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A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line). The group velocity of a collection of waves is defined as =.
When R ∗ > 100, the data asymptotically approach a horizontal line; they are independent of Re, f D, and ε / D . The intermediate range of 5 < R ∗ < 100 constitutes a transition from one behavior to the other. The data depart from the line B(R ∗) = R ∗ very slowly, reach a maximum near R ∗ = 10, then fall to a constant value.
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
In this article, the following conventions and definitions are to be understood: The Reynolds number Re is taken to be Re = V D / ν, where V is the mean velocity of fluid flow, D is the pipe diameter, and where ν is the kinematic viscosity μ / ρ, with μ the fluid's Dynamic viscosity, and ρ the fluid's density.
The observed line shape is a convolution of the intrinsic line shape with the instrument transfer function. [3] Each of these mechanisms, and others, can act in isolation or in combination. If each effect is independent of the other, the observed line profile is a convolution of the line profiles of each mechanism. Thus, a combination of ...
The left plot, titled 'Concave Line with Log-Normal Noise', displays a scatter plot of the observed data (y) against the independent variable (x). The red line represents the 'Median line', while the blue line is the 'Mean line'. This plot illustrates a dataset with a power-law relationship between the variables, represented by a concave line.
Uncountable ordinals also exist, along with uncountable epsilon numbers whose index is an uncountable ordinal. The smallest epsilon number ε 0 appears in many induction proofs, because for many purposes transfinite induction is only required up to ε 0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem).
Calculate RMS emittance of the original beam: = If the length of the quadrupole is short compared to its focal length f = 1 / K {\displaystyle f=1/K} , where K {\displaystyle K} is the field strength of the quadrupole, its transfer matrix Q {\displaystyle Q} can be approximated by the thin lens approximation: