The question of pressure fluctuations within a fluid is a complex one, often requiring a deep dive into the principles of fluid dynamics. While initial estimations might simplify the picture, a more rigorous approach, such as that developed by Landau and Lifshitz, offers a comprehensive understanding. This exploration delves into the nature of these fluctuations, contrasting them with the behavior of electromagnetic waves and focusing on the fundamental motion of molecules within the fluid.
At a basic level, pressure fluctuations in a fluid arise from the constant, random motion of its constituent molecules. This kinetic energy at the molecular level translates into macroscopic pressure variations. A simplified approach, like the Rayleigh-Jeans estimate, can offer an initial approximation of these fluctuations. However, for a more accurate and detailed description, particularly in the context of fluid dynamics, we turn to more sophisticated models.
Landau and Lifshitz, in their seminal work on fluid dynamics, provide a framework rooted in the linearized Navier-Stokes equations, augmented with noise terms that are dictated by fluctuation-dissipation relations. This leads to a formula describing the pressure correlator in Fourier space, represented as:
$$ langle delta Pdelta Prangle_{omega k} = 2 ,{Re}, frac{k^2rho T u^4(i+gamma_Tomega/(uk^2))} {omega(omega^2-k^2u^2)+2iomega ugamma}. $$
This equation, although intricate, encapsulates the interplay of various factors influencing pressure fluctuations. Here, $langledelta Pdelta Prangle_{omega k}$ represents the pressure correlator, $Re$ denotes the real part, $u$ is the speed of sound within the fluid, and $gamma$ is the sound attenuation rate, given by:
$$ gamma = frac{k^2}{2rho u}left[zeta +frac{4}{3}eta +frac{chi u^2rho^2}{T} left(frac{partial P}{partial T}right)_s^2right], $$
where $(zeta,eta,chi)$ are the bulk viscosity, shear viscosity, and thermal conductivity, respectively. $gamma_T$ specifically refers to the thermal conductivity component of $gamma$. This formulation highlights that pressure fluctuations are not merely random noise but are structured by the fundamental modes of fluid motion, including propagating sound waves ($omega^2=u^2k^2$) and the diffusion of heat ($w=ichi k^2$).
It’s crucial to distinguish this molecular motion-driven pressure fluctuation from phenomena associated with electromagnetic waves. Electromagnetic waves are disturbances in electric and magnetic fields that propagate through space. While electromagnetic radiation can exert pressure (radiation pressure), and interact with matter at the molecular level, the pressure fluctuations described by the Landau-Lifshitz formula are fundamentally rooted in the thermal and kinetic motion of fluid molecules. They are not a direct manifestation of electromagnetic wave propagation within the fluid medium itself.
Integrating the pressure correlator over all possible wave vectors ($k$), focusing on the sound peaks and using a Lorentzian approximation, leads to:
$$ int d^3k , langle delta Pdelta Prangle_{omega k} = frac{8pi^2rho Tomega^2}{u} $$
This result bears a resemblance to the Rayleigh-Jeans formula, particularly in the audible frequency range, suggesting that for sound waves, simpler approximations can hold. However, it’s vital to remember that this simplification arises under specific conditions and that the underlying physics is governed by the more complex dynamics captured in the Landau-Lifshitz formulation.
In conclusion, pressure fluctuations in fluids, as described by advanced fluid dynamics, are primarily a consequence of molecular motion and the inherent thermal energy within the system. While electromagnetic waves are a distinct form of energy propagation, the pressure fluctuations discussed here are rooted in the kinetic behavior of fluid molecules and are accurately modeled by equations derived from the Navier-Stokes framework, offering a far more nuanced picture than simple white or Brownian noise models might suggest. The power spectrum is shaped by the fluid’s eigenmodes, emphasizing the continuous spectrum of fluid motion beyond simplistic particle models.
