says every point of wavefront emit wavelet in all directions. Then why does a back ward wave not exist? Can any expert tell real answer? On different sites I get different and contradictory answer.  I am asking question from my class 12 book. It says " Huygens argued that the amplitude of the secondary wavelets is maximum in forward direction and zero in the backward direction; by making this adhoc assumption, Huygens could explain the absence of backwave. However, this adhoc assumption is not satisfactory and the absence of the backwave is really justified from more rigorous wave theory."  What justification is given by wave theory?  answer explained by Marty green on   is different from answer given by Acuriousmind on     Which is correct?
I was reading French's Vibrations &amp; Waves where he discusses Huygens-Frensel Principle.   The principle talks about how secondary sources give rise to secondary wavelets to form the displaced wavefront. However, any secondary source can form two wavelets one moving forward &amp; other towards the original source as pointed by French:     [...] The Huygen's construction would define two subsequent wavefronts , not one. In addition to a new wavefront farther away from the source, there would be another one corresponding to a wavefront back toward the surface, But we know this does not happen.    Then he writes:     If the Huygens way of visualizing wave propagation is to be acceptable, it must introduce the unidirectional property of a traveling wave. This can be achieved by requiring that the disturbance  starting out from a given point in the medium at a given instant is not equal in all directions. Specifically, if $O$ is the true original source ,&amp; $S$ is the origin of Huygens wavelet, &amp; $P$ is the point at which the disturbance is being recorded, then the effect at $P$ due to the region near $S$is a function of $f(\theta)$ of the angle $\theta$ between $OS$ &amp; $SP$.     I am not understanding how his reasoning actually averts the possiblity of formation of back-waves. Can anyone help me visualise what he is talking? How does his reasoning maintain the unidirectional propagation of waves? Can anyone explain his argument?
POSTFIX main.cf permission denied Is there a simple way to reduce the amount of white space around the TikZ in the MWE pasted below (in particular in the x direction)?  \documentclass{article} \usepackage{xcolor} \usepackage{tikz}  \begin{document} VRCRCRCRCC \tikz[baseline] \node[fill=yellow!30, anchor=base,inner sep=0.7pt, outer xsep=-2pt] {ABCDE}; VBZBZBBBBG \end{document}     PS: This question may be a duplicate of . I posted it nevertheless since 1) my example is much simpler (and hence may have a much simpler solution) and 2) I don't see how the answers given there could help in my case. If you disagree feel free to mark my question as "exact (!) duplicate of ...". Derivative of $f(x)=|x|$ How to present a vertical multiplication/addition How do I present vertical (columnar) multiplication and addition in LaTeX. Like this:       3 8 4 x     5 6 ---------   2 3 0 4 1 9 2 0 --------- 2 1 5 0 4   I used the "array" environment. But i have right align problem. Any other best ideas? If I just render the default cube it will look like this:   It is rendered with color depth 16. I can see there are artefacts consisting of visible "line strokes" on some of the cube faces.  The bands seems to not go darker and darker, but alternates between light and dark, or is my eye fooled?  Why?  Edit  Here comes a gradient created by gimp without banding, using the same computer: How to map stock ticker symbols to ISIN (International Securities Identification Number)? How can I can convert ticker symbols (which include information on the exchange) to an International Securities Identification Number (ISIN) (which contains no information on the exchange)? Also I am interested in the other way round — given an ISIN and a market identification code (MIC) how can I determine the ticker symbol? Any databases around? MacPorts fails to find 'make' I've installed Xcode and MacPorts on a brand new mid-2013 MacBook Air, but MacPorts doesn't work, saying that it cannot find anything.  mba:~ cnst$ sudo port install wget Warning: The Command Line Tools for Xcode don't appear to be installed; most ports will likely fail to build. Warning: See http://guide.macports.org/chunked/installing.xcode.html for more information. ---&gt;  Computing dependencies for wgetError: Unable to execute port: can't read "build.cmd": Failed to locate 'make' in path: '/opt/local/bin:/opt/local/sbin:/bin:/sbin:/usr/bin:/usr/sbin' or at its MacPorts configuration time location, did you move it? mba:~ cnst$   And, indeed, there is no make or clang in the path, but it is definitely installed, though:  mba:~ cnst$ find /Applications -name "clang" -or -name "make" /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/clang /Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang /Applications/Xcode.app/Contents/Developer/usr/bin/make mba:~ cnst$   Why is there no make or clang in the path?  Why MacPorts doesn't work?  I notice that  mentions that you have to install something from ADC, Apple Command Line Developer Tools, but ADC doesn't work since last Thursday or Friday due to their security breach, so, the developer tools are nowhere to be found.  What's the proper solution here? Here is what I tried doing:  #!/bin/sh res=`df | awk 'FNR == 2 { print $5 }'` res2=$((res+0))   Here is my error:     -bash: 53%: syntax error: operand expected (error token is "%")   I Googled 'string to int bash' which showed me $((res+0)) but it doesn't work due to the % in the string (53%). How do I convert it in such a way I can do an if statement such as the one below (but that works)?  if [ "$res2" -ge 50 ] then     echo "Your disk &gt;=50% full!" fi How to find effects of magical items? Similar to this , but specific to D&amp;D 5e, how are players meant to find out the effects of items/potions/scrolls/weapons/armor?  Say, they find a potion in some chemist's chest. It is red and smells nice. Should they perform an investigation/perception check? Or an arcana/nature check? If I tell the players specifically to do an Arcana check, then I am already hinting at the fact that this potion will have some magical effect. It could just be a natural elixir concocted by Druids to help your health regen.  Same goes for weapons. They find the legendary bow of X, which has a +3 to hit and a 10% chance of striking other enemies near the target. I describe the bow as unique, glowing, made from dragon scales and hovering on air. Would they investigate their properties? Or just I just let them fire at a wall until something happens?   Our playstyle so far has been to use active checks for a quick understanding of the item (drink a little bit of Potion as an Investigation check to find whether it heals or not, sensing magical properties of the bow with Arcana checks). When time is available, and experimentation is necessary, allow the PCs to do some creative testing and, based on their ideas, "After Y time, you realize that the bow sometimes hits several points of the wall instead of just your target".  According to the RAW, is there a specific way of checking the stats/effects of such items? How many points in RSA, such that $m^e = m \bmod n$ For every RSA-cryptosystem, there exist some messages $m$, for which it holds that $m^e \equiv m \pmod n$ As to the question, how many such messages exist, this question has already been asked and answered here:  However, I do only understand the answer up to a certain point Suppose that $n$ is only 1 prime. Then it holds that the number of unconcealable messages is $\gcd(e-1, p-1)+1$ This is because in $m^e \equiv m \pmod  p$ , either $m = 0$ or $m \not= 0$. In the latter case $m$ can be expressed as $g^t$, where $g$ is a Generator in $\mathbb{Z_p}$ $g^{t*e} \equiv g^t \pmod p$ $g^{t(e-1) \bmod p -1 } \equiv g^{0 \bmod p -1} \pmod p$ $t(e-1) \equiv 0 \pmod {p-1} $ has exactly $\gcd(e-1, p-1)$  solutions, so the total number of unconcealable messages is, together with $m=0$ , $\gcd(e-1, p-1)+1$ When we now look at real RSA, which has two prime factors in its modulus, it holds that the number of unconcealable messages is $(\gcd(e-1, p-1)+1)\cdot(\gcd(e-1, q-1)+1)$ Why is that so? I suppose it has to do with CRT, but I just cant understand why. Install 18.04 server without network connection The latest version of Ubuntu server (18.04) seems to demand network configuration and refuse to boot without it. I have no desire to connect the server in question to the internet for anything, and have installed previous versions of Ubuntu server without network no questions asked.  Is a network connection an absolute requirement for the latest version of Ubuntu server? Or is there something I'm missing?  The installation cannot proceed past the network configuration page unless an ethernet connection is active, by my current estimates. I have no ethernet network within range of the machine so can't test it on an internet-less network. I only have WiFi within immediate range (which does have internet access) but it won't recognise the device and it should be simpler to get the installation to happen without any updates than to go to the effort of finding the drivers to support the WiFi adapter.  Note that this is NOT a duplicate of  since the new installer seems to be the real problem here. Creating centerline from lines using QGIS How do I create a centerline (vector data) from two parallel polylines (vector data) in QGIS? For example, if the distances between two parallel lines are 14m, then I want to create a centerline at 7m from both lines. What exactly is a distribution? I know very little of Probability and Statistics, and am wishing to learn.  I see the word "distribution" used all over the place in different contexts.  For example, a discrete random variable has a "probability distribution."  I know what this is. A continuous random variable has a probability density function, then for $x\in\mathbb{R}$, the integral from $-\infty$ to $x$ of the probability density function is the cumulative distribution function evaluated at $x$.   And apparently just "distribution function" is synonymous with "cumulative distribution function," at least when talking about continuous random variables (question: are they always synonyms?).  Then there are many famous distributions.  $\Gamma$ distribution $\chi^2$ distribution, etc.  But what exactly is a $\Gamma$ distribution?  Is it the cumulative distribution function of a $\Gamma$ random variable?  Or the probability density function of a $\Gamma$ random variable?  But then a frequency distribution of a finite data set appears to be a histogram.  Long story short: in Probability and Statistics, what is the definition of the word "distribution"?    I know the definition of distribution in Mathematics (an element of the dual space of the collection of test functions equipped with the inductive limit topology), but not Probability and Statistics. When is "Y" a vowel? In school we are taught the vowels: A, E, I, O, U, and sometimes Y.    got me thinking about when the letter Y is considered to be a vowel.  I understand (perhaps incorrectly) that in words like bicycle and why it is a vowel. What about the word voyeur (as mentioned in the XKCD alt-text)?   If I've got this backwards, and Y is almost always a vowel, how can I tell when it is a consonant? Thinking back, I don't think my education ever covered the difference between them, we just memorized which letters were which. Version Control: Entire site or just pieces? I've been using git to track changes in the plugins and themes I've developed for different sites. Each plugin or theme has their own repository in whichever local install of WordPress I happen to be using for that project. This works fine when I'm just working on one solitary plugin or theme.  But when I have a local version of an entire site which is a collection of different plugins, themes, WordPress core, etc. Should I be keeping that entire directory under version control?  Is there a best practice to always have a collection of repositories for each theme/plugin, or for each client site to have one monolithic repository that includes all the code used in the project (including the latest copy of WordPress via svn.) tense and time and composite sentence If the eating meal starts at 7:00 AM and finishes at 8:00 AM. the drinking liquor starts at 7:00 AM and finishes at 7:50 AM.  can I say that :   I drank the liquor when I ate the meal. I ate the meal when I drank the liquor.   Is I need to use progressive tense ? Can I prepay a Credit Card to make a Large Purchase greater than my credit limit? I want to make a large purchase at a retail store to the sum of $10,000 but the credit limit on my card is set at $2000.  If I send $10K to my credit card will the purchase be allowed to go through?  Not sure if the credit limit is a hard limit or not?  I don't want to be left embarrassed if my card is declined in the store lol.  I called my bank but the answer they gave wasn't too convincing. It was kind of "Errr yes i think its OK!"  Does any one know first hand? What is the net charge of the Earth? Many years ago I read a book or story where a group of people discover a dark city - maybe on another planet - which appears devoid of life but is in very good condition. As they gradually explore the city one of the them falls into a trap and is cut to pieces by something mechanical. This detail I can remember as it was a pretty gruesome idea. I don't know the ending but my guess is that none of them make it out alive. In newer Google Maps I can make area available offline.  Do other applications that rely on Google Maps (utilize Google Maps Android API) use offline maps?