|WikiProject Physics||(Rated Start-class, High-importance)|
...so it means you just add up the results?22.214.171.124 04:13, 22 January 2006 (UTC)
- It means that if you add two solutions, it is also a solution. Hypothetically, if F[x]=x and F[x]=y AND if the superposition principle holds, THEN F[x] also = x+y Fresheneesz 10:24, 11 March 2006 (UTC)
- Now see, this is a clear explanation. Why can't it be stated in this way in the article? Currently the article is only comprehensable when you've taken higher math in university. One or two sentences in the introduction describing the principle to laymen would be appreciated. 126.96.36.199 10:56, 17 June 2007 (UTC)
The superposition principle is often cited in mathematics. It would be very helpful to note in what ways the superposition principle is used in math. I don't know enough about it to add it. Fresheneesz 10:24, 11 March 2006 (UTC)
This article is far too user-unfriendly. I am a great fan of calculus, though I am more enthusiastic about referencing - but this physics article does not have enough physis. Diagrams of superpositioning between two sound waves of same frequency would be great. I know this is the principle of superposition but there is no article titled for linear superposition and what little physics is out there about this is well... little. Considering how useful linear superpositioning is with everyday lifes, its amazing that there isn't a FA about it. Tourskin 04:59, 8 March 2007 (UTC)
Principle and property
As far as I know, the superposition is not only a principle, but also (and mainly) one of the properties of any linear system, by definition. I suggest to use the word in the article and create a page named "superposition property" redirected to this article. Paolo.dL (talk) 18:16, 28 June 2008 (UTC)
- I haven't looked it up in a dictionary, so I don't know what the difference is supposed to be between a principle and a property :-). But I do believe that the people who call it the "superposition principle" are talking about exactly the same thing as the people who call it the "superposition property". Anyway, Google says that the phrase "superposition principle" is 20 times as common as the phrase "superposition property". I'd say Wikipedia should stick with the overwhelmingly more-common term, whether or not it's linguistically and pedagogically ideal. But maybe I'm misunderstanding what you're proposing.
- Anyway, since "superposition property" appears to be a synonym for "superposition principle" (less common but not unheard-of), it seems to me that it's perfectly sensible to create a redirect page from there to here. :-) --Steve (talk) 05:51, 29 June 2008 (UTC)
I just rewrote a substantial proportion of the article. The most unorthodox thing I put in, I suspect, is splitting the superposition principle into the "first version" and "second version". I'm well aware that these are closely related, and that there are uses of the superposition principle that can't neatly be classed as one or the other. But I think it's important to keep it in there, if for no other reason than the sociological fact that people talking about the superposition principle are sometimes imagining waves passing through each other and interfering, and sometimes imagining some sort of stimulus-response (like a voltage being applied to a linear circuit), but they're rarely thinking about both at once. Moreover, this distinction seems to corresponds relatively cleanly with whether the most natural form of the equation describing the phenomenon is homogeneous or inhomogeneous. Thoughts? --Steve (talk) 05:04, 25 November 2007 (UTC)
- I just tagged this page as needing some reorganization, and then came here to discuss it, so I suppose most of my thoughts on it are a response to your rewrite, Steve. I'm not really comfortable with the idea of your two "versions", mostly because it is not a standard term used in explaining the concepts and I've never thought of it that way. Perhaps all that's needed is a mention that the concept is more fully explained at interference. But I may be missing the point, so I'm just tossing my concern "out there" for now. --Qrystal (talk) 20:00, 26 February 2008 (UTC)
- I cannot see the application of the first (simplified) version. You apply it to wave interference, but wave interference typically has an output different from zero. I suggest to remove the first version, unless you can give a primary reference that uses the same definition, and explain the application. Paolo.dL (talk) 18:07, 28 June 2008 (UTC)
- The application is wave interference. For example, here's the classic wave equation:
- (see the article wave equation). The relevent linear operator is a functional:
- Since F is linear, then if u and v satisfy the wave equation (i.e., F(u)=F(v)=0), so does u+v (i.e., F(u+v)=0). Ergo, two waves can pass through each other. Does that make sense?
- The application is wave interference. For example, here's the classic wave equation:
- I'm fine with your terminology of "simplified" and "complete". I would also be fine editing the article to make it clearer how the "simplified" equation applies to waves passing through each other, as in the above paragraph. It's not obvious, and I can't blame you for (seemingly) not seeing the connection. --Steve (talk) 05:20, 29 June 2008 (UTC)
The simplified version is useless and makes article less readable
Honestly, I cannot see the advantage of a simplified version which is almost equivalent to the complete one. The complete version is already simple enough. And it is obviously valid in the singular case when the output is zero. So we really don't need to simplify it. Paradoxically, even though your intent is to simplify, you don't get what you want: yes, the first version is slightly (and uselessly) simpler, but the article becomes markedly more complex. The overall effect is definitely negative.
I remind you that unfortunately Wikipedia cannot be used to publish the original ideas and interpretations of the editors, even though in some cases these are helpful. But yes, if and only if the simplified version is used in the literature by some authoritative mathematician of physicist, you should edit the article and explain the application, as you proposed. Otherwise, I propose to delete the first version.
- Hmm, well I see your point about it being harder to read, and I also have no interest in going to the library to find sources. So I'm rewriting it without the simplified version. I'm halfway done, and I'll finish tonight, if that's okay. :-) --Steve (talk) 18:23, 29 June 2008 (UTC)
Thank you. Being in a hurry I forgot to tell you, in my previous comment, that I appreciated your intention to make the concept accessible. And I believe that only good authors are willing to rewrite their own articles. This reminds me this posting, written by one of the most brilliant and passionate editors in WikiProject Mathematics. You might enjoy reading it. Paolo.dL (talk) 20:39, 29 June 2008 (UTC)
- Thanks! The post is inspiring...it (almost) makes me want to try to make good articles into great ones, rather than my usual interest of making horrible articles into mediocre ones. Anyway, I tried again, without mentioning the simplified version as such, and compensated by explaining everything in greater detail. Of course, there's plenty of room for improvement, and feel free to try to do so, or let me know if you think I did anything particularly objectionable. :-) --Steve (talk) 04:01, 30 June 2008 (UTC)
Good job. I am busy with a massive discussion on Talk:Centrifugal force. I did too many changes and people was not able to digest them. Only a suggestion for you: this is also called, in mathematics, the property or homogeneity (of first degree). "Superposition principle" is used mainly in physics and system theory. This article starts with "In mathematics"... This is perhaps something worth to explain, if you agree. Paolo.dL (talk) 10:46, 30 June 2008 (UTC)
- I just added that linear is synonymous with homogeneous of degree 1, but I'm actually not sure that's right. For example, f: R^2 --> R, with f(x,y)= either x (if xy>0) or 0 (otherwise) is homogeneous of degree one, but not linear, right? --Steve (talk) 16:48, 30 June 2008 (UTC)
- I am very sorry. I wrote that too much in a hurry. There are two conditions for linearity of a function or transformation or map: additivity and homogeneity of first degree. And additivity (not homogeneity) is synonimous of superposition property. I apologize. See if you like my edits. I am not 100% sure, but that's what I deduce from linear and linear map. Feel free to undo or correct. Paolo.dL (talk) 22:04, 2 July 2008 (UTC)
The strength of superposition in finding a general solution to a BVP is best illustrated with several non-zero boundary conditions. I suggest that this is illustrated - but, I'm too busy right now. I could do it maybe in a week or two if no one else takes interest. Tparameter (talk) 04:13, 4 July 2008 (UTC)
Additivity does not imply homogeneity
This page seems to imply that superposition is identical to linearity. However, a linear map must have both additivity and homogeneity (of degree 1). That is,
If all you've shown is,
then you have not shown linearity.
- The references to linearity should be removed entirely.
- The references to linearity should be augmented with a note about the scaling property.
- The scaling property should be added here (is "superposition" another name for linearity? I don't think so, but this page has scant sources, and so all I can do is guess).
- Homogeneity of degree 1 is for all practical purposes a consequence of additivity. For example, say you have an additive function in one variable f(x). Since f(x+...+x)=f(x)+...+f(x), it follows pretty easily that f(ax)=af(x) for any rational number a. What about irrational a? Well, if you assume the axiom of choice, then there are bizarre, incredibly discontinuous, impossible-to-write-down functions that are linear but non-homogeneous. But if there's even one nonzero point at which f is continuous, then it follows that f is homogeneous. People talking about the superposition principle are usually talking about reasonably continuous and well-behaved systems, and in those contexts, homogeneity over real numbers is a consequence of additivity. But you're right that the article could use clarification. --Steve (talk) 19:07, 30 January 2009 (UTC)
- In pages 2 through 5 of Thomas Kailath's "Linear Systems" (Prentice-Hall, 1980), it's argued that superposition can fail for a linear system. But a discontinuous, nonhomogeneous function is used in the argument. To me this further emphasizes the need to include homogeniety in the definition of linearity. Mebden (talk) 09:22, 5 May 2009 (UTC)
I like how this article is evolving, thanks! A quick semantic point: Schrodinger's equation does not "govern" wave behavior. It "describes" it as stated in the introduction to the main article. I will change it next week, unless someone wishes to defend the claim that calculation of Schrodinger's equation is the a priori physical cause of wave behavior. Cheers, Wolfworks (talk) 20:09, 17 November 2011 (UTC)