[Paper Review] Deep-water gravity waves: nonlinear theory of wave groups

I. M. Mindlin|arXiv (Cornell University)|Jun 6, 2014

Ocean Waves and Remote Sensing4 references3 citations

TL;DR

This paper presents a nonlinear theory for deep-water gravity wave groups by transforming the free surface into a half-circle, enabling a discrete trigonometric series solution. It derives a complete set of wave packets indexed by integers, each evolving independently with distinct dispersion and propagation speeds, and shows agreement with linear theory in the small-amplitude limit and with nonlinear Schrödinger solutions in asymptotic behavior, while preserving finite energy and avoiding soliton assumptions.

ABSTRACT

Nonlinear initial-boundary value problem on deep-water gravity waves of finite amplitude is solved approximately (up to small terms of higher order) assuming that the waves are generated by an initial disturbance to the water and the horizontal dimensions of the initially disturbed body of the water are much larger than the magnitude of the free surface displacement. A numerable set of specific free surface waves is obtained in closed form and it is shown that free surface waves produced by an arbitrary initial disturbance to the water is a combination (not superposition: the waves are nonlinear) of the specific waves. A set of dispersive wave packets is found with one-to-one correspondence between the packets and positive integers, say, packet numbers, such that any initial free surface displacement gradually disintegrates into a number (limited or unlimited, depending on initial conditions) of the wave packets. The greater the packet number, the shorter the wavelength of the packet's carrier wave component, the slower the packet travels, the slower the packet disperses; evolution of any of the packets is not influenced by evolution of any other one. It is found that in case of infinitesimal wave amplitude the present theory is in agreement with linear wave theory. On the other hand, the behaviour of wave packets of large numbers and asymptotic behaviour of solutions of Schrödinger equation for weakly nonlinear waves are found to be similar, except for dispersion The theory is tested against experiments performed in a water tank.

Motivation & Objective

• To develop a nonlinear initial-boundary value theory for deep-water gravity waves generated by finite initial disturbances.
• To overcome limitations of the nonlinear Schrödinger equation approach by avoiding envelope soliton assumptions and preserving finite energy.
• To construct a complete, numerable set of wave packets that describe the disintegration of arbitrary initial surface displacements.
• To ensure the solution remains physically consistent by requiring finite energy and vanishing at infinity.
• To validate the model against experimental data and show agreement with linear and weakly nonlinear wave behavior.

Proposed method

• Transform the free surface trace into a half-circle via a nonlinear coordinate mapping to enable discrete harmonic analysis instead of continuous Fourier transforms.
• Assume initial disturbances have horizontal scales much larger than vertical displacements, enabling a small-parameter expansion.
• Expand initial conditions and solution components in trigonometric series convergent on the mapped domain.
• Solve the transformed equations order-by-order in a small parameter (ratio of horizontal to vertical scales), retaining only small higher-order terms.
• Use Laguerre polynomials and integral transforms to derive exact solutions for coefficient evolution in time.
• Apply boundary conditions requiring finite energy and vanishing at infinity, which restricts solutions to a discrete set of wave packets.

Experimental results

Research questions

• RQ1 How can deep-water gravity waves from an arbitrary initial disturbance be decomposed into fundamental wave components without assuming wave envelopes or solitons?
• RQ2 What is the role of dispersion and nonlinearity in the long-time evolution of wave groups when energy is finite and solutions vanish at infinity?
• RQ3 How does the proposed discrete wave packet model compare with linear wave theory and weakly nonlinear Schrödinger equation solutions?
• RQ4 Can the disintegration of an initial wave packet into multiple independent wave packets be described analytically with a one-to-one correspondence to positive integers?
• RQ5 Under what conditions does the leading-order solution approximate the full nonlinear problem with bounded error?

Key findings

• A complete, numerable set of wave packets is derived, each indexed by a positive integer, with one-to-one correspondence between packet number and physical properties such as wavelength and group velocity.
• Wave packets evolve independently; the evolution of one does not affect others, and each maintains its shape and velocity over time.
• For large packet numbers, the asymptotic behavior of the wave packets closely resembles that of solutions to the nonlinear Schrödinger equation, except for differences in dispersion.
• In the limit of infinitesimal wave amplitude, the theory reduces to linear wave theory, confirming consistency with established results.
• The model predicts wave group behavior consistent with experimental observations, including pulse travel time and length estimates (e.g., ~27.5 s travel time, ~3.1 m length for a 2.5 Hz packet).
• Higher-order approximations remain bounded due to the structure of the forcing terms, which are expressed as trigonometric series with summable coefficients, ensuring the solution remains a reasonable approximation for small parameters.