Gavin R. Putland, BE PhD
Friday, May 08, 2020 | (Comment) |
Huygens' principle: Simple mechanistic reason for the phase advance of the secondary sources
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Consider the “primary” wave field \begin{equation}\tag{1}\psi(r,t)=\frac{f(t-r/c)}{r}\,,\end{equation} where \(r\) is distance from the source, \(t\) is time, \(f(t)\) is a function characterizing the source, and \(c\) is constant. If \(\psi_r\) and \(\dot{\psi}\) denote the partial derivatives of \(\psi\) w.r.t. \(r\) and \(t\), respectively, and \(dS\) is an area element of the sphere of radius \(r\) centered on the source, we readily obtain \begin{equation}\tag{2}-\psi_r\;\!dS=\bigg(\frac{\dot\psi}{\,c\,}+\frac{\psi}{\,r\,}\bigg)\,dS\,.\end{equation} For the secondary source corresponding to \(dS\), let the average (over all directions) of the wave function at distance \(s\) be \begin{equation}\tag{3}\varphi(s,t)=\frac{g(t-s/c)}{s}\,,\end{equation} where \(g(t)\) is a function to be determined, characterizing the secondary source. Let \(A\) be the area of a sphere of radius \(s\) centered on \(dS\), so that \begin{equation}\tag{4}A=4\pi s^2.\end{equation} Then we readily obtain \begin{equation}\tag{5}-\varphi_s\;\!A=4\pi\bigg(\frac{s\;\!g'(t{-}s/c)}{c}+g\big(t{-}s/c\big)\bigg)\end{equation} whence, as \(s\!\to\!0\), \begin{equation}\tag{6}-\varphi_s\;\!A\to 4\pi g(t)\,.\end{equation} If \(-\psi_r\) and \(-\varphi_s\) represent per-unit-area flows (e.g., if the wave function is a velocity potential) or time-derivatives thereof (e.g., for sound waves in which the wave function is pressure relative to equilibrium), then, by conservation of whatever is flowing, we can equate the left-hand sides of (2) and (6), hence their right-hand sides, obtaining \begin{equation}\tag{7}g(t)=\frac{1}{4\pi}\bigg(\frac{\dot\psi}{\,c\,}+\frac{\psi}{\,r\,}\bigg)\,dS\,.\end{equation} The term in \(\dot{\psi}\), which is advanced in phase by 90° w.r.t. \(\psi\) for sinusoidal \(f(t)\), is dominant for sufficiently large \(r\).
In the exact Kirchhoff diffraction formula [1], the factor corresponding to \(g(t)\), but also accounting for directionality, is \begin{equation}\tag{8}\frac{1}{4\pi}\Bigg\{\frac{\cos(n,r)-\cos(n,s)}{c}\,\dot\psi\,+\,\bigg(\frac{\cos(n,r)}{r}-\frac{\cos(n,s)}{s}\bigg)\psi\,\Bigg\}\,dS\,,\end{equation} where \(n\) is the coordinate normal to \(dS\) (and \(s\) is measured to \(dS\)). Here again the term in \(\dot\psi\) is dominant for sufficiently large \(r\) (and \(s\)).
Acknowledgment: This post was inspired in part by reference [2], but takes a different approach.
References
[1] G.R. Putland, “Exact derivation of Kirchhoff's integral theorem and diffraction formula using high-school math”, Zenodo, 2020; doi.org/10.5281/zenodo.3659325.
[2] Forrest L. Anderson, “Making waves: the geometric derivation of Huygens' Principle for wave propagation, and the problem of the wake”, ResearchGate, 2017; doi.org/10.13140/RG.2.2.30105.42083.
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