Timeline for What happens to waves when they hit smaller apertures than their wavelenghts?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 7, 2016 at 0:28 | comment | added | Faiz Khairil | You can access the bethe journal here: web.stanford.edu/class/ee349/Handouts/Bethe_PR1944.pdf | |
| Oct 17, 2014 at 6:22 | comment | added | Luboš Motl | Huygens' principle is a more specific result going well beyond the dimensional analysis and I haven't used it. We haven't even used the dispersion relations (e.g. the constancy of the speed of waves as a function of the frequency). This simple interference of the points is similar to but simpler than Hyugens' principle, and it works both in odd and even dimensions. | |
| Oct 17, 2014 at 6:20 | comment | added | Luboš Motl | Dear Ben, the factors of $g,\rho,\lambda$ and their powers, whatever is needed for a given type of wave, may always be added uniquely by dimensional analysis. There is some energy flux for $d\sim\lambda$ and relatively to this energy flux $E_0$, if $d\lt \lambda$, the energy flux for a smaller hole is comparable to $E_0\cdot (d/\lambda)^4$ where $E_0/\lambda^4$ was simply inserted to get the correct result (boundary conditions) for $d=\lambda$. It's only the $d^4$ that is "nontrivial" in the result. | |
| Oct 16, 2014 at 22:42 | comment | added | user4552 | I think I understand the $(a/\lambda)^4$ result now. I wrote out an explanation here: physics.stackexchange.com/a/141713/4552 | |
| Oct 16, 2014 at 16:03 | comment | added | user4552 | @LubošMotl: A second thing about your argument also doesn't make sense to me, which is that it seems to be wrong dimensionally, unless the transmission is defined as I described in my preceding comment. | |
| Oct 16, 2014 at 15:49 | comment | added | user4552 | @LubošMotl: From online discussions I found by googling, it looked to me like the transmission was defined as the ratio of the diffracted power to the power incident on the hole. But I don't have access to the Bethe paper, and maybe the online sources were wrong or I was misinterpreting them. Doesn't Huygens' principle not even work in odd dimensions? | |
| Oct 16, 2014 at 7:46 | comment | added | Luboš Motl | It's trivial to see why it scales like $(d/\lambda)^4$. One gets constructive interference (nearly the same phases) from $O(d^2)$ points (area) in the hole, so the amplitude goes like $d^2$ which is why the intensity goes like $d^4$. Note that this is for propagation in 3+1 dimensions. For 2+1 dim, like water waves, one only gets $d$ for the amplitude and $d^2$ for the intensity. | |
| Oct 16, 2014 at 2:39 | comment | added | user4552 | The original paper by Bethe seems to be "Theory of Diffraction by Small Holes," Phys. Rev. 66, (1944) 163. I don't have access to the paper, but from descriptions online it appears that for a circular hole with diameter $d<\lambda$, the transmission is $(d/\lambda)^4$. Apparently there is also a calculation presented in Jackson. | |
| Oct 16, 2014 at 2:18 | history | answered | hyd | CC BY-SA 3.0 |