Theory of Oscillations of Gravitational Waves

Studying in this paper the theory of oscillations and stability of flows bounded by rigid walls. Oscillations of a type – gravitational waves-are produced in flows with free boundaries in the presence of a gravity force. Owing to the presence of the free boundary. Showing below those gravitational waves may turn out to be unstable even if the flow velocity profile of an ideal liquid has no inflection points.


I. Introduction
Gravitational waves are hard to detect [1][2][3][4][5].These waves are actually faint ripples in space-time, the four-dimensional world that Einstein created in his theories of special and general relativity.As a gravitational wave passes by, objects would change their length, but by only about one part in 10 21 , which for the distance from the sun to Earth is about one atomic diameter.This is an extremely small effect.To increase the magnitude of the oscillation to be measured, these detectors are interferometers with arms several kilometers long.A laser beam is split and travels several kilometers in perpendicular directions to suspended mirrors.The reflected beams are combined and interfere, providing a pattern of light and dark that shows the relative phases of the two beams.The Laser Interferometer Gravitational Wave Observatory (LIGO), with one detector in Washington and the other in Louisiana, is currently being constructed and tested.The two detectors will be operated together to reduce the chances that noise at one or the other would mimic a gravitational wave.The photograph shows the Washington detector [6,7].Land-based interferometers like LIGO are vulnerable to the effects of seismic noise, especially at low frequencies.This frequency dependence is important, because super massive black holes and compact binary stars are expected to produce gravitational waves at low frequencies, below 1 cycle per second (Hz).To avoid this noise, the

B. M. Dakhel
interferometer can be flown in space on three well-separated satellites.Note that, since a gravitational wave changes space itself, no matter is required between the mirrors [8].
Marginally stable power recycling cavities are being used by nearly all interferometric gravitational wave detectors.With stability factors very close to unity the frequency separation of the higher order optical modes is smaller than the cavity bandwidth.As a consequence these higher order modes will resonate inside the cavity distorting the spatial mode of the interferometer control sidebands.Without losing generality we study and compare two designs of stable power recycling cavities for the proposed 5 kilometer long Australian International Gravitational Observatory (AIGO), a high power advanced interferometric gravitational wave detector.The length of various optical cavities that form the interferometer and the modulation frequencies that generate the control sidebands are also selected [9].The theory of oscillations and stability of continuous media (liquid, gravitating medium) has been the subject of many studies.Most touch upon, and in many cases consider in detail, a continuous medium moving with variable velocity in space [10][11][12][13][14][15][16][17][18][19][20][21].The present paper differs in that it employs consistently a point of view according to which the change (growth or damping) of the perturbation amplitudes in a nonuniformly moving medium is due to resonant interaction with the motion of the medium.

Basic Equations
Considering gravitational waves in an immobile liquid.Introducing gravity acceleration .)( in the R. H. S. of the equation of motion: Let the level 0 = x correspond to the unperturbed surface of the liquid.When it becomes wavy, the perturbation of the pressure on the level is the vertical displacement of the liquid surface in the oscillations.On the other hand, from the y component of the linearized equation of motion then get In the half space ) 0 ( < x filled by the liquid the perturbation of the stream function satisfies the Rayleigh equation, from which found that 0 . This solution describes a wave localized on the surface.To satisfy the boundary condition (2), the oscillation frequency should be Where x .With this circumstance taken into account, it is necessary to use for the quantity 0 / dV df indicative of the ratio of the number of overtaking and lagging particles the expression . (Recall that the liquid is assumed homogeneous, .0 const ≡ ρ ).Resonant buildup of gravitational waves in flow of an ideal liquid of finite depth h was considered in [22].Analysis of the oscillations with not too large values of ) ( when the oscillations can be described by the Rayleigh equation supplemented by the Landau rule for by passing the resonance point.(This is exactly how the instability of gravitational waves was considered in [22]).For low Reynolds numbers and large wavelengths along y 0 the .This case was analyzed in [23,24].
Considering now the buildup of gravitational waves by wind [12,25].Assume that the air moves above the water surface, i. e., in the region 0 > x , along − y axis with velocity ) ( 0 x V .The water is assumed to be at rest.For slow oscillations with phase velocity much lower than the sound velocity neglecting the compressibility of the air.Assuming also the air to be homogeneous, weightless, and nonviscous ) 1 (Re >> , describing these oscillations by the Rayleigh equation.
Establishing the conditions for joining the solutions on the water-air interface.Since the displacements of the air and water particles on the interface are having ; here and below the " "+ and " signs label quantities to the state of the air and of the water, respectively.From the y component of the equation of motion of the air then the other side of the interface, the pressure perturbation is ) Analyzing in conclusion the buildup of gravitational waves by a stratified air flow, i.e., a flow in which not only the velocity but also the density of the air varies with altitude.The differential equation describing the oscillations of such flow can be easily obtained from the linearized motion and continuity equations.In the approximation in which air is incompressible, it takes the form [15,17] From this equation obtained the following equation By integrate (6) using the following relation ) The growth rate of the wind instability, calculated with the aid of (7), is equal to: The first term in the brackets describes the buildup (damping) of the oscillations through change of the kinetic energy of the resonant liquid layer.The oscillations, however, alter not only the velocity but also the positions of the liquid particles.In the presence of a gravitational field, the particle displacements along − x axis, which lead to a change of the potential energy, also influence the dynamics of oscillation development.

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Now taking the motion of the liquid into account.In real flows the velocity of liquid relative to the bottom increases in the direction towards the surface.When gravitational (surface) waves are considered it is natural to use a reference frame connected with the surface of the liquid.In this frame the flow velocity is negative and increases in absolute value towards the interior of the liquid.Since interesting in resonance effects, considering oscillations with 0 < k and having, just as the liquid , a negative phase velocity.Assume that the condition k is met.In this case the influence of the liquid motion on the oscillations can be analyzed by successive approximations.Considering Rayleigh equation after multiplying it by * 1 ψ and subtract from product and after that integrate it from one boundary of the flow to the other, in which putting is as before the resonance point at which the phase velocity of the gravitational waves ) flow velocity.An example of an unstable velocity when the numbers of resonant particles overtaking and lagging the wave are compared, it must be recognized that in the assumed reference frame both the flow velocity and the oscillation velocity are negative ] particles are located below the point s

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The resonant interaction with the wind can introduce into the frequency a small imaginary part.To determine ω Proceeding next as in the first part of the present section obtaining [of the lower efficiency of stirring up the heavy liquid by the light air.