Yeunwoo Cho

On the stability of lumps and wave collapse in water waves

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg–de Vries solitary waves, the present study is concerned with a distinct class of gravity–capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney–Roskes–Davey–Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev–Petviashvili equation, a model for weakly nonlinear gravity–capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.

Resonantly forced gravity–capillary lumps on deep water. Part 1. Experiments

The wave pattern generated by a pressure source moving over the free surface of deep water at speeds, U , below the minimum phase speed for linear gravity–capillary waves, c min , was investigated experimentally using a combination of photographic measurement techniques. In similar experiments, using a single pressure amplitude, Diorio et al . ( Phys. Rev. Lett. , vol. 103, 2009, 214502) pointed out that the resulting surface response pattern exhibits remarkable nonlinear features as U approaches c min , and three distinct response states were identified. Here, we present a set of measurements for four surface-pressure amplitudes and provide a detailed quantitative examination of the various behaviours. At low speeds, the pattern resembles the stationary state ( U = 0), essentially a circular dimple located directly under the pressure source (called a state I response). At a critical speed, but still below c min , there is an abrupt transition to a wave-like state (state II) that features a marked increase in the response amplitude and the formation of a localized solitary depression downstream of the pressure source. This solitary depression is steady, elongated in the cross-stream relative to the streamwise direction, and resembles freely propagating gravity–capillary ‘lump’ solutions of potential flow theory on deep water. Detailed measurements of the shape of this depression are presented and compared with computed lump profiles from the literature. The amplitude of the solitary depression decreases with increasing U (another known feature of lumps) and is independent of the surface pressure magnitude. The speed at which the transition from states I to II occurs decreases with increasing surface pressure. For speeds very close to the transition point, time-dependent oscillations are observed and their dependence on speed and pressure magnitude are reported. As the speed approaches c min , a second transition is observed. Here, the steady solitary depression gives way to an unsteady state (state III), characterized by periodic shedding of lump-like disturbances from the tails of a V-shaped pattern.

Resonantly forced gravity–capillary lumps on deep water. Part 2. Theoretical model

A theoretical model is presented for the generation of waves by a localized pressure distribution moving on the surface of deep water with speed near the minimum gravity–capillary phase speed, c min . The model employs a simple forced–damped nonlinear dispersive equation. Even though it is not formally derived from the full governing equations, the proposed model equation combines the main effects controlling the response and captures the salient features of the experimental results reported in Diorio et al . ( J. Fluid Mech. , vol. 672, 2011, pp. 268–287 – Part 1 of this work). Specifically, as the speed of the pressure disturbance is increased towards c min , three distinct responses arise: state I is confined beneath the applied pressure and corresponds to the linear subcritical steady solution; state II is steady, too, but features a steep gravity–capillary lump downstream of the pressure source; and state III is time-periodic, involving continuous shedding of lumps downstream. The transitions from states I to II and from states II to III, observed experimentally, are associated with certain limit points in the steady-state response diagram computed via numerical continuation. Moreover, within the speed range that state II is reached, the maximum response amplitude turns out to be virtually independent of the strength of the pressure disturbance, in agreement with the experiment. The proposed model equation, while ad hoc , brings out the delicate interplay between dispersive, nonlinear and viscous effects that takes place near c min , and may also prove useful in other physical settings where a phase-speed minimum at non-zero wavenumber occurs.

Semi-analytical Karhunen–Loeve representation of irregular waves based on the prolate spheroidal wave functions

Abstract A new semi-analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in Karhunen–Loeve (K–L) representation of random data such as irregular ocean waves. Instead of direct numerical approach to this matrix eigenvalue problem, which may suffer from the computational inaccuracy for big data, a pair of integral and differential equations are considered, which are related to the so-called prolate spheroidal wave functions (PSWF). First, the PSWF is expressed as a summation of a small number of the analytical Legendre functions. After substituting them into the PSWF differential equation, a much smaller size matrix eigenvalue problem is obtained than the direct numerical K–L matrix eigenvalue problem. By solving this with a minimal numerical effort, the PSWF and the associated eigenvalue of the PSWF differential equation are obtained. Then, the eigenvalue of the PSWF integral equation is analytically expressed by the functional values of the PSWF and the eigenvalues obtained in the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K–L integral equation for the representation of exemplary wave data such as ordinary irregular waves. It is found that, with the same accuracy, the required memory size of the present method is smaller than that of the direct numerical K–L representation and the computation time of the present method is shorter than that of the semi-analytical method based on the sinusoidal functions.