Spontaneous thermal fluctuations occur at microscopic level in liquids. It is said that hydrodynamic description is valid in the long wavelength and low frequency limit. So, to depict the thermal fluctuations occurring at microscopic level an extension of hydrodynamics is done by retaining the basic structure of hydrodynamic equations which is known as Molecular Hydrodynamics([https://www.sciencedirect.com/science/article/pii/B0122274105.004580#a0005])[1]. Are the wavelength and frequency that is being talked about here of the thermal fluctuations ? If that is so, what is actually a thermal fluctuation ? Is it a visible quantity in itself, or simply a manifestation of fluctuation of temperature through other quantities like velocity or number density of the liquid ? How can the wavelength and frequency of thermal fluctuation be determined ?
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$\begingroup$ Are you perhaps referring to the potential flow calculations around ships, operating at the air-water interface? In that case, linearized methods assuming a flat interface are possible at the low-speed & high-speed limits, but not in between. $\endgroup$– D. HalseyCommented Nov 14, 2020 at 16:09
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$\begingroup$ No sir, I am not referring to these kind of calculations. $\endgroup$– bubucodexCommented Nov 15, 2020 at 7:45
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$\begingroup$ You should probably edit your question to provide more context. The hydrodynamic description of what? $\endgroup$– D. HalseyCommented Nov 15, 2020 at 17:40
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The linearised hydrodynamic equations for a fluid (no internal dofs), include the continuity equation, the Navier-Stokes equation and the energy transport equation. Respectively, they represent conservation of mass, momentum and kinetic energy. I assume you have seen these equations and in the second of these equations there contains the term
$$\frac{{c_0}^2}{\gamma} \nabla \rho_1(x,t)$$
where the constant $c_0$ is called the adiabatic sound speed.
Since you are dealing with pressure wave propagation through the fluid, the long wavelength or small frequency condition refers to that for sound waves (travelling in the fluid as pressure waves at the speed of sound).
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$\begingroup$ What about thermal fluctuations ? Can I say that this description is valid for long wavelength and low frequency thermal fluctuations ? Or, Am I talking about the same thing as you ? Because sound waves are caused by thermal fluctuations only ? Again linearised hydrodynamics is valid for small amplitude oscillations . Does this assumption of small amplitude have any connection with long wavelength, small frequency limit ? $\endgroup$ Commented Nov 14, 2020 at 6:28
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$\begingroup$ My expertise is not in fluid mechanics. I answered the question you asked. It seems like you have another question regarding these equations. I would suggest you post another question but please include the equations so that others can learn as well. It also makes the question easier to answer. Thanks. $\endgroup$– joseph hCommented Nov 14, 2020 at 8:52
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