Wikipedia:Reference desk/Archives/Mathematics/2009 August 31

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August 31

Empty set and phi

So, in my topology class last year, a student referred to the empty set as phi. Then, this past week, one of my professors did the same thing. It seems to me that $\emptyset$ is more like a 0 with a line through it than $\phi$, and the 0 would match up symbolically as well. Is it related at all to $\phi$ or are these two people incorrect in calling it $\phi$? StatisticsMan (talk) 01:04, 31 August 2009 (UTC)

I've not heard "phi" for the empty set, and calling it that is incorrect or at least not standard. However $\varnothing \$ is a common variant on $\emptyset \$ that looks a good bit like $\phi \$. Eric. 98.207.86.2 (talk) 01:16, 31 August 2009 (UTC)

Nor have I. People mis-reading the symbol me thinks. ~~ Dr Dec (Talk) ~~ 01:21, 31 August 2009 (UTC)

Ditto. It is an easy mistake to make, but I've always heard it pronounced as "the empty set" or just "empty". --Tango (talk) 01:26, 31 August 2009 (UTC)

This is my thought when it was just a student in the class saying it last year. When a professor says it, then it makes me wonder, though it still seems wrong. Thanks for your thoughts. StatisticsMan (talk) 02:42, 31 August 2009 (UTC)

From the page Earliest Uses of Various Mathmatical Symbols:

The null set symbol (Ø) first appeared in N. Bourbaki Éléments de mathématique Fasc.1: Les structures fondamentales de l'analyse; Liv.1: Theorie de ensembles. (Fascicule de resultants) (1939): "certaines propriétés... ne sont vraies pour aucun élément de E... la partie qu’elles définissent est appelée la partie vide de E, et designée par la notation Ø." (p. 4.)

André Weil (1906-1998) says in his autobiography that he was responsible for the symbol:

Wisely, we had decided to publish an installment establishing the system of notation for set theory, rather than wait for the detailed treatment that was to follow: it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval. Much later, my own part in these discussions earned me the respect of my daughter Nicolette, when she learned the symbol Ø for the empty set at school and I told her that I had been personally responsible for its adoption. The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar.

So no relation to phi. Tesseran (talk) 03:35, 31 August 2009 (UTC)

Indeed. The symbol originated as an Ø and is completely unrelated to phi. Gandalf61 (talk) 07:50, 31 August 2009 (UTC)

Wow. This is the first time I've heard of the symbol for the empty set not being phi. For the past 10, maybe 15 years I was sure it was. Our article really should state explicitly that it's not. -- Meni Rosenfeld (talk) 20:37, 31 August 2009 (UTC)

Done - it now states that explicitly. Gandalf61 (talk) 21:56, 31 August 2009 (UTC)

So, how about the way some computer scientists write zero, that is again Ø? Why do they need it? Did it follow the notation of the empty set ?(via von Neumann construction?) or where did it originate? And is it still used? How they distinguish zero from the empty set? :( --pma (talk) 22:22, 31 August 2009 (UTC)

The slash through a zero is to distinguish it from the letter O. You don't usually need to distinguish between zero and the empty set - it should be clear from context. --Tango (talk) 22:43, 31 August 2009 (UTC)

Method of Characteristics to solve PDEs

First, I do admit that this is a homework problem. The question is to solve
$2tu_{t}+xu_{x}=0$
using the method of characteristics. I have done that and the solution is
$u(x,t)=g\left({\frac {x}{\sqrt {2t}}}\right)$. Then the initial condition $u(x,0)=f(x)$ is given but how does this work? I can't just plug in t=0 in my solution and figure out what g(z) is. How can I resolve this? Any ideas/hints? Did I do something wrong? I checked my solution by plugging it into the PDE and it works fine. Thanks!97.118.56.41 (talk) 03:52, 31 August 2009 (UTC)

The method of characteristics requires the characteristics to be transverse to the initial manifold (here it's $\scriptstyle \mathbb {R} \times \{0\}$, the x-axis). On the contrary here the initial manifold is itself a characteristic line (the support of a ch.). As a consequence, the initial values of a solution can't be prescribed on it, and, indeed, the equation already tells you that any solution defined on $\scriptstyle \mathbb {R} \times \{0\}$ has to be constant there. --pma (talk) 06:57, 31 August 2009 (UTC)

Sequence limit

Resolved

--Shahab (talk) 17:22, 31 August 2009 (UTC)

I want to find the limit of the sequence $(an+b)^{1/n}$ where a and b are constants? I know that $n^{1/n}\rightarrow 1$ but how should I use this result. Thanks--Shahab (talk) 10:10, 31 August 2009 (UTC)

Lim x->0 (1+x)^1/x is e. Use this result, and come back if you have any problems. Rkr1991 ^{(Wanna chat?)} 10:24, 31 August 2009 (UTC)

That limit cannot be used, since we are taking a limit as n goes to infinity here.

However, notice that:

$an\leq an+b\leq (a+b)n$

$(an)^{1/n}\leq (an+b)^{1/n}\leq [(a+b)n]^{1/n}$

for n≥1. That should get you your answer. --COVIZAPIBETEFOKY (talk) 13:08, 31 August 2009 (UTC)

(of course, I assumed that b≥0, here; if it's not, simply reverse the signs) --COVIZAPIBETEFOKY (talk) 13:09, 31 August 2009 (UTC)

Thanks. However I can't resolve all cases still. What will happen if $a+b<0$ or $a<0$? Then I can't use the fact that $a^{1/n}\rightarrow 1$. Is the sequence divergent in those cases?--Shahab (talk) 14:49, 31 August 2009 (UTC)

The expression an + b is just a finite scalling and finite translation of n. Since x⁰ = 1 for any real number x, and you already know that n^1/n → 1 as n → ∞. I would say that (an + b)^1/n → 1 as n → ∞. Of course, depending on your choices of a and b you may have complex values for your sequence. This doesn't matter though because you will always have a real limit. For a complex number z we see that |z^1/n| → 1 as n → ∞ and (more importantly) arg(z^1/n) → 0 as n → ∞, i.e. z^1/n tends to a number with modulus 1 and argument 0, i.e. z^1/n → 1. ~~ Dr Dec (Talk) ~~ 16:10, 31 August 2009 (UTC)

I don't understand how you conclude (an + b)^1/n → 1 from n^1/n → 1. COVIZAPIBETEFOKY's approach made sense to me because I used the results related to algebra of sequences and the sandwitch theorem. Perhaps you can quote which result you are using. Thanks--Shahab (talk) 16:24, 31 August 2009 (UTC)

Indeed, COVIZAPIBETEFOKY's response gives a water-tight argument, and will lead you to a solution. I was trying to give some intuative comments as to why the limit is what it is. It is simple to see that the limit is 1, and I have shown that above. But intuition is not a proof, and of course you should use something along the lines of COVIZAPIBETEFOKY's rigorous analytical argument. If you can't see why my comments might lead you to conclude (and then try to prove) that the limit is one then maybe spend some time thinking it over. It's very natural and very intuative. The bottom line is that x⁰ = 1 for all fixed real numbers x. Basically, the only problem you get is from n^1/n. Once you know that it doesn't pose any problems then you're in the clear. ~~ Dr Dec (Talk) ~~ 17:21, 31 August 2009 (UTC)

My first solution didn't have accurate bounds in the general case; this one works (as long as a≥0, which has to be, since you can't always take an n^th root of a negative number), by taking the absolute value of b.

$a\leq an+b\leq (a+|b|)n$

$a^{1/n}\leq (an+b)^{1/n}\leq [(a+|b|)n]^{1/n}$

Where both equations are true for sufficiently large n.

This is pretty easy to generalize to any polynomial of arbitrary degree, by the way. --COVIZAPIBETEFOKY (talk) 17:16, 31 August 2009 (UTC)

Great. Thanks COVIZAPIBETEFOKY and Dr Dec!--Shahab (talk) 17:22, 31 August 2009 (UTC)

COVIZAPIBETEFOKY, there does seem to be one problem this the inequality

$a^{1/n}\leq (an+b)^{1/n}\leq [(a+|b|)n]^{1/n}.$

The sequence can take complex values, for example a = –1 and b = 0. When n is even we get imaginary numbers; so the inequality signs don't make such sense. Besides (–1)^1/4 has four possible values! Surely there should be some modulus signs? ~~ Dr Dec (Talk) ~~ 17:29, 31 August 2009 (UTC)

COVIZAPIBETEFOKY specified that his solution is only valid for a≥0 (although when a=0 b must be positive). In those cases we can assume we're talking about the positive real root. Otherwise the sequence diverges. And if we allow complex values then it's not particularly well defined and definitely doesn't converge in general although I guess you could set up a scheme to choose roots so that it would converge to any value on the unit circle you wanted. However in all cases (except a=b=0) it can be said that |(an + b)^1/n| goes to 1. Rckrone (talk) 18:09, 31 August 2009 (UTC)

I know, I was just prompting him to leave a revised inequality that involved muduli. ~~ Dr Dec (Talk) ~~ 18:13, 31 August 2009 (UTC)

p.s. the sequence does not diverge. It's limit is 1 for all real numbers a and b. As my post above showed: for any complex number z we have that |z^1/n| → 1 and arg(z^1/n) → 0 as n → ∞. This implies that the modulus of the sequence converges to 1 for all real number a and b. Find some a and b so that the sequence diverges. ~~ Dr Dec (Talk) ~~ 18:16, 31 August 2009 (UTC)

arg(z^1/n) doesn't go to 0 in general since z^1/n is ambiguous. The nth roots of z are evenly spaced around the circle. You could pick a value of arg(z^1/n) for each n to make a sequence that went to 0. You could also choose a sequence that goes to any value you like, or one that diverges. Just as an example, if for real a and b with a<0 we take the "obvious" root for each n odd, I would think that would be the real root (which is negative), in which case the odd terms go to -1. Rckrone (talk) 18:32, 31 August 2009 (UTC)

Really? Well, if $z=Re^{i(\theta +2\pi k)}\,$ where k is an integer then $z^{1/n}=R^{1/n}e^{i(\theta +2\pi k)/n}\,$ and clearly, for a fixed choice of k, $i(\theta +2\pi k)/n\to 0\,$ as $n\to \infty \,$. ~~ Dr Dec (Talk) ~~ 19:31, 31 August 2009 (UTC)

k is not fixed for all n. I can choose k_n to be any sequence of integers I want and it'll still produce a sequence of nth roots of z. Going to my example before: Let z = |z|e^iπ and choose k_n = n/2 for even n and k_n = (n-1)/2 for odd n. The nth term of my sequence is |z|^1/ne^iπ(n+1)/n for even n and |z|^1/ne^iπ for odd n. This sequence goes to -1 (the arguements go to π). It's not hard to see how you can create sequences that have lots of different behaviors depending on how you pick the nth root, and in the general complex case there's not an obvious "preferred" scheme for picking roots. Rckrone (talk) 19:58, 31 August 2009 (UTC)

(@dec)I would not worry about complex numbers; that exercise clearly refers to real numbers; it's quite implicit in the notation that a>0 and b is real. One could have asked the same question in a more general setting (complex numbers/matrices/linear operators/Banach algebras or whatever), but since nobody did, COVIZ's is, I think, exactly the right answer to the OP. --pma (talk) 18:18, 31 August 2009 (UTC)

The original question simply says that a and b are constant. It doesn't mention the positivity of a or b. COVIZ has the right solution. With a few moduli signs it would work even when a and b were taken to be complex. ~~ Dr Dec (Talk) ~~ 18:22, 31 August 2009 (UTC)

So, in my opinion the additional information on complex numbers may be of some interest, indeed; but in this and similar situations, it should be better to add it as a further remark or side remark, not as a correction to a totally satisfying answer. Otherwise, the OP will get confused, the first person who answerred will give you an explanation, you'll reply, I'll no help adding my boring remarks, a useless debeate will start, and this desk becames a mess again. It's not to criticize; I'm just looking with great admiration at those fellows here that are able to give short, concise, yet complete and illuminating explanations; let's try to learn how they do. --pma (talk) 21:38, 31 August 2009 (UTC)

Agreed! ~~ Dr Dec (Talk) ~~ 16:54, 1 September 2009 (UTC)

Millimeters

I speak of the probe on the left.

After reading the metre article, I still have a question regarding measurements -- as a dentist, I measure periodontal pockets (the sulci of gum tissue surrounding teeth in patients with periodontal disease) with a periodontal probe to the nearest mm. Because we (dentists in general) round to the nearest mm, and because there is some inherent intra- and interexaminer discrepancy due to probing force, velocity, angulation and the degree of inflammation present at the base of the sulci, any measurement is expected to be within 1 mm of what another competent dentist would record. My question -- since a mm is 1/1000 of a meter and a meter is arbitrary (in that there is no reason to have chosen the half-meridian for the original calculation), had a mm been 0.9 or 0.8 of our actual mm, wouldn't our measurements be more accurate? DRosenbach ^{(Talk | Contribs)} 12:53, 31 August 2009 (UTC)

With the smaller units, if you were still able to reliably measure within one "short" mm, then your measure would be more accurate. But I assume there would be a tradeoff between the this gain in accuracy and the loss in accuracy due to the examiner discrepancies. If the millimeter were smaller, it would be even more difficult for examiners to measure accurately. Staecker (talk) 13:06, 31 August 2009 (UTC)

True, but it's perfectly possible that the optimal trade-off between the stated precision, and the ability to reliably make measurements with that precision, is at an increment less than a mm. However, in other applications the optimum may be higher, so it's not universally preferable to choose a smaller meter. If this is significant you (where "you" is a guild of some sort) can always define your own optimal unit of measurement for your application. -- Meni Rosenfeld (talk) 20:25, 31 August 2009 (UTC)

There are two sources of error: the various sorts of individual variation that you mentioned, and the rounding to the nearest mm. If the definition of mm were changed, then the second source of error would be reduced (because you're rounding less) but the first would not be changed. If the first source of error is the dominant source of error (e.g., dentists can't be more accurate than 2 or 3mm because of measurement error), then changing the definition of mm will have negligible effect, whereas if the second source of error is the dominant source of error (e.g., dentists can measure down to 0.01 mm but this precision is lost because of rounding to the nearest 1mm), then rounding to a smaller unit would significantly reduce error. Eric. 216.27.191.178 (talk) 20:34, 31 August 2009 (UTC)

Octahemioctahedron

This is the octahemioctahedron, with tetrahedral symmetry... 3/2 3 | 3	...but what about this? Can it have octahedral symmetry?
Cantellated tetrahedron 3 3 | 2	Cuboctahedron 2 | 3 4

Does the octahemioctahedron have tetrahedral or octahedral symmetry? Since it's a facetting of the cuboctahedron with octahedral symmetry. Thanks. Professor M. Fiendish, Esq. 12:58, 31 August 2009 (UTC)

The article you linked - infobox - symmetry group. The answer is in there. It seems correct.83.100.250.79 (talk) 15:06, 31 August 2009 (UTC)

If you consider the eight triangular faces as being identical and indistinguishable (right hand diagrams) then it has octahedral symmetry. If you consider the triangular faces as separated into two families of four, distinguished by whether you are seeing the "top" of "bottom" surface of the triangle (red and blue triangles in the left hand diagrams) then the octahedral symmetry is broken and what remains is tetrahedral symmetry. Gandalf61 (talk) 18:21, 31 August 2009 (UTC)

Usually by "symmetries of a polyedron", or more generally "symmetries of a metric space", people mean its group of isometries... so I'd say the former you said is the natural choice. I suddenly had a suspect, and googoled "colored polyhedra" and "colored symmetries": they are there. Everything that may be thought by a human mind is already there. --pma (talk) 21:59, 31 August 2009 (UTC)

I contributed to the article and images. The Wythoff symbol 3/2 3 | 3 has tetrahedral symmetry, two triangle colors. There may be a secondary reflective construction with octahedral symmetryLooking at Schwarz triangle options, I don't see a simple form, maybe as a composite form?! I'm asking around by email too. Tom Ruen (talk) 22:33, 31 August 2009 (UTC)

OK, what about the tetrahemihexahedron? It's a facetting of the octahedron, which has octahedral symmetry.

Tetrahemihexahedron.

Especially since the dual, the tetrahemihexacron, is made up of 3 intersecting infinite square prisms (octahedral symmetry! There's a cube in the central intersection.)

Tetrahemihexacron.

Professor M. Fiendish, Esq. 04:42, 2 September 2009 (UTC)

Oh, maybe we should check the symmetries of the octahemioctacron.

Octahemioctacron.

Professor M. Fiendish, Esq. 04:42, 2 September 2009 (UTC)

Thank you, math reference desk

Hi. I'm just posting here to say that I passed my Complex Analysis qualifying exam, and you guys helped me a lot while I was studying for that. This page is one heck of a resource, and that wouldn't be true if not for the patient and friendly mathematicians who answer questions here. You guys rock, individually and as a group, and I wish a thousand blessings upon all your houses. :) -GTBacchus^(talk) 19:19, 31 August 2009 (UTC)

I second the above totally.--Shahab (talk) 05:09, 1 September 2009 (UTC)

Weighted sum of distance from a curvy line

Consider a busy road which is a curvy line on a flat plane. The sound intensity at every point on the curvy line is the same - call it a sound intensity level of 1, say. The sound intensity at an X,Y point on the plane is I think the sum of the noise from each point on the curvy line, except that the noise from each point is weighted by the inverse of its distance squared, since sound energy decreases with distance.

My question is - what would be the best practical way to calculate the sound intensity at an X,Y point on the plane?

All I have is a paper map which includes the road, and my maths is sub-calculus. Ideally I would like to have contour lines of the sound intensity over the plane, but just being able to calculate the sound intensity at two different points will do. I realise that this is an abstraction, but its a start. The calculations could be done manually or is there any suitable mapping software that could be used? Thanks 92.27.79.62 (talk) 20:30, 31 August 2009 (UTC)

Well, if $\textstyle \gamma :I\to \mathbb {R} ^{2};$ (or $\textstyle \mathbb {R} ^{3}$ if you like) is the arc-length parametrization of your road, the intensity at the point x it is proportional to $\textstyle \int _{I}|x-\gamma (s)|^{-2}ds$. here |v| denote the euclidean norm or v. Then, depending on the curve, it may be possible to compute the integral, or just evaluate it numerically. In any case I hope it is just an ideal calculus, and you do not have to face a real traffic noise! --pma (talk) 21:08, 31 August 2009 (UTC)Warning: In fact, the above integral was a mathematical representation of a quantity induced with a inverse square law by a source uniformly disributed on the curve γ, provided one assume a superposition principle. But I do not see why it should be true here. --pma (talk) 11:51, 2 September 2009 (UTC)

If it's a real road map, one way you could get an estimate manually for a specific point is to draw concentric circles around your point on the map with a compass. Then for each ring between consecutive circles estimate the length of road in the ring and take the distance away for all that road to be the average of the radii of the two circles. So if you have length d of road between circles of radius r and s, then the contribution of that part of the road is d/((r+s)/2)². Then sum up the contributions of all the rings. This is going to be least accurate for road that's very close to your point, so you would probably want to draw your circles closer together as you get closer to your point. Depending on how accurate you want to be, this could be a lot of work. Rckrone (talk) 21:11, 31 August 2009 (UTC)

Yes, but the sample points should be equally spaced on the road, to avoid having to weight each individually, (in what is already a time consuming process..) 83.100.250.79 (talk) 22:09, 31 August 2009 (UTC) Ignore this - had a better idea..:

I have a better idea - if you can get the x,y coordinates of a set of points along the road (choosing points so that the 'zig-zag' straight line is fairly close to the original road) - then enter the points into a spreadsheet (or specially prepared program) - the program could then calculate the contour lines for the sound levels - using the distance between adjacent points as a weighting factor, and the inverse square rule as well.

This way you only need to measure points once - but you can get an entire contour map made.

Would you be able to make a program to do that (any programming experience?) - it would be quite easy to write - someone would probably volunteer to do that on the computer desk, as it is not too difficult (I could write that if necessary).83.100.250.79 (talk) 22:30, 31 August 2009 (UTC)

Yeah, this method is much better than mine since you can get a computer to do most of the work without having to feed it too much data, and it's more accurate. Rckrone (talk) 22:41, 31 August 2009 (UTC)

Assuming there is no equation for the road - one solution would be to scan the map - and then use software to convert the road into a line of points (tricky bit - but possible) - then the sums of intensities to give contours would be actually trivial for the computer.

If you want to follow this route (pun!) then the hardest step is finding software which will take an image (assume you have scanned or photographed the map, and removed any stuff that is not the road) and convert it to a set of points describing the line/curve of the road. You should ask for this if this sounds useful to you. I would guess the maths desk would be better for the algorthym, but the computer desk would be more likely to know of software.

If you don't do this I think the solution will be a lot of hand measuring, and calculations.83.100.250.79 (talk) 22:06, 31 August 2009 (UTC)

Thanks. Remember as I said my maths is beneath calculus, so the first responce means nothing to me. I remember I have a computer program somewhere that is designed to digitise lines from images of graphs, so that would probably help. Or I could do it manually. This is a practical real problem, by the way, not an exercise. You can assume as a starting point that I have coordinates for points along the road. I have not done any amateur computer programming for a long time - the only language I was fluent with was GWBasic. I will try and find the digitisation program and get back to you later. 78.147.28.17 (talk) 10:20, 1 September 2009 (UTC)

There's a side-issue which has occurred to me - since there are an infinite number of points in a line, won't the sum of the sound from these points along the road be in theory infinate? Or at least, wont the calculated intensity of the sound at X,Y compared with that at the road be heavily dependant on how many points along the road I decide to use? If I use for example twice the number of road-points, then the total sound intensity is approximately doubled. 78.144.246.12 (talk) 14:44, 1 September 2009 (UTC)

If you sample the curve with 2x points, you divide the result with 2. You need to sample relatively evenly in space, eg with a rectangular grid. --91.145.89.58 (talk) 15:21, 1 September 2009 (UTC)

Yes using N points divide the total by N, also bear in mind to weight each point by the approximate length it represents. - using the weights - you then divide by the sum of the weighting lengths at the end, instead of N.83.100.250.79 (talk) 18:38, 1 September 2009 (UTC)

I'm not sure about the two comments above - if for example the road formed a hairpin bend, and you went and stood so that you had road noise on either side of you, then you would expect the intensity of the noise to add and be greater where you stood than on the road itself. Dividing the sum of the noise by the number of points, or the total of the inverse squared distance would not give this effect I think. 78.146.3.82 (talk) 20:38, 3 September 2009 (UTC)

It will. Noise intensity is directly proportional to the inverse square of distance from the source because it spreads in 3d space. In this case, it is also directly proportional to the length of the road you have, especially when the road is short. However if you have a long and curvy road the idea of distance is a bit vague as the road isn't exactly point-like and its length isn't always trivial to measure either. So you model the road as n shorter roads, ie you divide your original road to n pieces. The larger you make the n, the more point-like your pieces of road get and you get more accurate results. You compute the final sound intensity as the sum of the intensities of the n pieces. The intensity at a point by a piece is C * x / r**2, where C is some road-specific constant ("base intensity"), x the length of the piece, and r**2 distance from the road squared. If you use 2n pieces instead, x will be two times smaller on average and the result won't double. So ideally the whole sum is not divided, but in practise you might not want to bother find out the length of every piece and would take a shortcut approach instead. --194.197.235.240 (talk) 00:10, 4 September 2009 (UTC)

But I really I have some doubts about the validity of a superposition principle here. What is your definition of "sound intensity"? --pma (talk) 04:59, 4 September 2009 (UTC)

Sound intensity. I don't see why superposition wouldn't apply unless there are weird phase differences and resonance, but my physics are quite basic. --194.197.235.240 (talk) 05:58, 4 September 2009 (UTC)

Well, I would be a bit puzzled by a scalar physical quantity that obeys to both an inverse square law and to a superposition principle. The reason is that 1/|x|² is not a solution of any physically reasonable linear PDE in 2D or 3D. Said in another way, a superposition principle does hold for sound waves, what reflects their mathematical description as solution of the linear wave equation $\scriptstyle (\partial _{tt}-\Delta )w=0$; but the sound intensity as a scalar quantity, seems related in a nonlinear way to the sound waves, so that the additivity should be destroyed. In fact, your link clarifies this a bit: it defines sound intensity as a vector, which in the case of point source is radially directed and decays as 1/r². If the sound intensity defined that way also admits a (vectorial) superposition principle, what makes sense (but it is still unclear to me) then it would be very analogous to a Newtonian force, and it could be treated in terms of potential theory. --pma (talk) 14:49, 4 September 2009 (UTC)

It looks like from what has been written that I should divide the road up into small segments and take both the segment length and the noise intensity into account. So I could say the noise intensity is 1 per 1 metre of road. This would give a formula of I(x,y)=I(road) x L / (D**2) where I is the noise intensity, L is the length of the segment, and D is the distance between the road and the point. But this formula does not seem to be dimensionally balanced: the L / (D**2) part should be dimensionless for the two sides to balance, but it is not. Where am I going wrong please? 89.240.207.84 (talk) 14:51, 4 September 2009 (UTC)

It does not seem wrong; I(x,y) and I(road) in the formula just are not homogeneous quantities, as e.g. are not homogeneous the force and the mass in the analogous Newton law. --pma (talk) 14:58, 4 September 2009 (UTC)

Sorry I cannot believe that. I'm sure any true physical law would be dimensionally consistant. 78.146.70.127 (talk) 19:13, 4 September 2009 (UTC)

I mean, quoting for instance the above link: "For a spherical sound source, the intensity in the radial direction as a function of distance r from the centre of the source is: I_r=P_ac/4πr²..."; as you see there is no "sound intensity" term on the righ-hand side; correspondingly, the term denoted "I(road)" in your equation is not an intensity sound. But that is quite clear; the main point remains the fact that (I think) there is no superposition principle for this scalar sound intensity. --pma (talk) 20:04, 4 September 2009 (UTC)

No-no-no! The intensity we hear is proportional to the average of the pressure disturbance squared. We may very well take your “vector intensity” and call its module the scalar one I will subsequently be concerned about. Two different points/cars on the road certainly produce sound totally independently, that is, their phases and frequencies are different. Add a wavepacket $\scriptstyle p_{1}=A_{1}\sin(\omega _{1}t)$ and another one, $\scriptstyle p_{2}=A_{2}\sin(\omega _{2}t+\varphi )$. Let's choose such units that the intensity is the time average of the amplitude squared ($\scriptstyle p^{2}$). Then the intensities of the packets are, of course, $\scriptstyle I_{1}=\langle p_{1}^{2}\rangle =\lim \limits _{\tau \to \infty }{\frac {1}{\tau }}\int _{0}^{\tau }p_{1}^{2}={\frac {A_{1}^{2}}{2}}$ and similarly $\scriptstyle I_{2}=\langle p_{2}^{2}\rangle ={\frac {A_{2}^{2}}{2}}$ (just notice that both $\scriptstyle p^{2}$-s are sinusoids). Now find the total intensity you hear:

$I=\langle (p_{1}+p_{2})^{2}\rangle ={\begin{cases}I_{1}+I_{2}&{\text{if }}\omega _{1}\neq \omega _{2}\\I_{1}+I_{2}+2{\sqrt {I_{1}I_{2}}}\cos \varphi &{\text{if }}\omega _{1}=\omega _{2}{\text{.}}\end{cases}}$

The last formula is easily provable imagining the $\scriptstyle p$-s as a projections of phasors and using the cosine law (and almost being wrong like me, when I first misplaced the $\scriptstyle \varphi$ in my mental diagram $\scriptstyle {\ddot {\smile }}$). So the superposition principle does not hold only in an unphysical special case! The wavepackets to be added have, in reality, always different frequencies and thus are incoherent. Even if they had the same frequency $\scriptstyle \omega$, but random phases, superposition of intensities would still hold if we average over the phase shift (and the wavepackets are short if they're not being specifically generated, so the averaging is what our brain does): then

$I=\left\langle {\frac {1}{2\pi }}\int _{0}^{2\pi }\left[A_{1}\sin(\omega t)+A_{2}\sin(\omega t+\varphi )\right]^{2}d\varphi \right\rangle _{t}=$

$=\left\langle I_{1}+I_{2}-I_{1}\cos(2\omega t)\right\rangle _{t}=I_{1}+I_{2}{\text{.}}$

Oh my, if it were otherwise: we would be maddened about all that interference, especially visually, since all the theory applies also in optics… I know that the electric field $\scriptstyle {\vec {E}}$ is a vector there, however, interference is most easily discussed between waves with the same polarization, so they still add up scalarly. In optics, Poynting vector is the “vector intensity” and its time-averaged magnitude the usual intensity AKA irradiance. I am trying to think about the PDE $\scriptstyle I$ follows, but here I am as lost as pma was. What about an integro-differential equation? I don't say I already have one, though. Also, a superposition principle that does not hold in the most general case (just like here) does not imply linearity at all: we have solitons etc, although, in turn, non-interaction of solitons does not imply their linear superposition. — Pt _(T) 20:57, 7 September 2009 (UTC)

On another thought, this conditionality of the superposition is irrelevant here: it concerns only the way intensities arise from pressure waves, not the intensities themselves: you can always change the frequency of any source to make it incoherent with everything else, thus restoring the superposition. Thus, the question now is: if we know that the intensity obeys both superposition law and inverse-square fall-off (because of symmetry and energy conservation), then which PDE (or alike) does it follow? May it be nonlinear? (Sorry, original poster, this has become interesting in its own right! Actually, this could give us a ready-made mathematical machinery just for you.)  — Pt _(T) 21:16, 7 September 2009 (UTC)

Actually, electromagnetism gives us a PDE! Coulomb's law is an inverse-square vector equation that after integration gives rise to a inverse-first-power potential law, a scalar equation written as a Poisson equation as $\scriptstyle \Delta \varphi \propto -\rho$, where $\scriptstyle \rho$ is the density of sources and the coefficient of proportionality can be made 1 by choice of units. Now, this inverse-square dependence comes from the fact that in 3 space dimensions the area of a sphere is proportional to its radius squared, and Gauss' law. In 4 dimensions (not Minkowski spacetime, but just 4 Euclidean dimensions), the sphere's area $\scriptstyle \propto r^{3}$, so the Gauss' law ($\scriptstyle \operatorname {div} {\vec {E}}\propto \rho$) gives now rise to an inverse-cube vector equation and integrating (going from force to work) gives us a inverse-square linear scalar equation! It is simply a 4-dimensional Poisson equation, which we now write for acoustics again ($\scriptstyle \varphi \mapsto I$):

$\sum _{i=1}^{4}\partial _{x_{i}x_{i}}I\propto -\rho {\text{.}}$

Hence we have to put in a known source density $\scriptstyle \rho$ (which can be non-zero only where $\scriptstyle x_{3}=x_{4}=0$, since our original problem is 2D), solve the equation in 4 dimensions while assuming the solution to vanish at infinity (is this boundary condition enough?) and finally restrict it to the 2D plane we are interested in. If the road is infinitely thin (and in the two extra dimensions it has to be!), $\scriptstyle \rho$ is no longer an ordinary function, but a distribution involving the Dirac delta function, but it should not be a problem. Finally, someone $\scriptstyle {\overset {\_\,.}{\smile }}$ should explain a good numerical Poisson equation method to the original poster! — Pt _(T) 00:57, 8 September 2009 (UTC)