Physical Oceanography, Vol. 15, No. 5, 2005
CAPILLARY-GRAVITATIONAL WAVES OF FINITE AMPLITUDE
ON THE SURFACE OF A HOMOGENEOUS LIQUID
A. E. Bukatov and A. A. Bukatov
The method of multiple scales is used to deduce equations for three nonlinear approximations of
the capillary-gravitational disturbances of the free surface of a layer of a homogeneous liquid of
constant depth. In these equations, the space-time variations of the wave profile in the expres-
sion for the velocity potential on the liquid surface are taken into account. On this basis, we con-
struct asymptotic expansions up to the quantities of the third order of smallness for the velocity
potential and elevations of the liquid surface induced by running periodic waves of finite ampli-
tude. Furthermore, we analyze the dependences of the amplitude-phase characteristics of wave
disturbances on the surface tension, depth of the liquid, and the length and steepness of waves of
the first harmonic.
The influence of surface tension on the wave processes in a homogeneous liquid is studied in the linear case
in [1–3]. The theoretical analysis of the development of low-amplitude waves in a two-layer liquid with regard
for the capillary forces acting on the free surface and the interface of layers is carried out in . The capillary-
gravitational surface periodic running waves of finite amplitude are investigated in  by the method of multi-
scale expansions without analyzing the changes in the amplitude-phase characteristics caused by the dependence
of the potential of the velocity of motion of liquid particles on the free surface on its space and time deforma-
In the present work, the method of multiscale asymptotic expansions is used to deduce equations for non-
linear approximations taking into account the space and time variations of the wave profile in the expression for
the velocity potential on the liquid–air interface.
These equations are used to analyze the influence of surface tension on the propagation of periodic waves
of finite amplitude. We also estimate the changes introduced in the structure of disturbances as a result of taking
into account the space and time deformations of the liquid surface in the expression for the velocity potential in
deducing the kinematic and dynamic boundary conditions for nonlinear approximations.
Statement of the Problem.
Consider the influence of surface tension on the propagation of periodic waves of finite amplitude in a ho-
mogeneous perfect incompressible liquid of finite depth H. Under the assumption that the motion of liquid is
potential, in dimensionless variables x = k
, z = k
, and t = kg t
, where k is the wave number, the anal-
yzed problem is reduced to the solution of the Laplace equation
ϕ = 0, –
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
25–34, September–October, 2005. Original article submitted February
0928-5105/05/1505–0289 © 2005 Springer Science+Business Media, Inc. 289