Gravitational waves (hydrodynamics)

Gravitational waves on water are a type of waves on the surface of a liquid at which the force returning the deformed surface of the liquid to the state of equilibrium is simply the force of gravity associated with the difference in height of the crest and depression in the gravitational field .

The trajectories of conventional water particles at a not very large (comparable to wavelength) depth are an ellipse, as illustrated by the swinging of the bottle

Content

1 General properties
2 The dispersion law for weak waves
- 2.1 Gravitational waves in shallow water
- 2.2 Gravity waves in deep water
- 2.3 Gravitational waves in the general case
3 Some problems of the theory of gravitational waves on water
4 See also
5 Literature

General Properties

Gravitational waves on water are nonlinear waves . Accurate mathematical analysis is possible only in a linearized approximation and in the absence of turbulence . In addition, usually we are talking about waves on the surface of an ideal fluid . The results of the exact solution in this case are described below.

Gravitational waves on the water are not transverse and not longitudinal . During oscillations, fluid particles describe some curves, that is, they move both in the direction of motion and across it. In the linearized approximation, these trajectories have the form of circles. This leads to the fact that the wave profile is not sinusoidal, but has characteristic pointed ridges and more gentle dips.

Nonlinear effects occur when the wave amplitude becomes comparable to its length. One of the characteristic effects in this mode is the appearance of kinks at the tops of the waves. In addition, there is the possibility of tipping the wave. These effects are not yet amenable to accurate analytical calculation.

The law of dispersion for weak waves

The behavior of small-amplitude waves can be described with good accuracy by the linearized equations of fluid motion . For the validity of this approximation, it is necessary that the wave amplitude is substantially less than both the wavelength and the depth of the pond.

There are two limiting situations for which the solution to the problem has the simplest form - these are gravitational waves in shallow water and deep water.

Shallow gravity waves

The approximation of waves in shallow water is true in those cases when the wavelength significantly exceeds the depth of the reservoir. A classic example of such waves is the tsunami in the ocean: until the tsunami reaches the shore, it is a wave with an amplitude of the order of several meters and a length of tens and hundreds of kilometers, which, of course, is significantly greater than the depth of the ocean.

The law of dispersion and wave velocity in this case has the form:

$\omega ={\sqrt {gH}}\cdot k\,;\quad v_{ph}=v_{gr}={\sqrt {gH}},$ ${\ displaystyle \ omega = {\ sqrt {gH}} \ cdot k \,; \ quad v_ {ph} = v_ {gr} = {\ sqrt {gH}},}$

Where

H

{\ displaystyle H}

- depth of the reservoir (distance to the bottom from the surface),

g

{\ displaystyle g}

- gravitational field strength ( acceleration of gravity ).

\omega

{\ displaystyle \ omega}

- the angular frequency of oscillation in the wave,

k

{\ displaystyle k}

- wave number (the reciprocal of the wavelength ),

v_{ph},v_{gr}

{\ displaystyle v_ {ph}, v_ {gr}}

- phase and group speeds, respectively.

This dispersion law leads to some phenomena that can be easily seen on the seashore.

Even if the wave in the open sea was at an angle to the shore, then when approaching the shore, wave crests tend to unfold parallel to the shore. This is due to the fact that near the coast, when the depth begins to gradually decrease, the wave velocity drops. Therefore, the oblique wave slows down on the approach to the shore, unfolding at the same time.
Due to a similar mechanism, when approaching the coast, the longitudinal size of the tsunami decreases, while the wave height increases.

Deep Water Gravitational Waves

The approach of the wave in deep water is true when the depth of the pond significantly exceeds the wavelength. In this case, an infinitely deep body of water is considered for simplicity. This is justified, because when the surface vibrates, not all the water column actually moves, but only a surface layer with a depth of the order of the wavelength.

The law of dispersion and wave velocity in this case has the form:

$\omega ={\sqrt {gk}}\,;\quad v_{ph}=2v_{gr}={\sqrt {g \over k}}.$ ${\ displaystyle \ omega = {\ sqrt {gk}} \,; \ quad v_ {ph} = 2v_ {gr} = {\ sqrt {g \ over k}}.}$

It follows from the written law that the phase and group velocities of gravitational waves in this case are proportional to the wavelength. In other words, long-wave oscillations will propagate through water faster than short-wave ones, which leads to a number of interesting phenomena:

Throwing a stone into the water and looking at the circles formed by it, you can notice that the wave boundary does not expand evenly, but approximately equally accelerated . Moreover, the larger the boundary, the more long-wave oscillations it forms.
A beautiful consequence of the written law of dispersion are ship waves .

Gravitational Waves in General

If the wavelength is comparable with the pool depth H , then the dispersion law in this case has the form:

$\omega ={\sqrt {gk\cdot th(kH)}}\,.$ ${\ displaystyle \ omega = {\ sqrt {gk \ cdot th (kH)}},,}$

Some problems of the theory of gravitational waves on water

The mechanism of formation and stability of the so-called killer waves - sudden waves of extreme amplitude is still not understood.

Literature

Graz, Yu.V. Lectures on hydrodynamics.-M., Lenand, 2014