Gravity-Capillary Lumps Generated by a Moving Pressure Source

Abstract

The nonlinear wave pattern generated by a localized pressure source moving over a liquid free surface at speeds below the minimum phase speed (c min ) of linear gravity-capillary waves is investigated experimentally and theoretically. At these speeds, freely propagating fully localized solitary waves, or ''lumps,'' are known theoretically to be possible. For pressure-source speeds far below c min , the surface response is a local depression similar to the case with no forward speed. As the speed is increased, a critical value is reached c c % 0:9c min where there is an abrupt transition to a wavelike state that features a steady disturbance similar to a steep lump behind the pressure forcing. As the speed approaches c min , a second transition is found; the new state is unsteady and is characterized by continuous shedding of lumps from the tips of a V-shaped pattern. DOI: 10.1103/PhysRevLett.103.214502 PACS numbers: 47.35.Fg, 47.10.AÀ, 47.20.Ky Solitary waves are prominent nonlinear features of dispersive wave systems in various areas of physics [1][2][3]. Many key advances in understanding solitary waves were initiated by studies of waves on the free surface of a liquid. Most attention has been paid to solitary waves of the Korteweg-De Vries type [4,5], which bifurcate in the shallow-water limit. A different class of solitary waves, however, is possible on deep water [6][7][8][9] in the presence of gravity and surface tension; they bifurcate from linear sinusoidal waves at the minimum gravity-capillary phase speed, c min ¼ ffiffiffi 2 p ð g= Þ 1=4 , and propagate at speeds less than c min (g is the gravitational acceleration, the fluid density, and the surface tension). In dimensionless form, using L ¼ ð = gÞ 1=2 as the length scale and L=c min as the time scale, the dispersion relation for waves with frequency ! and wave number k isand the minimum of the phase speed !=k occurs at k ¼ k min ¼ 1. [14,15] and are stable at finite steepness [16]. On the experimental side, 2D steep-depression solitary waves have been generated by using an air jet from a narrow slit as forcing over a current with speed below c min [17]; however, persistent unsteadiness of the response due to cross-stream disturbances was also noted. In this Letter, we report on an experimental and theoretical investigation of gravity-capillary lumps generated by a localized pressure source moving at speeds below c min .The experiments were carried out in a water tank 7.3 m long and 76 cm wide with water depth of approximately 60 cm. The tank has glass walls and bottom for optical access. The surface tension was measured in situ with a Willhelmy plate and remained at ¼ 73 dyn=cm throughout the experiments. A pipe with inner diameter D ¼ 2:5 mm was mounted vertically to a carriage that rides on top of the tank. The open end of the pipe was positioned 1 cm above the water surface. The carriage was towed at various speeds by a servomotor with a precision of better than 0.3%. We define a speed parameter, ¼ c=c min , where c is the speed of the pipe-...

Investigation of symmetry breaking in periodic gravity–capillary waves

“…Three-dimensional fully localised solitary waves, which are commonly referred to as 'lumps', were observed recently in the experiments by Diorio et al (2009Diorio et al ( , 2011. Therefore the results presented in this paper naturally bring up the question of the possible existence of non-symmetric progressive waves in three dimensions.…”

Three-dimensional waves beneath an ice sheet due to a steadily moving pressure

“…In recent decades, many analytical and numerical results have been obtained for gravity-capillary waves (see [9] for a review). Some experiments were also recently performed (see [10] for two dimensions and [11] for three dimensions) for gravity-capillary waves. However, it should *Author for correspondence (e.parau@uea.ac.uk).…”

Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water

“…This fact is related to the existence of genuinely three-dimensional solitary waves, which are commonly referred to as 'lumps' in the literature. Gravity-capillary lumps were numerically found by Pȃrȃu, Vanden-Broeck & Cooker (2005) in the full Euler equations, and experimentally observed by Diorio et al (2009Diorio et al ( , 2011. The stability and dynamics of lumps, such as lump generation either by a localised forcing moving steadily or via transverse instability of a plane wave, long-time evolutions of unstable lumps, and collisions between stable lumps, have not been truly justified with the fully nonlinear water-wave equations in three dimensions, and are therefore of great interest for future studies.…”

“…The existence of stable solitary waves brings up the question of the excitation of these steady solutions. In laboratories, depression gravity-capillary solitary waves were generated under controlled conditions by Longuet-Higgins & Zhang (1997), Diorio et al (2009Diorio et al ( , 2011, among others. The experimentalists excited the depression waves by blowing air towards the surface of the fluid and moving the air source with a speed close to the phase speed minimum.…”

“…On the experimental side, the depression gravity-capillary solitary waves can be excited by applying a localised pressure distribution moving with a constant subcritical speed; once the pressure is removed, a free depression solitary wave is shed and then quickly dissipates due to the viscous effect (see Longuet-Higgins & Zhang (1997), Diorio et al (2009Diorio et al ( , 2011 for more details). Two-dimensional gravity-capillary travelling-wave solutions under a fully localised, constant-moving forcing were first computed by Vanden-Broeck & Dias (1992) in deep water.…”