588 research outputs found
On the absence of trapped water waves near a cliffed cape
The water wave problem is considered for a class of semi-infinite domains
each having its shore shaped as a cliffed cape. In particular, the free surface
of a water domain is supposed to be an infinite sector whose vertex angle is
greater than , whereas the water layer lying under the free surface is of
constant depth. Under these assumptions, it is shown that there are no trapped
mode solutions of the problem for all values of a non-dimensional spectral
parameter; in other words, no point eigenvalues are embedded in the problem's
continuous spectrum.Comment: 8 pages, 1 figur
The floating-body problem: an integro-differential equation without irregular frequencies
The linear boundary value problem under consideration describes time-harmonic
motion of water in a horizontal three-dimensional layer of constant depth in
the presence of an obstacle adjacent to the upper side of the layer (floating
body). This problem for a complex-valued harmonic function involves mixed
boundary conditions and a radiation condition at infinity. Under rather general
geometric assumptions the existence of a unique solution is proved for all
values of the nonnegative problem's parameter related to the frequency of
oscillations. The proof is based on the representation of solution as a sum of
simple- and double-layer potentials with densities distributed over the
obstacle's surface, thus reducing the problem to an indefinite
integro-differential equation. The latter is shown to be soluble for all
continuous right-hand side terms for which purpose S.~G. Krein's theorem about
indefinite equations is used.Comment: 12 pages, 1 figur
Two-dimensional water waves in the presence of a freely floating body: conditions for the absence of trapped modes
The coupled motion is investigated for a mechanical system consisting of
water and a body freely floating in it. Water occupies either a half-space or a
layer of constant depth into which an infinitely long surface-piercing cylinder
is immersed, thus allowing us to study two-dimensional modes. Under the
assumption that the motion is of small amplitude near equilibrium, a linear
setting is applicable and for the time-harmonic oscillations it reduces to a
spectral problem with the frequency of oscillations as the spectral parameter.
It is essential that one of the problem's relations is linear with respect to
the parameter, whereas two others are quadratic with respect to it.
Within this framework, it is shown that the total energy of the water motion
is finite and the equipartition of energy holds for the whole system. On this
basis, it is proved that no wave modes can be trapped provided their
frequencies exceed a bound depending on cylinder's properties, whereas its
geometry is subject to some restrictions and, in some cases, certain
restrictions are imposed on the type of mode.Comment: 11 pages, 1 figur
Mean value properties of harmonic functions and related topics (a survey)
Results involving various mean value properties are reviewed for harmonic,
biharmonic and metaharmonic functions. It is also considered how the standard
mean value property can be weakened to imply harmonicity and belonging to other
classes of functions.Comment: 21 page
A Faber--Krahn inequality for indented and cut membranes
In 1960, Payne and Weinberger proved that among all domains that lie within a
wedge (an angle whose measure is less than or equal to ), and have a given
value of a certain integral the circular sector has the lowest fundamental
eigenvalue of the Dirichlet Laplacian. Here, it is shown that an analogue of
this assertion is true for domains with a cut and for indented domains; that
is, for those located in a reflex angle (its measure is between and ).Comment: 6 page
On Delusive Nodes of Free Oscillations
Two theorems and one conjecture about nodal sets of eigenfunctions arising in
various spectral problems for the Laplacian are reviewed. It occurred that all
these assertions are incorrect or only partly correct, but their analysis has
brought better understanding of the corresponding area of mathematical physics.
The contribution made by V. I. Arnold is emphasized.Comment: 14 pages, 4 figure
Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents
The two-dimensional free-boundary problem of steady periodic waves with
vorticity is considered for water of finite depth. We investigate how flows
with small-amplitude Stokes waves on the free surface bifurcate from a
horizontal parallel shear flow in which counter-currents may be present. Two
bifurcation mechanisms are described: for waves with fixed Bernoulli's constant
and fixed wavelength. In both cases the corresponding dispersion equations
serve for defining wavelengths from which Stokes waves bifurcate. Sufficient
conditions guaranteeing the existence of roots of these equations are obtained.
Two particular vorticity distributions are considered in order to illustrate
general results.Comment: 42 pages, 2 figures. New material and references are added, the
presentation of old material is amende
A comparison theorem for super- and subsolutions of and its application to water waves with vorticity
A comparison theorem is proved for a pair of solutions that satisfy in a weak
sense opposite differential inequalities with nonlinearity of the form
with belonging to the class . The solutions are assumed to have
non-vanishing gradients in the domain, where the inequalities are considered.
The comparison theorem is applied to the problem describing steady, periodic
water waves with vorticity in the case of arbitrary free-surface profiles
including overhanging ones. Bounds for these profiles as well as
streamfunctions and admissible values of the total head are obtained.Comment: 15 pages, 1 figur
No steady water waves of small amplitude are supported by a shear flow with still free surface
The two-dimensional free-boundary problem describing steady gravity waves
with vorticity on water of finite depth is considered. It is proved that no
small-amplitude waves are supported by a horizontal shear flow whose free
surface is still in a coordinate frame such that the flow is time-independent
in it. The class of vorticity distributions for which such flows exist includes
all positive constant distributions, as well as linear and quadric ones with
arbitrary positive coefficients.Comment: 12 page
On freely floating bodies trapping time-harmonic waves in water covered by brash ice
A mechanical system consisting of water covered by brash ice and a body
freely floating near equilibrium is considered. The water occupies a half-space
into which an infinitely long surface-piercing cylinder is immersed, thus
allowing us to study two-dimensional modes of the coupled motion which is
assumed to be of small amplitude. The corresponding linear setting for
time-harmonic oscillations reduces to a spectral problem whose parameter is the
frequency. A constant that characterises the brash ice divides the set of
frequencies into two subsets and the results obtained for each of these subsets
are essentially different.
For frequencies belonging to a finite interval adjacent to zero, the total
energy of motion is finite and the equipartition of energy holds for the whole
system. For every frequency from this interval, a family of motionless bodies
trapping waves is constructed by virtue of the semi-inverse procedure. For
sufficiently large frequencies outside of this interval, all solutions of
finite energy are trivial.Comment: 13 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1503.0219
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