 𝒜 inoptics, the velocity and displacement fields 𝒱 and ℛ are directly observable in acoustic and water-surface waves[11, 16, 37–39]. Second, the vector representation ofsound and water-surface waves can be relevant for wave-matter interactions, such as interactions with dipole par-ticles coupled to the vector velocity field [7, 40, 41]. a r X i v : . [ phy s i c s . c l a ss - ph ] F e b FIG. 1. Random 2D acoustic field obtained by the interference of 𝑁 = 7 plane waves with the same frequency and amplitudebut with random directions within the ( 𝑥, 𝑦 ) plane and random phases. (a) Wavevectors of the interfering waves (with color-coded phases) and distribution of the intensity of the velocity field | V | (greyscale). (b) Color-coded distribution of the phaseof the quadratic field V · V . The phase singularities (vortices) of this field correspond to C-points of purely circular polarizationin the field V . L-lines correspond to purely linear polarizations. (c) Distribution of the normalized polarization of V . Red,blue, and green colors correspond to right-handed, left-handed, and near-linear polarizations, respectively. The C-points in a2D polarization field can be labeled by two independent topological numbers [27]: (i) 𝑛 𝐷 = 1 / 𝑛 𝐷 = − / V · V ; (ii) 𝑛 𝐶 = 1 / 𝑛 𝐶 = − / II. POLARIZATION SINGULARITIES IN A 2DACOUSTIC FIELD
We consider monochromatic sound waves in a homoge-neous fluid or gas, which are described by the equations[13]: 𝑖 𝜔 𝛽 𝑃 = ∇ · V , 𝑖 𝜔 𝜌 V = ∇ 𝑃 . (1)Here, 𝜔 is the frequency, 𝜌 and 𝛽 are the density andcompressibility of the medium, whereas 𝑃 ( r ) and V ( r )are the complex pressure and velocity fields. The realtime-dependent fields are 𝒫 ( r , 𝑡 ) = Re[ 𝑃 ( r ) exp( − 𝑖𝜔𝑡 )]and 𝒱 ( r , 𝑡 ) = Re[ V ( r ) exp( − 𝑖𝜔𝑡 )].The plane-wave solution of Eqs. (1) is 𝑃 = 𝑃 exp( 𝑖 k · r ) , V = 𝑉 ¯ k exp( 𝑖 k · r ) , (2)where ¯ k = k /𝑘 , k is the wavevector, 𝑘 = 𝜔/𝑐 is thewavenumber, 𝑐 = 1 / √ 𝜌𝛽 is the speed of sound, and 𝑃 = √︀ 𝜌/𝛽 𝑉 . Sound waves are longitudinal because V ‖ k , but still have a vector nature described by thevelocity field V [4, 6–10]. In what follows, we will focuson the polarization properties of this vector wave field:the real velocity field 𝒱 ( r , 𝑡 ) at a given point r traces a polarization ellipse .We first examine a random speckle-like sound-wavefield in 2D. Namely, we consider the interference of 𝑁 plane waves (2), V = ∑︀ 𝑁𝑗 =1 V 𝑗 , with wavevectors k 𝑗 , 𝑗 =1 , ..., 𝑁 randomly distributed over the circle 𝑘 𝑥 + 𝑘 𝑦 = 𝑘 ( 𝑘 𝑧 = 0), and with equal amplitudes | 𝑉 𝑗 | but randomphases 𝜑 𝑗 = Arg( 𝑉 𝑗 ), as shown in Fig. 1(a). Due tothe longitudinal character of sound waves, 𝑉 𝑧 = 0, andthe polarization ellipses of such field all lie in the ( 𝑥, 𝑦 ) plane. Figure 1 shows an example of such random fieldincluding its intensity and polarization distributions.The distributions in Fig. 1 are similar to the corre-sponding distributions in random paraxial electromag-netic fields, with wavevectors directed almost along the 𝑧 -axis and polarization ellipses approximately lying inthe ( 𝑥, 𝑦 ) plane [17, 25, 27]. The only difference is thatparaxial electromagnetic fields have a typical inhomo-geneity scale of ( 𝜃𝑘 ) − , where 𝜃 ≪ 𝑧 -axis,while in the acoustic case 𝜃 = 𝜋/ 𝑘 − . Polarization singularities ofgeneric 2D polarization fields are the C-points of purely-circular polarization and
L-lines of purely-linear polar-ization [17, 22, 24–27], as shown in Figs. 1(b,c).The C-points correspond to phase sigularities (vor-tices) in the scalar field Ψ( r ) = V ( r ) · V ( r ) [17, 24, 27],Fig. 1(b). Notably, these points generically coincide nei-ther with zeros of the scalar pressure field 𝑃 ( r ), nor withzeros of | V ( r ) | . Furthermore, each C-point in a 2D po-larization field can be characterized by two half-integertopological numbers [27]. The first, 𝑛 𝐶 , corresponds tothe number of turns of the major semiaxis of the polar-ization ellipse along a closed contour including the C-point. The second, 𝑛 𝐷 , is half the topological charge ofthe corresponding phase singularity in the field Ψ. Inthe generic (non-degenerate) case, singularities have thetopological numbers 𝑛 𝐶 = ± / 𝑛 𝐷 = ± / FIG. 2. Random 3D acoustic field obtained by the interference of 𝑁 = 7 plane waves with the same frequency and amplitudebut with random directions within the 𝑘 𝑧 > | V | in the 𝑧 = 0 plane. (c) Color-coded distributionof the phase of the quadratic field V · V in the 𝑧 = 0 plane. Phase singularities (vortices) of this field correspond to C-pointsof purely circular polarization in the field V . (d) Distribution of the normalized polarization ellipses of V in the 𝑧 = 0 plane.(e) Continuous evolution of polarization ellipses with their major semiaxes along a contour encircling a nondegenerate C-pointexhibits a M¨obius-strip structure [28, 29, 31]. (f) For a contour encircling an even number of C-points (including zero), thereis no polarization M¨obius strip [27]. III. C-POINTS AND POLARIZATION M ¨OBIUSSTRIPS IN 3D ACOUSTIC FIELDSA. Random fields
Akin to nonparaxial 3D electromagnetic fields, genericsound-wave fields have polarization characterized by theellipses traced by the velocity field 𝒱 ( r , 𝑡 ) at every point r , which can be arbitrarily oriented in 3D space. To showan example of such field, we consider an interference of 𝑁 plane waves with equal amplitudes | 𝑉 𝑗 | , wavevectors k 𝑗 , 𝑗 = 1 , ..., 𝑁 , with directions randomly distributed overthe hemisphere 𝑘 𝑧 > 𝑘 𝑥 + 𝑘 𝑦 + 𝑘 𝑧 = 𝑘 ), and randomphases 𝜑 𝑗 = Arg( 𝑉 𝑗 ), see Fig. 2(a).The distributions of the resulting intensity | V | andof the phase of the quadratic field V · V over the 𝑧 = 0plane are shown in Figs. 2(b,c). Similar to the 2D case,the phase singularities of the quadratic field correspondto the C-points (polarization singularities) in the polar-ization distribution, Fig. 2(d). However, in the 3D case,the circular polarizations in these points do not generi-cally lie in the ( 𝑥, 𝑦 ) plane [17, 22, 24, 27].Furthermore, distributions of the 3D polarization el-lipses in the vicinity of C-points have remarkable topo- logical properties. Namely, continuous evolution of themajor semiaxes of the polarization ellipse along a contourencircling a non-degenerate C-point traces a 3D M¨obius-strip -like structure [27–29, 31–36], Fig. 2(e). Notably,the number of turns of the polarization ellipse aroundthe contour is not topologically stable: continuous defor-mations of the contour (without crossing C-points) canresult in the change of the number of turns by an integernumber [30, 42]. However, the number of turns modulo1/2, which distinguish the ‘M¨obius’ (half-integer num-ber of turns) and ‘non-M¨obius’ (integer number of turns)cases is topologically stable. It directly corresponds tothe number of C-points enclosed by the contour modulo2 [27], see Figs. 2(d,e,f).Recently, polarization M¨obius strips attracted greatattention in optics [27, 31–36]. We argue that entirelysimilar polarization structures naturally appear in inho-mogeneous sound-wave fields. In addition to the randomfield shown in Fig. 2, below we consider examples of reg-ular sound-wave fields with polarization singularities andM¨obius strip.
FIG. 3. The wavevectors k 𝑗 , distributions of the intensity of the velocity field, | V | , and of the phase of the quadratic field V · V in the 𝑧 = 0 plane, for the three-wave superpositions (3) with 𝑁 = 3, 𝜃 = 𝜋/ ℓ = 0 (a,b,c) and ℓ = 1 (d,e,f). B. Three-wave interference
We now consider examples of regular (non-random) 3Dacoustic fields with polarization singularities and M¨obiusstrips. In optics, such singularities are often generatedin vector vortex beams [17, 27, 31, 32, 35, 36]. Here wealso consider a superposition of 𝑁 acoustic plane waveswith wavevectors evenly distributed within a cone of po-lar angle 𝜃 = 𝜃 and with an azimuthal phase differencecorresponding to a vortex of order ℓ : V = 𝑉 𝑁 ∑︁ 𝑗 =1 ¯ k 𝑗 exp[ 𝑖 k 𝑗 · r + 𝑖 ℓ 𝜑 𝑗 ] , (3)where k 𝑗 = 𝑘 (sin 𝜃 cos 𝜑 𝑗 , sin 𝜃 sin 𝜑 𝑗 , cos 𝜃 ) and 𝜑 𝑗 =2 𝜋 ( 𝑗 − /𝑁 . In the limit of 𝑁 ≫
1, this superpositiontends to an acoustic Bessel beam [6].The minimal number of plane waves to generate po-larization singularities is 𝑁 = 3. Figure 3 shows thewavevectors k 𝑗 , distributions of the intensity of the ve-locity field, | V | , and of the phase of the quadratic field,Arg( V · V ), for the three-wave superpositions (3) with 𝜃 = 𝜋/ ℓ = 0 and ℓ = 1. One can see a number of first-order C-points, i.e., phase singularities in the quadraticfield V · V . Accordingly, 3D polarization ellipses alonga contour enclosing an odd number of C-points form po-larization M¨obius strips. Importantly, the spacing be-tween the C-points in Fig. 3 is controlled by the polarangle 𝜃 . When 𝜃 ≪ even-order C-points with no M¨obiusstrips around them. In particular, the four C-points atthe center of Fig. 3(f) with the integer total topologicalcharge 𝑛 𝐷 = 3 / − / 𝑛 𝐷 = 1) C-point in the paraxial limit. This isin sharp contrast to the electromagnetic (optical) waves,where isolated first-order C-points can appear even in theparaxial case. C. Vortex beams
Consider now the large- 𝑁 limit of the superposition(3), which generates acoustic vortex (Bessel) beams. Dueto the cylindrical symmetry, such beams can have an iso-lated C-point at the center. However, in contrast to opti-cal vectorial vortex beams, the C-point at the center of anacoustic vortex beam always has an even order (integer 𝑛 𝐷 = ℓ ) [6] (see Fig. 4). This does not allow one to gen-erate an acoustic polarization M¨obius strip in a symmet-ric vortex configuration as in optics [27, 31, 32, 35, 36].However, breaking the cylindrical symmetry of the beamresults in splitting of the even-order C-point at the centerinto a number of the first-order C-points ( 𝑛 𝐷 = ± / V = V vortex + 𝑉 ′ ¯ k ′ exp( 𝑖 k ′ · r ) , (4) FIG. 4. The wavevectors, distributions of the intensity of the velocity field, | V | , and of the phase of the quadratic field V · V in the 𝑧 = 0 plane, for the Bessel-beam superposition (3) with 𝑁 = 20, 𝜃 = 𝜋/ ℓ = 2 (a,b,c) and the same Bessel beaminterfering with an additional plane wave, Eq. (4), with 𝑉 ′ /𝑉 = 2 (d,e,f). One can see the splitting of the even-order C-point(integer 𝑛 𝐷 = ℓ ) (c) into 2 | 𝑛 𝐷 | first-order C-points (f) when breaking the cylindrical symmetry of the vortex beam. where V vortex is the vortex-beam field, such as Eq. (3)with 𝑁 ≫
1, whereas k ′ = 𝑘 (1 , , 𝑘 − ,and this spacing can decrease in the paraxial regime andin the presence of additional symmetries. This also de-termines the typical subwavelength size of the acousticpolarization M¨obius strips. IV. POLARIZATION SINGULARITIES ANDM ¨OBIUS STRIPS IN WATER-SURFACE WAVES
One of the key differences between electromagnetic andacoustic fields is that the electric and magnetic fields arevectors in abstract spaces of the field components (thereis no ‘ether’ and nothing moves in a free-space electro-magnetic wave), while the velocity field corresponds tothe motion of the medium particles (atoms or molecules)in real space. Moreover, instead of the velocity field, onecan consider the displacement field ℛ ( r , 𝑡 ): 𝒱 = 𝜕 𝑡 ℛ ,or, for a monochromatic field, V = − 𝑖 𝜔 R . The dis-placement field can be regarded as a ‘vector-potential’ for the velocity field [10], it has the same polarization, butnow the polarization ellipses traced by ℛ correspond to real-space trajectories of the medium particles.This opens an avenue to the direct observation of polar-ization ellipses and more complicated structures [12]. Insound waves, typical displacement amplitudes are smalland their direct observations are challenging [37, 38].However, similar medium displacements can be easily ob-served in another type of classical waves, namely, water-surface (e.g., gravity or capillary) waves [13], with typ-ical displacement scales ranging from milimeters to me-ters. Recently, there were several studies on polarizationproperties of structured water waves [11, 12, 39], andhere we show that these waves naturally reveal genericpolarization singularities.For the sake of simplicity, we consider deep-water grav-ity waves on the unperturbed water surface 𝑧 = 0 [13].The equations of motion for the complex displacementfield R = ( 𝑋, 𝑌, 𝑍 ) of the water-surface particles in amonochromatic wave field can be written in a form sim-ilar to the acoustic equations (1) [11, 12]: 𝜔 𝑍 = − 𝑔 ∇ ⊥ · R ⊥ , 𝜔 R ⊥ = 𝑔 ∇ ⊥ 𝑍 . (5)Here 𝑔 is the gravitational acceleration, R ⊥ = ( 𝑋, 𝑌 ),and ∇ ⊥ = ( 𝜕 𝑥 , 𝜕 𝑦 ). Making the plane-wave ansatz ∇ ⊥ → 𝑖 k , k = ( 𝑘 𝑥 , 𝑘 𝑦 ), in Eqs. (5), we obtain the dis-persion relation 𝜔 = 𝑔𝑘 . FIG. 5. Random 3D water-surface wave field obtained by the interference of 𝑁 = 7 plane waves with the same frequency andamplitude but with random directions in the ( 𝑥, 𝑦 ) plane and random phases. (a) Wavevectors of the interfering waves withcolor-coded phases and their circular polarizations. (b) Color-coded distribution of the phase of the quadratic field R · R in thewater-surface 𝑧 = 0 plane. Phase singularities (vortices) of this field correspond to C-points of purely circular polarization in thefield R . (c) Distribution of the normalized polarization ellipses of the R field in the 𝑧 = 0 plane. These ellipses are trajectoriesof water-surface particles. The instantaneous water surface at 𝑡 = 0 is shown in gray. (d) Akin to Fig. 2, the continuousevolution of polarization ellipses with their major semiaxes along a contour encircling a C-point exhibits a polarization M¨obiusstrip. The plane-wave solution of Eqs. (5) is: 𝑍 = 𝑍 exp( 𝑖 k · r ) , R ⊥ = 𝑖 𝑍 ¯ k exp( 𝑖 k · r ) . (6)These relations show that deep-water gravity waveshave equal longitudinal ( k -directed) and transverse ( 𝑧 -directed) displacement components phase-shifted by 𝜋/ circularly polarized in the meridional (prop-agation) plane including the wavevector and normalto the water surface [16]. Such (generically, ellipti-cal) meridional polarization is a common feature of sur-face and evanescent waves in different physical contexts[4, 5, 14, 15]. Therefore, interfering plane water waveswith wavevectors lying in the ( 𝑥, 𝑦 ) plane results ingeneric 3D polarization structures with all three com-ponents of the displacement field R .To show that such polarization distributions generi-cally posses polarization singularities, we consider theinterference of 𝑁 plane waves (6): R = ∑︀ 𝑁𝑗 =1 R 𝑗 ≡ ∑︀ 𝑁𝑗 =1 R 𝑗 exp( 𝑖 k 𝑗 · r ⊥ ), with R 𝑗 = 𝑍 𝑗 ( 𝑖 ¯ 𝑘 𝑗𝑥 , 𝑖 ¯ 𝑘 𝑗𝑦 , k 𝑗 randomly distributed over the circle 𝑘 𝑥 + 𝑘 𝑦 = 𝑘 , and equal amplitudes | 𝑍 𝑗 | but random phases 𝜑 𝑗 = Arg( 𝑍 𝑗 ), as shown in Fig. 5(a) (cf. Fig. 1). Fig-ure 5(b) shows the phase distribution of the quadraticfield Ψ = R · R ; it clearly exhibits phase singularitiescorresponding to C-points of the vector field R . Thedistribution of the polarization ellipses, i.e., trajectoriesof the water-surface particles, over the 𝑧 = 0 plane isshown in Fig. 5(c). Tracing orientation of the polariza-tion ellipses along a contour encircling a C-point revealsthe generic M¨obius-strip structure, Fig. 5(d). Animatedversions of Figs. 5(c,d) are given in the SupplementalMovies, where one can see motion of the water surfaceand separate water particles. In particular, the animatedversion of Fig. 5(d) shows the temporal evolution of thedisplacement vectors ℛ ( r , 𝑡 ) along the contour, whichcan form ‘twisted ribbon carousels’ [43].Thus, by tracing 3D trajectories of water particles in arandom (yet monochromatic) water-surface wavefield onecan directly observe generic polarization singularities of3D vector wavefields. V. CONCLUDING REMARKS
This work was motivated by recent strong interest in (i)polarization M¨obius strips in 3D polarized optical fields[27–36] and (ii) vectorial spin properties of acoustic andwater-surface waves [4–12]. We have shown that theseresearch directions can be naturally coupled, and thatpolarization singularities, such as C-points and polar-ization M¨obius strips, are ubiquitous for inhomogeneous(yet monochromatic) acoustic and water-surface waves.The vector velocity or displacement of the medium parti-cles provide complex-valued elliptical polarization fieldsvarying across the space. We have considered variousexamples of random and regular interference fields con-sisting of multiple (three or more) plane waves, whichexhibit polarization singularities and M¨obius strips.In contrast to well-studied electromagnetic polariza-tions associated with the motion of abstract field vectors,acoustic and water-wave polarizations correspond to real-space trajectories of the medium particles. In particu-lar, these are readily directly observable for water-surfacewaves [11, 39]. Also, while optical vectorial-vortex beamscan bear an isolated first-order C-point and a M¨obiusstrip around it [27, 31, 32, 35, 36], acoustic C-points typ-ically appear in clusters with subwavelength distance be-tween the points.Analyzing wave-field singularities is useful because oftheir topological robustness; they provide a ‘skeleton’ ofan inhomogeneous field [44]. So far, only phase singu-larities of the scalar pressure field 𝑃 were considered insound-wave fields. The vector velocity field V ∝ ∇ 𝑃 andits polarization singularities provide an alternative rep- resentation and can be more relevant, e.g., in problemsinvolving dipole wave-matter coupling. Note that thevectorial representation of a gradient of a scalar wave-field was previously considered in Ref. [45].For water-surface waves, the scalar representation isbased on the vertical displacement field 𝑍 . Tidal oceanwaves were also studied in terms of the 2D polarizationfield of the horizontal current [19–21, 46, 47] associatedwith the velocity components ( 𝑉 𝑥 , 𝑉 𝑦 ). We argue thatthese scalar and 2D vector fields can be regarded as com-ponents of a single 3D vector displacement R or velocity V field. Moreover, we have considered gravity deep-waterwaves, which are much more feasible for experimentallaboratory studies than tidal waves [11, 39].Notably, our arguments are not restricted to purelysound and water-surface waves. They can be equally ap-plied to any fluid/gas or fluid/fluid surface waves as wellas internal gravity waves in stratified fluid or gas me-dia. We hope that our work will stimulate further stud-ies and possibly applications of 3D polarization texturesand topological vectorial properties of various waves inacoustics and fluid mechanics. ACKNOWLEDGMENTS
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