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September 15, 2015, 08:43 
Should translation change eigenvalues?

#1 
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Jingchang.Shi
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Hello, everyone!
I'm working on flow instability and just a newbie. Now, I try to employ spectral method  collocation method to solve OrrSommerfeld(OS) equation. I've already solved the eigenvalue problem of Poiseuille flow. And I want to move to Boundary layer instability., during which I come across a problem concerning coordinates transformation. The problem is mainly a mathematical one. Specificlly, OS equation is as follows. Mean velocity profile of Poiseuille flow is . What I concerns is c, wave speed, also eigenvalue. I treat OS equation as a general eigenvalue problem, which leads OS equation to be of the form , where both A and B are operators. By some eigenvalue solver, eigenvalues can be solved. Mathematically, eigenvalues are determined by operators A and B. Ok, I finish the above. Then I consider coordinate transformation. Take a simple example as translation: . It translates field from to . Since the field is translated, operators A and B should be changed to be defined on the new field. Specifically, partial differential operator and velocity profile should be changed. In this case, coordinate transformation is just translation. So, partial differential operator is unchanged and velocity profile becomes . All in all, what changes in this eigenvalue problem is only velocity profile. It seems operator indeed changes, which results in different eigenvalues from original ones. Then there exists contradiction. I mean, mathematically, different operators in eigenvalue problem will lead to different eigenvalues. Meanwhile, physically, coordinate translation leads to the same eigenvalues. In fact, I do get same eigenvalues numerically by spectral method  collocation method. So, where I am wrong from the above? Last edited by aerosjc; September 15, 2015 at 20:26. 

September 18, 2015, 19:49 

#2  
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Lucky
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Quote:
Your eigenvectors will look different in the new basis, but it is the same vector (you still have the same velocity profile, but it is expressed differently in your new variables). Scalars are not transformed so your eigenvalues in the new basis are the same as the old eigenvalues. During the coordinate transformation, the A and B will change along with the eigenvectors but the eigenvalues will remain the same. Last edited by LuckyTran; September 18, 2015 at 21:34. 

September 18, 2015, 20:07 

#3 
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Jingchang.Shi
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September 18, 2015, 20:57 

#4  
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Lucky
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Quote:
Under linear transformation theory with this coordinate transformation, the vectors become stretched and so the eigenvalues do change. Recall that the eigenvalues tell you how much the eigenvector is stretched by the transformation. Before the eigenvalues were constant because a translation cannot stretch a vector, but stretching the coordinates can stretch a vector and hence change an eigenvalue (the eigenvalue is a measure of stretch). However, if you generalize the definition of eigenvalues to nonlinear systems, then you can find these nonlinear eigenvalues which are invariant under nonlinear coordinate transformations. 

September 18, 2015, 21:05 

#5 
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Jingchang.Shi
Join Date: Aug 2012
Location: Hang Zhou, China
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Thank you!
Although I've read some math, I just forget about them when addressing some problem. I will review the invariance of eigenvalues. For "these nonlinear eigenvalues which are invariant under nonlinear coordinate transformations", I need to google it. 

Tags 
eigenvalues, orrsomerfeld, spectral method 
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