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The equation is named after Lord Rayleigh, who introduced it in 1880. [2] The Orr–Sommerfeld equation – introduced later, for the study of stability of parallel viscous flow – reduces to Rayleigh's equation when the viscosity is zero. [3] Rayleigh's equation, together with appropriate boundary conditions, most often poses an eigenvalue ...
Gaussian beam width () as a function of the axial distance .: beam waist; : confocal parameter; : Rayleigh length; : total angular spread In optics and especially laser science, the Rayleigh length or Rayleigh range, , is the distance along the propagation direction of a beam from the waist to the place where the area of the cross section is doubled. [1]
In fluid mechanics, the Rayleigh number (Ra, after Lord Rayleigh [1]) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free (or natural) convection. [ 2 ] [ 3 ] [ 4 ] It characterises the fluid's flow regime: [ 5 ] a value in a certain lower range denotes laminar flow ; a value in a higher range ...
The above equation can be used to plot the Rayleigh line on a Mach number versus ΔS graph, but the dimensionless enthalpy, H, versus ΔS diagram, is more often used. The dimensionless enthalpy equation is shown below with an equation relating the static temperature with its value at the choke location for a calorically perfect gas where the ...
Rayleigh distance in optics is the axial distance from a radiating aperture to a point at which the path difference between the axial ray and an edge ray is λ / 4. An approximation of the Rayleigh Distance is Z = D 2 2 λ {\displaystyle Z={\frac {D^{2}}{2\lambda }}} , in which Z is the Rayleigh distance, D is the aperture of radiation, λ the ...
The critical Rayleigh number can be obtained analytically for a number of different boundary conditions by doing a perturbation analysis on the linearized equations in the stable state. [16] The simplest case is that of two free boundaries, which Lord Rayleigh solved in 1916, obtaining Ra = 27 ⁄ 4 π 4 ≈ 657.51. [17]
Hence, the above formula can be used to estimate the noise variance in an MRI image from background data. [7] [8] The Rayleigh distribution was also employed in the field of nutrition for linking dietary nutrient levels and human and animal responses. In this way, the parameter σ may be used to calculate nutrient response relationship. [9]
By multiplying both sides of the equation by and dividing by the scalar , it is possible to express the eigenvalue problem as follow: = = for m = 1, 2, 3, ..., n. In the previous equation it is also possible to observe that the numerator is proportional to the potential energy while the denominator depicts a measure of the kinetic energy.