For physicists (again)
Offline Does anyone know how (or where) to derive the arbitrary n dimensional free space scalar wave equation of the following form?:
(∆ -c2 ∂2/∂t2)G=0 where ∆ is the Laplacian
I am not asking for the solution to the equation. That’s easy enough to derive. But I remember doing this quite well. When we studied the wave equation, we derived it in the same way d’Alembert did, by considering a 1D case for a transverse vibrating string. To derive the arbitrary n dimensional one is much, much harder. A few days ago I realized that I had never derived the equation. I just used it to solve problems in electrodynamics and quantum mechanics, and I hate using equations I have never derived.
At first I thought I could show it to be the case easily enough with a scalar transport equation of the following form for a conserved quantity given by the scalar field G(x,y,z...n) in n dimensions.
(∂/∂t)G=div cG+source term
It’s probably easiest to begin with a delta-dirac for the source term, but it doesn’t matter what it is (so long as it has no time derivative). However, if we treat the wave as a scalar field h where h is the amplitude function, where the phase velocity is c and substitute in, the wrong answer is delivered (it gives out the RHS as div(div) (that is, all of the second partial derivatives, including all the mixed ones) when it should give the Laplacian). At first I was confused by this until I realized that the following:
(∂h/∂t) =div c(h)+source term
Is invalid because h is not a quantity being transported by the wave. Thus it does not obey the scalar transport equation. Ever since then I’ve been trying to re-formulate a valid transport equation for the wave. At first I thought I had a breakthrough when I formed a vector field c(grad h) which gives the rays of the wave, which in source-less space must be conserved (that is, the closed surface integral of the vector field for source-less space must be zero), but I was unable to make progress (no way to link the vector field to the time derivative of the scalar field equation).
Does anyone know how to approach this or find a website that does? I’ve been looking all over, but all I can find (in textbooks, on web pages etc.) is derivations in terms of the 1D strings that d’Alembert studied (or 1D derivations using the scalar transport equation where it does work because there is no difference between div(div) and the Laplacian).
Note: For those of you confused as to why I have used the term div(div) when, strictly speaking, it makes no sense whatsoever (div is a vector operator that turns out a scalar), the abuse of notation comes from the loose way in which "c" is treated as a vector in this context (that is, we consider it constant and take it out of the derivative). This may be part of the problem in formulating a valid scalar transport equation.
EDIT: Also, in the post, G is NOT Green's Function, I just happened to use it to denote whatever (arbitrary) scalar wave equation could come to mind.
"Physical reality” isn’t some arbitrary demarcation. It is defined in terms of what we can systematically investigate, directly or not, by means of our senses. It is preposterous to assert that the process of systematic scientific reasoning arbitrarily excludes “non-physical explanations” because the very notion of “non-physical explanation” is contradictory.
-Me
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First and foremost: Assume the cow is a sphere
I'm going to bold vectors for this and X denotes the cross product.
The displacement function is u(x,y,t..)
Take the tensile force: FT=TotXn
where the vertical component is given by FT[dot product]k
By Newton's law the force is mass (given by mass density po multiplied by surface area A.) multiplied by the rate of acceleration (second deriviative of position with respect to time.. d2u/dt2)
Now take that over the whole area
[double integral]po[d2u/dt2 ]=[surface integral]TotXn[dotproduct]k
Now [surface integral]TotXn[dotproduct]k , by stoke's theorem is
[double integral]To[∆X(nXk.)][dot product]n DA
Since the region is arbitrary:
po[d2u/dt2 ]=To[∆X(nXk.)][dot product]n
where n is of course the normal to the virbrating membrane.
Now we can calculate the n as follows
n~-du/dx-du/dy+ k
Now we can take nXk
which is
-du/dy+du/dx
Now for
∆X[nXk]
which is clearly k[∆]
So sub back in to original equation
pod2u/dt2=To∆
Now
d2u/dt2=G∆
Where G=To/po
I think it's better to use c2 for the constant.
Excellent. Thanks very much.
Amazing that I couldn't find that anywhere. Not even in textbooks.
Hang on. A few things:
What is "t"?
2. In your derivation, you began with an expression for the surface integral of the vertical component of tensile stress across the membrane and turned out a double integral of the curl of nxk and projecting the vector onto the unit normal vector. This is still a surface integral, since ndA is the infnitesimal surface normal, but I'm not sure how you derive this from Stoke's Theorem, which states that the surface integral of the curl of a vector field is equal to the line integral of the vector field over the boundary.
3. You defined the normal to the membrane as:
n~-du/dx-du/dy+ k
This doesn't mean anything (you are adding scalars and vectors). Unless, of course, you are taking the gradient of u and forming the two resulting vector components (du/dx and du/dy and then subtracting them from their respective components of k). Even if this were the case, the notation is still suspect. k is the force needed to move the particle by one unit orthogonal distance, and du/dx (or dy) is just the rate of change of amplitude per unit change in one of the axes. So (1) how did you obtain this expression for the unit normal and (2) you are adding two quantities that don't have the same units (du/dx doesn't even have units).
"Physical reality” isn’t some arbitrary demarcation. It is defined in terms of what we can systematically investigate, directly or not, by means of our senses. It is preposterous to assert that the process of systematic scientific reasoning arbitrarily excludes “non-physical explanations” because the very notion of “non-physical explanation” is contradictory.
-Me
Books about atheism
Sorry, I kind of wrote this in a hurry and skipped a few steps.
't' is the tangent vector to the edge. There's a diagram in my text, but I can't draw it here obviously.
My book writes Stoke's theorem as:
[double integral]∆XB[dot product n] dA =[surface integral] B[dot product]t ds
Just sub in B=[nXk] [since the cross product always produces a vector]
Whoops, I kinda messed up there I kinda left out the unit vectors [except for the k lol]
My book was using the notation
A=xi+yj+zk
so A=[x,y,z]
So now
take z=u(x,y)
so gradient would be dz/dzk-du/dxi-du/dyj
I kinda left out the 'i' and 'j' for du/dx and du/dy respectivly in my original post.
so the gradient is du/dxi-du/dyj+1k
divide that by
SQRT[[du/dx]2+[du/dy]2+12]
since du/dx and du/dy are assumed small, SQRT[[du/dx]2+[du/dy]2+12]~1 so
n=-du/dxi-du/dyj+1k
n=[-du/dx,-du/dy,1]
and k=[0,0,1]
so nXk
skipped that step.
which is -du/dyi+du/dxj
I dropped the other unit vectors except k sorry about that.
Since I suck at explaining things here's a PDF from the University of British Columbia
http://www.math.ubc.ca/~feldman/apps/wave.pdf
Yeah, I've seen that PDF before, but I know that derivation. What you did (which I couldn't find anywhere, what book do you use?) was to extend the derivation into multiple dimensions.
"Physical reality” isn’t some arbitrary demarcation. It is defined in terms of what we can systematically investigate, directly or not, by means of our senses. It is preposterous to assert that the process of systematic scientific reasoning arbitrarily excludes “non-physical explanations” because the very notion of “non-physical explanation” is contradictory.
-Me
Books about atheism
http://www.amazon.com/Applied-Partial-Differential-Equations-4th/dp/0130652431