= Random-phase amplitude and the Rayleigh distribution
In an 1880 paper, <John William Strutt>, known as <Lord Rayleigh>, studied the resultant of many <oscillations> with a common <frequency> and arbitrary <oscillation phases>. His large-number calculation distinguished a mean intensity from a definite resultant <amplitude> and used planar coordinates followed by a polar integral. The resulting <amplitude> density is the <Rayleigh distribution> in a different scale convention. This is a large-number description, not an exact law for every finite collection of <oscillations>.<On the resultant of a large number of vibrations of the same pitch and of arbitrary phase by John William Strutt, 3rd Baron Rayleigh>[Philosophical Magazine, Series 5, volume 10 (August 1880), pp. 73–74: expected intensity versus definite resultant; pp. 75–77: planar construction and polar integration; primary-paper transcription inspected on Scribd]{ref}
To see the connection in a chosen model, take <independent> <oscillation phases> $\Theta_j$ with <uniform distribution> on $[0,2\pi)$ and equal component <amplitude> $a$. The <phasor> sum has coordinates
$$
X_n=a\sum_{j=1}^n\cos\Theta_j,\qquad Y_n=a\sum_{j=1}^n\sin\Theta_j.
$$
Both have <mean> zero, <variance> $na^2/2$ and zero <covariance>. The <multivariate central limit theorem> gives an approximately circular <Gaussian random vector> for large $n$, with $\sigma=a\sqrt{n/2}$. Its <amplitude> $\sqrt{X_n^2+Y_n^2}$ is therefore approximately Rayleigh. For every finite $n$, that <amplitude> is bounded by $na$, whereas a <Rayleigh distribution> has an unbounded tail. At $n=1$ it is exactly $a$. This simple endpoint check prevents the approximation from becoming a false exact identity.
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