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

Hello Daniel,

> -----Original Message-----
> From: maxima-bounces at math.utexas.edu
> [mailto:maxima-bounces at math.utexas.edu]On Behalf Of Daniel Lakeland
> Sent: Thursday, July 12, 2007 4:18 AM
> To: maxima at math.utexas.edu
> Subject: Re: [Maxima] [Newbie:] How to discard higher orders of
> variables?
> 
> 
> On Wed, Jul 11, 2007 at 09:54:23PM -0400, Stavros Macrakis wrote:
> > Another technique to consider for multiple variables <<1 is
> > 
> >          taylor( ..., [x,y,z], 0, 5)
> > 
> > which truncates at terms beyond x^a*y^b*z^c where a+b+c > 5
> 
> Good thought. In fact here's some interesting results from the
> original problem:
> 
> (%i23) det : determinant(Q);
> 
> (%o23) (kappa^2-Omega^2)*((1-nu)*(3*beta^2+1)*kappa^2/2-Omega^2)
> 			*(-Omega^2+beta^2*(kappa^4+1)+1)
> 	
> -(-beta^2*kappa^3-nu*kappa)^2*((1-nu)*(3*beta^2+1)*kappa^2/2-Omega^2)
> (%i24) factor(det);
> 
> (%o24) 
> -(2*Omega^2+3*nu*beta^2*kappa^2-3*beta^2*kappa^2+nu*kappa^2-kappa^2)
> 	*(Omega^4-beta^2*kappa^4*Omega^2-kappa^2*Omega^2-beta^2*Omega^2
> 		 
> -Omega^2-beta^4*kappa^6+beta^2*kappa^6-2*nu*beta^2*kappa^4
> 		 +beta^2*kappa^2-nu^2*kappa^2+kappa^2)
> 	/2
> 
> /* So det is a product of two factors... let's take it apart and solve
> each one seperately*/
> 
> (%i25) pickapart(%,3);
> 
> (%t25) 
> 2*Omega^2+3*nu*beta^2*kappa^2-3*beta^2*kappa^2+nu*kappa^2-kappa^2
> 
> 
> (%t26) 
> Omega^4-beta^2*kappa^4*Omega^2-kappa^2*Omega^2-beta^2*Omega^2-Omega^2
> 	      -beta^4*kappa^6+beta^2*kappa^6-2*nu*beta^2*kappa^4
> 	      +beta^2*kappa^2-nu^2*kappa^2+kappa^2
> 
> (%o26) -%t25*%t26/2
> (%i27) solve(%t25,kappa);
> 
> (%o27) [kappa = -sqrt(2)*Omega/(sqrt(1-nu)*sqrt(3*beta^2+1)),
> 	kappa = sqrt(2)*Omega/(sqrt(1-nu)*sqrt(3*beta^2+1))]
> 

These solutions correspond to the torsional vibration of the pipe, which
physically is decoupled from the other modes. See my previous mail for more
explanations.

> /* These are the simplest expressions for kappa that solve the
> determinant equation and come from the first factor, we know we can
> get several more solutions, but that they are extremely long, so
> instead lets use a taylor expansion of the remaining factor*/
> 
> 
> (%i29) taylor(%t26,[kappa,Omega,beta],0,4);
> 
> (%o29) -Omega^2+(-nu^2+1)*kappa^2+((-Omega^2+beta^2)*kappa^2
> 				  +Omega^4-beta^2*Omega^2)
> (%i30) solve(%,kappa);
> 
> (%o30) [kappa = -Omega*sqrt(Omega^2/(Omega^2-beta^2+nu^2-1)
> 			     -beta^2/(Omega^2-beta^2+nu^2-1)
> 			     -1/(Omega^2-beta^2+nu^2-1)),
> 	kappa = Omega*sqrt(Omega^2/(Omega^2-beta^2+nu^2-1)
> 			    -beta^2/(Omega^2-beta^2+nu^2-1)
> 			    -1/(Omega^2-beta^2+nu^2-1))]
> 
> /* These are two approximate solutions for kappa based on a truncated
> taylor series, so long as kappa, Omega, and beta are all relatively
> small, these should be close to the correct value. Let's 
> simplify this.*/
> 
> ratsimp(%o30);
> 
> (%o45) [kappa = 
> -Omega*sqrt((Omega^2-beta^2-1)/(Omega^2-beta^2+nu^2-1)),
> 	kappa = Omega*sqrt((Omega^2-beta^2-1)/(Omega^2-beta^2+nu^2-1))]
> 

With assuming Omega being small compared to nu (the Poisson ratio, typically
something like 0.3 for metals) and beta^2 small compared to 1 (one of
Fl?gges approximations), this gives the generally accepted  wave speed of
the n=o bending wave (somtimes also called longitudinal wave), which is a
travelling wave for all frequencies, at low frequencies (kappa = +/- Omega
/ sqrt(1 - nu^2) is the generally accepted low frequency approximation). So
the solution looks reliable (in terms of the physics of the problem). 

> So the 4 approximate expressions for kappa are:
> 
>  append(%o27,%o45);
> 
> (%o46) [kappa = -sqrt(2)*Omega/(sqrt(1-nu)*sqrt(3*beta^2+1)),
> 	kappa = sqrt(2)*Omega/(sqrt(1-nu)*sqrt(3*beta^2+1)),
> 	kappa = -Omega*sqrt((Omega^2-beta^2-1)/(Omega^2-beta^2+nu^2-1)),
> 	kappa = Omega*sqrt((Omega^2-beta^2-1)/(Omega^2-beta^2+nu^2-1))]
> 
> Dunno if those help, but the process was sufficiently entertaining,
> and hopefully informative for some readers :-)

Somehow (probably by discarding the higher powers of kappa), the 4 remaining
solutions (expected to be complex valued at low frequencies) got lost - I
will try some things to get them in... 

Thanks a lot for all ideas and help - and I will keep you informed if I
hopefully get some more insight and solutions...

Rolf