Unified analytical solution for group-induced infragravity waves based on Green’s function

Zhiling Liao, Qingping Zou, Ye Liu, Stephanie Contardo, Shaowu Li

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Short-wave group forcing is a major driving mechanism of infragravity waves. The subharmonic response to wave group forcing approaches resonance in shallow water where the group velocity is equal to the shallow-water wave-propagating speed. Currently, there is a lack of understanding of the connection between the free and bound components of group-induced infragravity waves and the consistency among existing solutions for off- and near-resonance conditions in intermediate and shallow water. Here, a unified solution of group-induced subharmonics is derived based on Green's function for the first time. The new solution is valid for any resonance intensity and is able to describe group-induced subharmonic behaviour at all water depths consistently from a new angle. The proposed solution reduces to existing solutions for intermediate depth (Longuet-Higgins & Stewart, J. Fluid Mech., vol. 13, 1962, pp. 481–504; Zou, Phys. Oceanogr., vol. 41, 2011, pp. 1842–1859), shallow water and/or over a plane sloping beach (Van Leeuwen, PhD thesis, TU Delft, 1992; Schäffer, J. Fluid Mech., vol. 247, 1993, pp. 551–588; Janssen et al., J. Geophys. Res., vol. 108, 2003, p. 3252; Contardo et al., J. Phys. Oceanogr., vol. 51, 2021, pp. 1465–1487; Liao et al., J. Phys. Oceanogr., vol. 51, 2021, pp. 2749–2765). Unlike previous solutions, the Green's function-based solution describes all subharmonics as free subharmonics continuously radiated away from each point source in the group-induced forcing field determined by wave radiation stress gradients. The superposition of all these free subharmonics yields so-called bound subharmonics by previous studies due to group-modulated emission of each free subharmonic through the source field bound to the wave group. Therefore, this solution provides theoretical evidence that the group-induced subharmonic at any observation point is dependent on the surrounding radiation stress field and topography. Under full-resonance conditions in shallow water, downwave-propagating subharmonics excited at all source locations interfere with each other constructively; therefore, their superposed amplitude is proportional to the travel distance of wave groups. Combined with the conventional moving-breakpoint forcing model, the predicted amplitude of the subharmonic in the surf zone by the present solution is in good agreement with laboratory observations.
Original languageEnglish
Article numberA37
JournalJournal of Fluid Mechanics
Early online date24 Jul 2023
Publication statusPublished - 25 Jul 2023


  • surface gravity waves

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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