Time-Reversed Water Waves Generated from an Instantaneous Time Mirror
Danming Peng, Yiyang Fan, Ruochen Liu, Xiasheng Guo, Sihui Wang
TTime-Reversed Water Waves Generated from an InstantaneousTime Mirror
Danming Peng, ∗ Yiyang Fan, and Ruochen Liu
Nanjing Foreign Language School, Nanjing 210008, China
Xiasheng Guo and Sihui Wang
School of Physics, Collaborative Innovation Center of Advanced Microstructure,Nanjing University, Nanjing 210093, China (Dated: August 19, 2020)
Abstract
An instantaneous time mirror (ITM) is an interesting approach to manipulate wave propagationfrom the time boundaries. In the time domain, the reversed wave is previously proven to bethe temporal derivative of the original pattern. Here, we further investigate into the relationshipbetween the wave patterns in the spatial domain both theoretically and experimentally. Therefraction of a square array of laser beams is used to determine the three-dimensional (3D) shapeof the water surface. The experimental results verify the theoretical prediction that the reversedpattern is related to the Laplacian of the initial wave field. Based on these findings, the behaviorsof the ITM activated in an inhomogeneous medium are discussed, and the phenomenon of totalenergy change is explained. ∗ [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] A ug . INTRODUCTION Devising different types of “time mirrors” for classical waves have attracted many inter-ests. In a traditional record-and-play back scheme, a wave field radiated from a source isusually detected by a group of antennas positioned in the far field, and then time-reversedand rebroadcasted by the same antenna array [1]. This kind of time-reversal mirror can con-vert divergent waves into convergent waves and focus back at the source, and is extensivelystudied for acoustic [2–5], electromagnetic [6], elastic [7], and water waves [8]. The re-transmitted waves have exactly the same profiles as the original ones, and the wave patternspropagate back as if the time is reversed. It is even possible that strongly nonlinear localizedwaves can be refocused in both time and space. This time, the reversed field is related tothe modulational instability [1]. Restoring the backward propagating wave pattern requiressufficient spatial sampling of the original field, and a large number of receiver/emitters isnecessary. Although it is possible to make use of the reverberating effect [8] and reduce thechannel number required for field reconstruction, time-reversal mirrors are still difficult toimplement in certain fields such as optic ones [9].It is also possible to achieve time reversal of through a sudden change in the system.Recently, Bacot et al. proposed an instantaneous time mirror (ITM) [10], which exerts atime-modulated perturbation on wave fields. Kadri then pointed out that, an analogy totime-reversal can be obtained using nonlinear acoustic-gravity wave theory [11]. An ITMis much easier to achieve than traditional time-reversal mirrors. For example, in a gravity-capillary wave field, a sudden change in the water wave speed (or gravity) can be easilygenerated by accelerating the water tank. Inspired by this new approach, many applicationshave emerged in different fields, e.g., spatiotemporal light control [12], signal filtration [13],full-duplex communication [14], wave scattering [15, 16], temporal control of graphene plas-mons [17], focusing beyond the diffraction limit [18, 19], and negative refraction [20, 21]among others [22, 23]. Recently, the ITM protocol is also introduced into quantum systems,generating wave function echoes with high fidelities [24].Bacot et al. pointed out that, in theory the reversed pattern due to an ITM does nothave a fully equivalent profile as the original wave, and they are related through a timederivative operator [10]. In this work, we push their work a step further. The reversedand the initial patterns are theoretically predicted to be Laplacian to each other in the2patial domain, which is then verified through experiment. In the experiment, a laser beamarray is adopted to work out the three-dimensional (3D) morphology of the gravity-capillarywave. The behaviors of the reversed waves such as their performances in an inhomogeneousmedium [25] is also discussed. We demonstrate that the ITM is effective in a gravity-capillarywave system where the wave velocity is uneven in space. Finally, the phenomenon of totalenergy change in the system is discussed.
II. TIME-REVERSED WATER WAVES DUE TO AN ITMA. The temporal relationship
Here we first summarize the theory of Bacot et al. [10] To realize an ITM, a mediumwhose wave velocity is easy to control is necessary. Water proves to be a good candidatesince water waves can be very dispersive. Specifically, the wave speed c = (cid:112) g/h in shallowwater [26] shows a dependence on the water depth h and the gravitational acceleration g .The wave velocity can then be easily changed by accelerating the water container, which inturn changes the effective gravitational acceleration inside the water domain. Meanwhile, awater wave pattern travels at a relatively low velocity and can be observed with naked eyes.Due to these reasons, an ITM can be created for the gravity-capillary wave by shaking awater tank in a short period of time.During the perturbation, the wave velocity increases from an initial c to c (cid:48) and quicklyreduces back to c , forming an impulse in the speed-time diagram. This process can beseparated into three stages, including a short high-speed period corresponding to the per-turbation and two long low-speed periods before and after that. The time boundaries joiningthese three stages can be characterized as two temporal interfaces, where the wave velocitychanges abruptly. Just like waves reflect and refract at a spatial interface, they also undergoreflection and refraction at these temporal interfaces. According to the d’Alembert waveequation and the continuity condition, the reflective coefficient R = ( c (cid:48) − c ) / c (cid:48) and therefractive coefficient T = ( c (cid:48) + c ) / c (cid:48) should apply at the first interface [10]. Likewise, thetemporal reflective coefficient R (cid:48) = ( c − c (cid:48) ) / c and the refractive coefficient T (cid:48) = ( c + c (cid:48) ) / c are valid when the wave velocity changes from c (cid:48) back to c at the second interface [10]. Forexample, after passing the first temporal interface at t = 0, the wave field changes from3 ( r , t ) to Rφ ( r , − t ) + T φ ( r , t ) due to temporal reflection and refraction, with r being theposition vector of a spatial point, while the wave vector remains unchanged.As there exists two temporal interfaces, the wave undergoes two temporal reflections andtwo refractions across each perturbation period. Consequently, the final reversed waveformis the superposition of several components. For convenience, set the first interface at t = 0and the second at t = τ . The wave evolves as φ ( r , t ) → Rφ ( r , − t ) + T φ ( r , t ) → RR (cid:48) φ ( r , t − τ ) + RT (cid:48) φ ( r , − t ) + T R (cid:48) φ ( r , τ − t ) + T T (cid:48) φ ( r , t ) , where each arrow represents the change of the waveform at a temporal interface. For sim-plicity, let the perturbation period τ be short enough such that the peak in the speed-timediagram can be approximated as an impulse described by a Dirac function. The reversedwave field ψ can then be written as [10], ψ ( r , t ) = α ∂φ∂t ( r , − t ) , (1)in which α is a coefficient indicating the amplitude of the perturbation. The minus signbefore t on the right side suggests a backward propagation, that the time is “reversed”.Therefore, the reversed wave is the time derivative of the original wave. There should existtwo other packets that propagate forwards in the wave field, which are − α∂φ ( r , t ) /∂t and φ ( r , t ), but they are unable to refocus back to the initial pattern. B. The spatial relationship
When the perturbation period is infinitesimally short, the reversed wave becomes thetime derivative of the original one. However, Eq. 1 actually describes the time domainrelationship between the waveforms ψ ( r , t ) and φ ( r , t ). Since wave shapes, especially thoseof water waves, are naturally mapped to the spatial domain when being observed or recorded,it is essential that the relationship between the shapes of the original and the reversed wavepatterns be further interrogated.As the experiment begins at t = − T , it is assumed that a wave pattern φ ( r , − T ) starts toemerge from the flat water surface, giving ∂φ ( r , t ) /∂t | t = − T = 0. Within a short period ∆ t ,the field becomes φ ( r , − T +∆ t ) and the wave pattern propagates outwards. After the pertur-bation at t = 0, the reversed wave ψ ( r , t ) emerges, and is proportional to the time derivative4f φ ( r , − t ). At the end of the experiment when t = T , ψ ( r , T ) = ∂φ ( r , t ) /∂t | t = − T = 0 isexpected. At this moment, the reversal water wave field completely subsides and becomesflat again provided that there are no boundary reflections. The final reversed pattern shouldappear at t = T − ∆ t as ψ ( r , T − ∆ t ). Integrating the d’Alembert wave equation yields, ∂φ ( r, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = − T +∆ t = ∂φ ( r, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = − T + c ∆ t ∇ φ ( r, − T ) , (2)where ∆ t is again a short period. According to Eq. 1, ψ ( r , T − ∆ t ) is proportional to the timederivative of φ ( r , − T + ∆ t ). By further considering ∂φ ( r , t ) /∂t | t = − T = 0 that is obtainedearlier, one can conclude that, ψ ( r, T − ∆ t ) = α ∂φ ( r, − T + ∆ t ) ∂t = αc ∆ t ∇ φ ( r, − T ) . (3)Since ∆ t is short, a Laplacian of the initial pattern should be a good measure of the finalreversed pattern ψ ( r , T − ∆ t ). III. EXPERIMENTAL VALIDATIONSA. The experimental protocol
An ITM is set up in the experiment. The 3D morphology of the water wave field ismeasured by a laser array device during the whole process. To verify the results of thetheoretical analysis, the wave profiles at the beginning and at the end of the experiment arereconstructed for comparison.As shown in FIGs. 1(a) and 1(b), a cuboid water tank (20 × ×
10 cm ) is supported byfour wooden planks, which are fixed at the corners of a vibrating table with strong double-sided tapes. Water drops are dripped into the tank from 10-cm above the water surfaceto produce ripples that spread out. At a certain moment, the vibrating table is shakenat an amplitude of − .
25 cm within a 100-ms period, allowing the effective gravitationalacceleration change from 10 to 24.1 m/s and then back to 10 m/s , which immediatelyresults in a wave that propagates backwards and refocuses at the source location. Thissudden perturbation serves as an instantaneous mirror and produces the time-reversed wavepattern. At the same time, the whole process is recorded with an iPhone X working in theslow-motion mode. 5hroughout the experiment, the 3D shape of the water wave field is determined throughdisplacement measurements. A 635-nm, 100-mW laser source emits light that passes through400 (20 ×
20) circular apertures on an aluminum baffle, forming an array of paralleled laserbeams going upwards, see FIG. 1(c). To prevent the laser source itself from vibrating, it isisolated from other parts of the system. The square array of beams penetrates the tank andcan be refracted at the water surface.
BaffleLensLaserVibratingTable
Laser
Source PlanksScreen Water
Tank (a) (b) (c) (d)
FIG. 1. The experimental setup. (a) Illustration of the front view. (b) A photo of the system. (c)The laser source. (d) The array of laser points.
In the absence of water waves, the laser beams can produce 400 red points evenly dis-tributed in a square on a light screen (which is a piece of A4-size paper), see FIG. 1(d).As the wave propagates, the water surface becomes wavy, deflecting the laser beams. Dueto the varying slope of the water surface, the red points on the light screen keep movingaround their origins. Therefore, by measuring the offsets of the points, local slopes of thewater surface can be determined according to the Snell’s Law. By integrating the slope as afunction of time, the global morphology of the water surface is obtained. In each experiment,a GNU software Tracker helps to extract the in-plane displacements of the 400 laser points.The shapes of the water wave field are in turn figured out at several moments before andafter the time mirror is activated. Using this method, an effective area of 4 × of thewater surface can be measured in a single experiment.6 . Characteristics of the refocused waves FIGURE. 2 shows several snapshots at equally distributed time intervals during thefocusing process of the reversed wave, including the recorded laser point arrays, the recon-structed 3D water surface, and the numerical simulation results based on the d’Alembertwave equation where the wave speed c is a piecewise function described as c ( t ) = αc / τ , − τ /2 < t < τ /2 c , otherwise . (4)and the parameters α = 0 .
25 s and τ = 100 ms are selected. Time x y xy xy xy (a)
20 3010 0 402010 30 0
20 3010 0 402010 30 0 4020 3010 0 402010 30 0 40
10 0 (b)
10 0 402010 30 0 4020
10 0 402010 30 0 4020 3010 0 40
30 0 4020 3010 0 402010 30 (c)
FIG. 2. Focusing of the reversed wave. (a) The recorded red points on the light screen. (b) Thereconstructed 3D morphologies of the water surface. (c) Simulated results of the time-reversalexperiment. The four columns correspond to t =0.344, 0.352, 0.360, and 0.369 s, while the initialpattern starts at t = − As the water drop contacts the surface at t = − .
510 s, behaviors of the reversal wavesat t =0.344, 0.352, 0.360, and 0.369 s are presented in columns 1-4 in FIG. 2, respectively.7n FIG. 2(a), the dense circle of points indicates a ripple, which propagates inwards withtime (from left to right), focusing towards the source. From the displacements of the redpoints, the 3D morphologies of the water surface are reconstructed and given in FIG. 2(b).It is clear to observe that, initially the center of the ripple is not disturbed by waves. As therefocused wave propagates inwards, a wave packet is raised at the center. In order to validatethe reconstructed patterns, the same refocusing process is simulated using a commercialsoftware Mathematica (v12.0, Wolfram). In the simulations, a standard Gaussian wavepacket is used to model the pattern generated by a wave droplet. Corresponding results inFIG. 2(c) show good consistency with the experimentally reconstructed patterns. Especially,in the last snapshots of FIGs. 2(b) and 2(c), the water wave automatically restores itself tothe original form of a water droplet. FIG. 3. Verification of the Laplacian relationship. The reconstructed 3D shapes of the (a) initialpattern at t = − .
510 s and (b) the final reversed wave at t = 0 .
505 s. The 1D plots along x = y of (c) the initial wave and (d) the final reversed wave and the Laplacian of the initial wave. The wave behaviors at the very beginning of the spreading process and the very end of therefocusing process deserve attention, since Eq. 3 actually relates wave patterns at these two8oments. To compare them explicitly, the corresponding reconstructed 3D morphologiesare first given in FIGs. 3(a) and 3(b), respectively. It is obvious that the ripple actually doesnot exist in the initial Gaussian wave at t = − .
510 s, but is observed in the final reversedpattern at t = 0 .
505 s. It is mentioned that, the final reversed pattern is observed slightlybefore t = T = 0 .
510 s, just as we predicted theoretically in Eq.3. One dimensional (1D)profiles along x = y are then plotted in FIGs. 3(c) and 3(d) for comparison, where the twowaveforms show distinct differences. To verify Eq. 3, the Laplacian of the initial wave profileis also presented in FIG. 3(d). Despite a small difference, the two wave profiles in FIG. 3(d)coincide well with each other, verifying the Laplacian relationship that we obtained. Thedifference might be ascribed to the reflections of the − α∂φ ( r , t ) /∂t and φ ( r , t ) packets fromthe tank boundaries. In fact, after t = T = 0 .
510 s, we were only able to observe complicatedgrid-like patterns, which are obviously due to the reflected sequences.
IV. DISCUSSIONSA. The medium inhomogeneity
The results in the above only reveals the situation in a homogeneous medium. It isdemonstrated by Fink that receiver-based time reversal can also be realized in a complexmedium [25], which is very useful for wave focusing and imaging in the presence of mediumscattering. For water waves studied here, the problem of medium inhomogeneity is alsoimportant because the wave velocity is dependent on the water depth. Specifically, if thewater tank has an uneven bottom, or is slightly tilted, the wave velocity should vary fromplace to place, and the pattern does not spread out at a unified speed. In this case, thed’Alembert wave equation becomes ∂ φ∂t = c ( r ) ∇ φ, (5)which is still linear with respect to t and satisfies time reversal symmetry. Therefore, φ ( r , − t )should be a valid solution to this wave equation. By taking a partial time derivative of theequation, one obtains ∂ ∂ ( − t ) (cid:18) − α ∂φ∂t (cid:19) = c ( r ) ∇ (cid:18) − α ∂φ∂t (cid:19) , (6)9ince the continuity condition at t = 0 is valid whether or not the wave velocity is homoge-neous, ψ ( r , t ) = α∂φ ( r , − t ) /∂t always satisfies Eq. 6 at any time after the perturbation. Inother words, no matter how the wave velocity is distributed, the instantaneous time mirroralways works, while the reversed wave profile is the time derivative of the original in thetime domain and the Laplacian relationship also holds in the spatial domain. t = −19 ms t = −10 ms t = −1 ms t = 19 ms t = 10 ms t = 1 ms TimeTime t = 0ITM ON FIG. 4. Simulated time-reversal of water waves in an inhomogeneous medium. First row: thenormal wave propagation; second row: the reversed wave propagation. Time runs clockwise in thefigure, so that the selected graphs in the first and the second rows are in one-to-one correspondence.
This conclusion is demonstrated through a numerical simulation with the results given inFIG. 4. Here, the wave velocity c is designed as c = | θ | / π + 0 .
3, where θ is the azimuthalangle. As a result, the wave spreads faster on the left side (the − x direction) than onthe right side. By setting the initial wave pattern as a symmetric flower, the unevenlydistributed c gradually breaks the symmetric pattern - the left half of the flower gets largerthan the right half. After the ITM is activated at t = 0, a reversed wave emerges and beginsto propagate backwards, finally restoring the flower pattern to its initial symmetric shape.Therefore, the ITM is effective in an inhomogeneous medium.10 . Change of the energy E / E Wave velocity (m/s) E / E Wave packet width (m)(a) (b)
FIG. 5. Parameter study on energy change. (a) The relative growth ratio of the kinetic energy∆
E/E is proportional to the square of wave velocity. ( α = 0 . σ = 0 .
1) (b) ∆
E/E isinversely proportional to the square of wave packet width. ( α = 0 . c = 1) Activation of the ITM is known to change the total energy [10], which makes the temporalreflection and refraction distinctly different from the spatial ones. For example, in spatialreflection and refraction, the total energy is conserved when the wave impinges onto aboundary. But here, energy of the gravity-capillary wave field is inconsistent before andafter the ITM works. It should be of interest to discuss why this happens and how differentparameters influence the total energy change. Firstly, the potential energy is conservedduring the perturbation, since the shape of the water wave field remains unchanged at theperturbation moment. The kinetic energy, however, is changed due to the contribution of thework done by the vibrating table. For simplicity, the wave can be considered as a collectionof harmonic oscillators, so the kinetic energy density is proportional to ( ∂φ/∂t ) . From thecontinuity condition, the difference of the total energy before and after the perturbationcan be obtained by integrating the kinetic energy density over the entire field. The ratiobetween the kinetic energy increment ∆ E and the initial kinetic energy E is∆ EE ∝ α c σ , (7)in which σ represents the width of the wave packet. From the results shown in FIGs. 5(a)and 5(b), it is quite unexpected that the ITM requires more energy input when the waveformgets narrower. This is actually true due to the fact that, if σ approaches infinity, the water11urface becomes completely flat and a perturbation cannot change the shape of the wavefield, and no energy could be transferred to the system. In a spatial reflection, the energyremains unchanged but the momentum changes. In a temporal reflection, the momentumremains unchanged but the energy changes, which reveals a beautiful symmetry in physics. V. CONCLUSION
Based on the recently developed ITM, behaviors of the time-reversed water wave arestudied. Through theoretical analysis, we find that the reversed wave is proportional tothe Laplacian of the original wave. Experiments are carried out to verify this prediction,where a quantitative measurement of the water wave field is achieved through detecting therefraction of a square array of laser beams. This displacement measurement method enablesvisualization of the 3D morphology of the water surface as the time goes by. The ITM effectis discussed and demonstrated in an inhomogeneous medium, i.e., where the wave velocityis unevenly distributed. Therefore, time-reversal of waves, especially that achieved throughITMs, can potentially be applied in more generalized situations. For better understandingof the findings obtained here, parameter studies are carried out to examine the change inthe wave energy, which involves the influence of the perturbation intensity, the wave velocityand the width of the initial wave packet. The results here can be helpful for applicationssuch as seismic wave detection and radio communication among others.
ACKNOWLEDGMENTS
D. P., Y. F, and R. L. express their gratitude to Mr. Zhiming Pan for his patient guidance.They are grateful to the S. T. Yau High School Science Award committee for offering theminvaluable comments, and to Prof. Yi Cao, Meta Test Corp and Prof. Qiang Chen for theirassistance on experimental equipment. They dedicate their first paper to their parents fortheir endless love and strong support. [1] A. Chabchoub and M. Fink, Time-reversal generation of rogue waves, Phys. Rev. Lett. ,124101 (2014).
2] M. Fink, Time reversal of ultrasonic fields. i. basic principles, IEEE Trans. Ultrason. Ferr.Freq. Control , 555 (1992).[3] M. Fink, D. Cassereau, A. Derode, C. Prada, P. Roux, M. Tanter, J.-L. Thomas, andW. Fran¸cois, Time-reversed acoustics, Rep. Prog. Phys , 1933 (2000).[4] M. Fink, G. Montaldo, and M. Tanter, Time reversal acoustics, in IEEE Ultrasonics Sympo-sium, 2004 (2004) pp. 850–859.[5] M. Fink, G. Montaldo, and M. Tanter, Time-reversal acoustics in biomedical engineering,Ann. Rev. Biomed. Eng. , 465 (2003).[6] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, Time reversal ofelectromagnetic waves, Phys. Rev. Lett. , 193904 (2004).[7] C. Draeger and M. Fink, One-channel time reversal of elastic waves in a chaotic 2D-siliconcavity, Phys. Rev. Lett. , 407 (1997).[8] A. Przadka, S. Feat, P. Petitjeans, V. Pagneux, A. Maurel, and M. Fink, Time reversal ofwater waves, Phys. Rev. Lett. , 064501 (2012).[9] J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Control of light trans-mission through opaque scattering media in space and time, Phys. Rev. Lett. , 103901(2011).[10] V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, Time reversal and holography withspacetime transformations, Nat. Phys. , 972 (2016).[11] U. Kadri, Time-reversal analogy by nonlinear acoustic-gravity wave triad resonance, Fluids ,91 (2019).[12] A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, Spatiotemporal light control withactive metasurfaces, Science , eeat3100 (2019).[13] T. T. Koutserimpas and R. Fleury, Electromagnetic waves in a time periodic medium withstep-varying refractive index, IEEE Trans. Antennas Propag. , 5300 (2018).[14] S. Taravati and G. V. Eleftheriades, Generalized space-time-periodic diffraction gratings: The-ory and applications, Phys. Rev. Applied , 204026 (2019).[15] Z.-L. Deck-L´eger, A. Akbarzadeh, and C. Caloz, Wave deflection and shifted refocusing in amedium modulated by a superluminal rectangular pulse, Phys. Rev. B , 104305 (2018).[16] Z.-L. Deck-L´eger, N. Chamanara, M. Skorobogatiy, M. G. Silveirinha, and C. Caloz, Uniform-velocity spacetime crystals, Advanced Photonics , 056002 (2019).
17] J. Wilson, F. Santosa, M. Min, and T. Low, Temporal control of graphene plasmons, Phys.Rev. B , 081411 (2018).[18] J. de Rosny and M. Fink, Overcoming the diffraction limit in wave physics using a time-reversalmirror and a novel acoustic sink, Phys. Rev. Lett. , 124301 (2002).[19] G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, Focusing beyond the diffraction limit withfar-field time reversal, Science , 1120 (2007).[20] J. B. Pendry, Time reversal and negative refraction, Science , 71 (2008).[21] V. Bruno, C. DeVault, S. Vezzoli, Z. Kudyshev, T. Huq, S. Mignuzzi, A. Jacassi, S. Saha,Y. D. Shah, S. A. Maier, D. R. S. Cumming, A. Boltasseva, M. Ferrera, M. Clerici, D. Fac-cio, R. Sapienza, and V. M. Shalaev, Negative refraction in time-varying strongly coupledplasmonic-antenna-epsilon-near-zero systems, Phys. Rev. Lett. , 043902 (2020).[22] G. Ducrozet, M. Fink, and A. Chabchoub, Time-reversal of nonlinear waves: Applicabilityand limitations, Phys. Rev. Fluids , 054302 (2016).[23] P. Reck, C. Gorini, and K. Richter, Quantum time mirrors for general two-band systems,Phys. Rev. B , 125421 (2018).[24] P. Reck, C. Gorini, A. Goussev, V. Krueckl, M. Fink, and K. Richter, Dirac quantum timemirror, Phys. Rev. B , 165421 (2017).[25] A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Controlling waves in space and time forimaging and focusing in complex media, Nat. Photonics , 283 (2012).[26] L. D. Landau and E. M. Lifshits, Course of Theoretical Physics: Fluid Mechanics (Butterworth-Heinemann, 1987).(Butterworth-Heinemann, 1987).