Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-03T00:59:47.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Time domain modelling of a Helmholtz resonator analogue for water waves

Published online by Cambridge University Press:  10 June 2021

Leo-Paul Euvé ,
Kim Pham ,
Philippe Petitjeans ,
Vincent Pagneux  and
Agnès Maurel Open the ORCID record for Agnès Maurel [Opens in a new window]
Leo-Paul Euvé*
Affiliation:
PMMH, ESPCI, Sorbonne Université, Université PSL, 1 rue Jussieu, 75005Paris, France Bluerium, Av. L. Philibert, 13100Aix-en-Provence, France
Kim Pham
Affiliation:
IMSIA, CNRS, EDF, CEA, ENSTA Paris, Institut Polytechnique de Paris, 828 Bd des Maréchaux, 91732Palaiseau, France
Philippe Petitjeans
Affiliation:
PMMH, ESPCI, Sorbonne Université, Université PSL, 1 rue Jussieu, 75005Paris, France
Vincent Pagneux
Affiliation:
Lab. d'Acoustique de l'Université du Mans (LAUM), Av. O. Messiaen, 72085Le Mans
Agnès Maurel
Affiliation:
Institut Langevin, ESPCI Paris, Université PSL, CNRS, 1 rue Jussieu, 75005Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the context of water waves, we consider a resonator with deep subwavelength resonance, analogue to the Helmholtz resonator in acoustics. In the shallow water regime, using asymptotic analysis, a one-dimensional model is derived in which the effect of the resonator is reduced to effective transmission conditions. These conditions clearly highlight two contributions. The first is associated with the dock on its own and it is responsible for a jump of the potential at the free surface. The second is due to the resonant cavity and it is responsible for a jump in the horizontal velocity. It involves as well the uniform amplitude within the resonant cavity with a transient dynamics explicitly given by the equation of a damped oscillator forced by the incident waves. The one-dimensional model is validated in the harmonic regime by comparison to direct two-dimensional numerics. It is shown to reproduce accurately the scattering coefficients and the amplitude within the resonator; interestingly, this remains broadly true for finite water depths. We further inspect the spatio-temporal behaviour of different types of wave packets interacting with the resonating and radiating cavity.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A22
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Archer, A.J., Wolgamot, H.A., Orszaghova, J., Bennetts, L.G., Peter, M.A. & Craster, R.V. 2020 Experimental realization of broadband control of water-wave-energy amplification in chirped arrays. Phys. Rev. Fluids 5 (6), 062801.CrossRefGoogle Scholar
Bartholomeusz, E.F. 1958 The reflexion of long waves at a step. Proc. Camb. Phil. Soc. 54 (1), 106–118.CrossRefGoogle Scholar
Bennetts, L.G., Peter, M.A. & Craster, R.V. 2018 Graded resonator arrays for spatial frequency separation and amplification of water waves. J. Fluid Mech. 854, R4.CrossRefGoogle Scholar
Bennetts, L.G., Peter, M.A. & Craster, R.V. 2019 Low-frequency wave-energy amplification in graded two-dimensional resonator arrays. Phil. Trans. R. Soc. Lond. A 377 (2156), 20190104.Google ScholarPubMed
Bobinski, T., Maurel, A., Petitjeans, P. & Pagneux, V. 2018 Backscattering reduction for resonating obstacle in water-wave channel. J. Fluid Mech. 845, R4.CrossRefGoogle Scholar
Caflisch, R.E., Miksis, M.J., Papanicolaou, G.C. & Ting, L. 1985 Effective equations for wave propagation in bubbly liquids. J. Fluid Mech. 153, 259–273.CrossRefGoogle Scholar
D'Alembert, J.L.R. 1747 Memoires de l'Academie royale des sciences et belles lettres. Classe de mathematique. (Paris), vol. 3, 214–219.Google Scholar
Dupont, G., Guenneau, S., Kimmoun, O., Molin, B. & Enoch, S. 2016 Cloaking a vertical cylinder via homogenization in the mild-slope equation. J. Fluid Mech. 796, R1.CrossRefGoogle Scholar
Dupont, G., Remy, F., Kimmoun, O., Molin, B., Guenneau, S. & Enoch, S. 2017 Type of dike using $c$-shaped vertical cylinders. Phys. Rev. B 96 (18), 180302.CrossRefGoogle Scholar
Evans, D.V. 1975 A note on the total reflexion or transmission of surface waves in the presence of parallel obstacles. J. Fluid Mech. 67 (3), 465–472.CrossRefGoogle Scholar
Evans, D.V. 1978 The oscillating water column wave-energy device. IMA J. Appl. Maths 22 (4), 423–433.CrossRefGoogle Scholar
Evans, D.V. & Morris, C.A.N. 1972 Complementary approximations to the solution of a problem in water waves. IMA J. Appl. Maths 10 (1), 1–9.CrossRefGoogle Scholar
Farhat, M., Guenneau, S., Alù, A. & Wu, Y. 2020 Scattering cancellation technique for acoustic spinning objects. Phys. Rev. B 101 (17), 174111.CrossRefGoogle Scholar
Helmholtz, H. & Ellis, A.J. 1954 On the Sensations of Tone as a Physiological Basis for the Theory of Music. Dover.Google Scholar
Iida, T. & Kashiwagi, M. 2018 Small water channel network for designing wave fields in shallow water. J. Fluid Mech. 849, 90–110.CrossRefGoogle Scholar
Ingard, U. 1953 On the theory and design of acoustic resonators. J. Acoust. Soc. Am. 25 (6), 1037–1061.CrossRefGoogle Scholar
Isaacs, J.D. & Wiegel, R.L. 1949 The measurement of wave heights by means of a float in an open-end pipe. EOS Trans. AGU 30 (4), 501–506.CrossRefGoogle Scholar
Jiménez, N., Romero-García, V., Pagneux, V. & Groby, J.-P. 2017 Rainbow-trapping absorbers: broadband, perfect and asymmetric sound absorption by subwavelength panels for transmission problems. Sci. Rep. 7 (1), 13595.CrossRefGoogle ScholarPubMed
Linton, C.M. 2011 Water waves over arrays of horizontal cylinders: band gaps and Bragg resonance. J. Fluid Mech. 670, 504–526.CrossRefGoogle Scholar
Linton, C.M. & Evans, D.V. 1990 Compressed air breakwater. In Proceedings of the 5th International Workshop on Water Waves and Floating Bodies, Manchester, UK, pp. 109–113.Google Scholar
Makwana, M.P., Laforge, N., Craster, R.V., Dupont, G., Guenneau, S., Laude, V. & Kadic, M. 2020 Experimental observations of topologically guided water waves within non-hexagonal structures. Appl. Phys. Lett. 116 (13), 131603.CrossRefGoogle Scholar
Maurel, A., Marigo, J.-J., Cobelli, P., Petitjeans, P. & Pagneux, V. 2017 Revisiting the anisotropy of metamaterials for water waves. Phys. Rev. B 96 (13), 134310.CrossRefGoogle Scholar
Maurel, A., Pham, K. & Marigo, J.-J. 2019 Scattering of gravity waves by a periodically structured ridge of finite extent. J. Fluid Mech. 871, 350–376.CrossRefGoogle Scholar
Mei, C.C., Stiassnie, M. & Yue, D. 1989 Theory and Applications of Ocean Surface Waves: Part 1: Linear Aspects Part 2: Nonlinear Aspects. World Scientific.Google Scholar
Mei, C.C., Stiassnie, M. & Yue, D.K.-P. 2005 Theory and Applications of Ocean Surface Waves: Nonlinear Aspects, vol. 2. World Scientific.Google Scholar
Mercier, J.-F., Marigo, J.-J. & Maurel, A. 2017 Influence of the neck shape for Helmholtz resonators. J. Acoust. Soc. Am. 142 (6), 3703–3714.CrossRefGoogle ScholarPubMed
Molin, B. 2001 On the piston and sloshing modes in moonpools. J. Fluid Mech. 430, 27–50.CrossRefGoogle Scholar
Molin, B., Zhang, X., Huang, H. & Remy, F. 2018 On natural modes in moonpools and gaps in finite depth. J. Fluid Mech. 840, 530–554.CrossRefGoogle Scholar
Monsalve, E., Maurel, A., Petitjeans, P. & Pagneux, V. 2019 Perfect absorption of water waves by linear or nonlinear critical coupling. Appl. Phys. Lett. 114 (1), 013901.CrossRefGoogle Scholar
Newman, J.N. 1974 Interaction of water waves with two closely spaced vertical obstacles. J. Fluid Mech. 66 (1), 97–106.CrossRefGoogle Scholar
Newman, J.N. 2014 Cloaking a circular cylinder in water waves. Eur. J. Mech. (B/Fluids) 47, 145–150.CrossRefGoogle Scholar
Parsons, N.F. & Martin, P.A. 1994 Scattering of water waves by submerged curved plates and by surface-piercing flat plates. Appl. Ocean Res. 16 (3), 129–139.CrossRefGoogle Scholar
Pham, K., Mercier, J.-F., Fuster, D., Marigo, J.-J. & Maurel, A. 2020 Scattering of acoustic waves by a nonlinear resonant bubbly screen. J. Fluid Mech. 906, A19.CrossRefGoogle Scholar
Porter, R. 2018 Cloaking in water waves. In Handbook of Metamaterials and Plasmonics (ed. S.A. Maier), vol. 2. World Scientific.Google Scholar
Porter, R. 2019 An extended linear shallow-water equation. J. Fluid Mech. 876, 413–427.CrossRefGoogle Scholar
Porter, R. & Porter, D. 2003 Scattered and free waves over periodic beds. J. Fluid Mech. 483, 129–163.CrossRefGoogle Scholar
Ravinthrakumar, S., Kristiansen, T., Molin, B. & Ommani, B. 2019 A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess. J. Fluid Mech. 877, 142–166.CrossRefGoogle Scholar
Romero-García, V., Jimenez, N., Theocharis, G., Achilleos, V., Merkel, A., Richoux, O., Tournat, V., Groby, J.-P. & Pagneux, V. 2020 Design of acoustic metamaterials made of Helmholtz resonators for perfect absorption by using the complex frequency plane. C. R. Phys. 21 (7–8), 713–749.Google Scholar
Tuck, E.O. 1975 Matching problems involving flow through small holes. In Advances in Applied Mechanics (ed. C.-S. Yih), vol. 15, pp. 89–158. Elsevier.CrossRefGoogle Scholar
Tuck, E.O. 1977 Some classical water-wave problems in varying depth. In Waves on Water of Variable Depth (ed. D.G. Provis & R. Radok), vol. 64, pp. 9–20. Springer.CrossRefGoogle Scholar
Zhao, W., Taylor, P.H., Wolgamot, H.A., Molin, B. & Taylor, R.E. 2020 Group dynamics and wave resonances in a narrow gap: modes and reduced group velocity. J. Fluid Mech. 883, A22.CrossRefGoogle Scholar