Skip to main content Accesibility Help

New gravity–capillary waves at low speeds. Part 1. Linear geometries

  • Philippe H. Trinh (a1) (a2) and S. Jonathan Chapman (a2)

When traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity–capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity–capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem.

Corresponding author
Email address for correspondence:
Hide All
Akylas, T. R. & Grimshaw, R. H. J. 1992 Solitary internal waves with oscillatory tails. J. Fluid Mech. 242, 279298.
Beale, J. T. 1991 Exact solitary water waves with capillary ripples at infinity. Commun. Pure Appl. Maths 44, 211257.
Berry, M. V. 1989a Stokes’ phenomenon; smoothing a Victorian discontinuity. Publ. Math. Inst. Hautes Études Sci. 68, 211221.
Berry, M. V. 1989b Uniform asymptotic smoothing of Stokes discontinuities. Proc. R. Soc. Lond. A 422, 721.
Boyd, J. P. 1998 Weakly Non-local Solitary Waves and Beyond-all-orders Asymptotics. Kluwer.
Boyd, J. P. 1999 The Devil’s invention: asymptotics, superasymptotics and hyperasymptotics. Acta Appl. 56, 198.
Chapman, S. J., King, J. R. & Adams, K. L. 1998 Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. R. Soc. Lond. A 454, 27332755.
Chapman, S. J. & Mortimer, D. B. 2005 Exponential asymptotics and Stokes lines in a partial differential equation. Proc. R. Soc. Lond. A 461, 23852421.
Chapman, S. J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Maths 62 (6), 18721898.
Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299326.
Costin, O. 2008 Asymptotics and Borel Summability, vol. 141. Chapman & Hall/CRC.
Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2 (6), 532540.
Debnath, L. 1994 Nonlinear Water Waves. Academic.
Dias, F., Menasce, D. & Vanden-Broeck, J.-M. 1996 Numerical study of capillary-gravity solitary waves. Eur. J. Mech. (B/Fluids) 15 (1), 1736.
Dias, F. & Vanden-Broeck, J.-M. 1993 Nonlinear bow flows with spray. J. Fluid Mech. 255, 91102.
Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic.
Forbes, L. K. 1983 Free-surface flow over a semi-circular obstruction, including the influence of gravity and surface tension. J. Fluid Mech. 127, 283297.
Grandison, S. & Vanden-Broeck, J.-M. 2006 Truncation approximations for gravity-capillary free-surface flows. J. Engng Maths 54, 8997.
Grimshaw, R. 2010 Exponential asymptotics and generalized solitary waves. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H.), pp. 71120. Springer.
Hunter, J. K. & Scheurle, J. 1988 Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 32, 253268.
Hunter, J. K. & Vanden-Broeck, J.-M. 1983 Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 134, 205219.
King, A. C. & Bloor, M. I. G. 1987 Free-surface flow over a step. J. Fluid Mech. 182, 193208.
King, A. C. & Bloor, M. I. G. 1990 Free-surface flow of a stream obstructed by an arbitrary bed topography. Q. J. Mech. Appl. Maths 43, 87106.
Lamb, H. 1932 Hydrodynamics. Dover.
Lustri, C. J., McCue, S. W. & Binder, B. J. 2012 Free surface flow past topography: a beyond-all-orders approach. Eur. J. Appl. Maths 1 (1), 127.
Maleewong, M., Asavanant, J. & Grimshaw, R. 2005a Free surface flow under gravity and surface tension due to an applied pressure distribution: I Bond number greater than one-third. Theor. Comput. Fluid Dyn. 19 (4), 237252.
Maleewong, M., Asavanant, J. & Grimshaw, R. 2005b Free surface flow under gravity and surface tension due to an applied pressure distribution: II Bond number less than one-third. Eur. J. Mech. (B/Fluids) 24, 502521.
Mortimer, D. B. 2004 Exponential asymptotics and Stokes lines in partial differential equations. PhD thesis, University of Oxford.
Ogilvie, T. F. 1968 Wave resistance: the low speed limit. Tech. Rep. Michigan University, Ann Arbor.
Olde Daalhuis, A. B. 1999 On the computation of Stokes multipliers via hyperasymptotics. Resurgent functions and convolution integral equations. Surikaisekikenkyusho Kokyuroku 1088, 6878.
Olde Daalhuis, A. B., Chapman, S. J., King, J. R., Ockendon, J. R. & Tew, R. H. 1995 Stokes phenomenon and matched asymptotic expansions. SIAM J. Appl. Maths 55 (6), 14691483.
Pomeau, Y., Ramani, A. & Grammaticos, B. 1988 Structural stability of the Korteweg-de Vries solitons under a singular perturbation. Physica D 31, 127134.
Rayleigh, Lord 1883 The form of standing waves on the surface of running water. Proc. Lond. Math. Soc. 15, 6978.
Russell, J. S. 1844 Report on waves. In 14th Meeting of the British Association for the Advancement of Science, pp. 311–390, plates XLVII-LVII. John Murray.
Schooley, A. H. 1960 Double, triple, and higher-order dimples in the profiles of wind-generated water waves in the capillary-gravity transition region. J. Geophys. Res. 65, 40754079.
Stoker, J. J. 1957 Water Waves. Interscience.
Sun, S. M. 1991 Existence of a generalized solitary wave solution for water with positive bond number less than 1/3. J. Math. Anal. Appl. 156, 471504.
Thomson, W. (Baron Kelvin) 1871 Hydrokinetic solutions and observations. Phil. Mag. 42 (4), 374.
Trinh, P. H. 2010 Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H.), Exponential Asymptotics and Stokes Line Smoothing for Generalized Solitary Waves, pp. 121126. Springer.
Trinh, P. H. & Chapman, S. J. 2013 New gravity–capillary waves at low speeds. Part 2. Nonlinear theory. J. Fluid Mech. 724, 392424.
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413439.
Tuck, E. O. 1991 Ship-hydrodynamic free-surface problems without waves. J. Ship Res. 35 (4), 277287.
Vanden-Broeck, J.-M. 2002 Wilton ripples generated by a moving pressure distribution. J. Fluid Mech. 451, 193201.
Vanden-Broeck, J.-M. & Dias, F. 1992 Gravity-capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240, 549557.
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Annotated Edn. Parabolic.
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688700.
Xie, X. & Tanveer, S. 2003 Rigorous results in steady finger selection in viscous fingering. Arch. Rat. Mech. Anal. 166 (3), 219286.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed