[Newbie:] How to discard higher orders of variables?

Hello Daniel,

thanks for your interest and support. I will try to give some comments and
physical interpretation.

> -----Original Message-----
> From: maxima-bounces at math.utexas.edu
> [mailto:maxima-bounces at math.utexas.edu]On Behalf Of Daniel Lakeland
> Sent: Wednesday, July 11, 2007 11:31 PM
> To: maxima at math.utexas.edu
> Subject: Re: [Maxima] [Newbie:] How to discard higher orders of
> variables?
> 
> 
> On Wed, Jul 11, 2007 at 09:46:12PM +0200, Schirmacher, Rolf wrote:
> > Hello Daniel,
> 
> Hello Rolf, 
> 
> Thanks for the more detailed description of your problem. I am
> actually quite interested in this type of problem (I have recently
> been doing some fluid mechanics problems involving similar algebraic
> simplification issues), but I am not very familiar with thin shell
> equations specifically, nor do I have your references available.
>

To be honest, I did not really expect that. I just mentioned them to give a
pointer in case ...
 
> It is extremely helpful to use maxima for reproducing other's
> algebra. or even for inventing your own so I welcome you to the group!
> 
> > So, what would the "real world" look like? I have attached 
> a wxMaxima input
> > file as a starting point. Everything looks quite simple so 
> far. The next
> > step would be to loosen the n:0 restriction and generate 
> cut-on frequency
> > formulas for arbitrary n. These formulas are given in the 
> literature, but at
> > the moment, reproducing them includes some manual 
> substitutions for me. You
> > might try, it is not that bad...
> > Even more interesting would be a general solution of the 
> det=0 equation
> > (might be bound to n=0 for the moment) for kappa, i.e. the 
> dispersion
> > equation. All you currently find in the literature (as far 
> as I know) are
> > plots and analytical approximations for small Omega 
> (limited to something
> > like Omega < 0.05 for n = 1). But be carefull, the terms of 
> the complete
> > solution will fill quite some screens on your computer... 
> So, simplifying /
> > approximating the det before solving for kappa should help ...
> 
> 
> > So, I hope, this is not too boaring to you and perhaps you 
> might have some
> > helpful hints?
> 
> I've taken a look at your file, and given that I have only a very
> limited idea of what you're trying to do, I have some observations as
> follows:
> 
> 
> kappa can be solved directly by maxima using solve(det,kappa) but it
> produces, as you say, many many terms in some of the solutions. The
> use of "ratsimp(det,kappa)" shows that the equation is an 8th order
> polynomial, so it should have 8 solutions. 

Yes, 8 solutions are expected. In fact, it are four solutions with + and
symmetrically four with - representing travelling waves in + / - axial
direction (standing waves).
 
Real valued solutions indicate travelling waves (i.e. power transport) while
imaginary / complex solutions represent evanescent waves (nearfields, etc.),
which are mostly out of interest. The cut on frequencies found by setting
kappa=0 are those frequencies where a mode type switches over from being
evanescent to travelling, i.e. the frequency where power flow starts for
some mode type.

> I'm impressed that maxima
> can find them all, given the general nontractability of 5th and higher
> order polynomials. However, perhaps some of these expansive solutions
> are uninteresting?  As I say, I am not familiar with the specifics of
> the theory. 

> Two of the solutions for kappa are quite simple, but I
> don't know how physically relevant they are.

The two simple solutions are the transvesal shear wave, for n=0 that is the
torsion of the pipe. Due to the modelling of the shells, they do not couple
with the other modes, so they can be easily separated and turn out to be
quite simple: depending on the shell material properties only, not on the
geometry of the shell/pipe (and therefore not containing any beta terms
...).

The other solutions are coupled by Poisson coupling and geometrical issues,
so they do not separate...

> 
> If I assume that kappa, Omega, beta are all smallish, and that
> therefore products of high powers of these are relatively unimportant,
> then I can simplify quite a bit...
> 

Sorry, this is a bit too simple for most applications (and might
nevertheless be helpful to get some insight). beta can be assumed to be
small (it is h / ( sqrt(12) * a), with h being the  thickness of the shell /
pipe wall and a being the radius of the shell / pipe, so typically it is
beta < 0.1).

Omega is some normalised frequency, the normalisation is done by the so
called "ring frequency", which is the frequency at which a compressional
wavelength equals the pipe circumference. At this frequency, the general
behaviour of the modes change, many additional come in (resulting in a high
modal density), etc. So, for acoustical issues (sound radiation, etc.) it
would be of interest not to be bound to small Omega, but be able to go up
to, say, Omega = 5 or so (depending on the problem of interest, the shell
geometry, etc.). For vibrational issues like pipe bending, Omega is
typically much smaller, say Omega < 0.1 or so. 

kappa is linked to Omega due to the nature of the problem. You might
understand the dispersion equation as some indication of the speed of a wave
as a function of frequency. kappa increasing linearly with Omega is a
constant speed over frequency (which will be related to shell material,
geometry, etc.), but kappa with other dependencies on Omega gives a more
complex behavour. For bending waves, you will typically find something like
kappa = const * sqrt(Omega).

As a result, discarding higher powers in kappa (as done by
ratsubst(0,beta^2*kappa^6,%) below) definitely leads to a "wrong"
simplification with regard to the physical problem. Cutting powers in beta
is o.k., cutting powers in Omega is acceptable for low frequency solutions.

> ratsimp(det);
> ratsubst(0,Omega^3,%);
> ratsubst(0,beta^3,%);
> ratsubst(0,beta^2*kappa^6,%);
> solve(%,kappa);
> 
> gives some almost tractable solutions... 
> 
> kappa = 0 or +- a largish expression...
> 
> if I assume in this expression that beta^2*Omega^2 = 0 I get:
> 
>                                                 2       2
> (%o109) [kappa = - Omega sqrt(- ((4 nu - 6) beta  + 2 nu  + nu - 3)
>                 2        3       2                 2     3     2
> /((nu - 1) Omega  + (3 nu  - 3 nu  - 4 nu + 4) beta  + nu  - 
> nu  - nu + 1)),
>                                      2       2
> kappa = Omega sqrt(- ((4 nu - 6) beta  + 2 nu  + nu - 3)
>                 2        3       2                 2     3     2
> /((nu - 1) Omega  + (3 nu  - 3 nu  - 4 nu + 4) beta  + nu  - 
> nu  - nu + 1)),
> kappa = 0]
> 
> Good luck though, those were a lot of assumptions... Let us know how
> it goes.
> 

Again, thanks for your support and interest in the equations.

Rolf