Chaos theory applied to waveguide optimization

[thadman]

If we wished to fully discern the transient behavior of waveguide coupled compression drivers for optimization purposes, a direct numerical simulation of the Navier Stokes second order partial differential equations would be required. However, the non-linearity of the Navier Stokes Equations means they form a fundamentally chaotic system which can not be predicted exactly.

I believe many have observed significant solution divergence at high frequencies (>10khz). Perhaps applying deterministic chaos theory towards the Navier Stokes equations may offer further improvements in understanding the behavior of waveguides.

Optimization could include restriction of the state space and minimization of the Lyapunov exponent.

Any thoughts?

[thadman]

C'mon, I know some of you have an interest in Chaos theory

[Jmmlc]

Hello,

In my institution we use dayly the Navier-Stockes equation to solves hydrogeological problems I don't see at the first glance where is relation between the Navier-Stockes equation and the chaos theory.

Non linearity doesn't mean randomness nor impredictability...

Best regards from Paris, France

[thadman]

Originally Posted by Jmmlc

Hello,

In my institution we use dayly the Navier-Stockes equation to solves hydrogeological problems I don't see at the first glance where is relation between the Navier-Stockes equation and the chaos theory.

Non linearity doesn't mean randomness nor impredictability...

Best regards from Paris, France

Jmmlc,

To solve the Navier-Stokes equations, we must do a numerical analysis as analytical solutions do not exist (except for highly idealized situations).

A Chaotic response can result from simple deterministic laws. We can show this with a simple equation.

x(t+1)=1.9 - x(t)^2

1.9-(1^2)=.9

1.9-(.9^2)=1.09

1.9-(1.09^2)=.712

1.9-(.712^2)=1.393

While at MIT, Lorenz conducted significant numerical analysis of weather patterns. He re-ran a print off from his previous data set and found the solution to be radically different from the initial solution. This radical difference was simply due to an incredibly small rounding error of the decimal points.

This phenomenon will permeate all numerical solutions as the initial conditions can only be described with a finite accuracy

Assuming the Finite Difference method is used, all terms of the third order or higher in the Taylor Series approximation are ignored. So right from the beginning, a numerical approximation is introduced and from sensitive dependence on initial conditions, this error can grow as iteration proceeds, producing a different result each time.

We can study the stability of these linear difference equations by using Neumann stability analysis, without considering chaos. Even with the Euler Explicit Form of the simplest one dimensional wave equation:

(∂u/∂t)+c(∂u/∂x)=0

the von Neumann stability analysis shows that this equation leads to an unstable solution no matter what the value of the time step. It is unconditionally unstable.

Originally Posted by Peitgen, Jurgens and Saupe

"The relation of the original differential equation to its numerical approximation is very delicate - the stability conditions show that. Changing over to a discrete approximation may change the nature of a problem significantly, a fact which has only entered the consciousness of numerical analysts quite recently. This is another merit of chaos theory".

[jaencer]

I dont understand the significance of this post...

[thadman]

Originally Posted by jaencer

I dont understand the significance of this post...

An analysis of chaos theory should help to define the trajectory of a system throughout the state space.

[rcw]

In a post about dsp correction of compression drivers in this forum, (I have forgotten the details), the Fourier transform of the pulse response did show indications of the kind of noise that is characteristic of the onset of chaotic behavior.

The typical compression driver does have what amounts to a large number of coupled oscillators that at some frequencies operate close to the region where chaotic behavior might well occur.

Running a study on this might be academically interesting but in the end all you have to do is to make a compression driver with a ring radiator and an annular cavity that has only one Helmholtz resonance, restricting the initial phase space vectors to a set that will not evolve into chaotic behavior.
rcw.

[JLH]

Originally Posted by rcw

In a post about dsp correction of compression drivers in this forum, (I have forgotten the details), the Fourier transform of the pulse response did show indications of the kind of noise that is characteristic of the onset of chaotic behavior.

The typical compression driver does have what amounts to a large number of coupled oscillators that at some frequencies operate close to the region where chaotic behavior might well occur.

Running a study on this might be academically interesting but in the end all you have to do is to make a compression driver with a ring radiator and an annular cavity that has only one Helmholtz resonance, restricting the initial phase space vectors to a set that will not evolve into chaotic behavior.
rcw.

So in other words we just need to use BMS compression drivers.

[Andrea:]

Chaos Theory? Utter speculative idiotic nonsense?

[thadman]

Originally Posted by rcw

In a post about dsp correction of compression drivers in this forum, (I have forgotten the details), the Fourier transform of the pulse response did show indications of the kind of noise that is characteristic of the onset of chaotic behavior.

The typical compression driver does have what amounts to a large number of coupled oscillators that at some frequencies operate close to the region where chaotic behavior might well occur.

Running a study on this might be academically interesting but in the end all you have to do is to make a compression driver with a ring radiator and an annular cavity that has only one Helmholtz resonance, restricting the initial phase space vectors to a set that will not evolve into chaotic behavior.
rcw.

Does such a model incorporate higher order modes as well as the non-linear viscous damping of the diaphragm's breakup modes? I would tend to think a high-dimensional full simulation, which includes all system non-linearities, would be significantly more complex and would benefit from a chaos analysis.

Regardless, Chaos theory should allow insight into what's going on beyond the limitation of the Lyapunov exponent (systems are only predictable to a finite time since infinite precision is not possible), where numerical errors sum to a gross error. It would also alleviate the issue with defining initial conditions. With a continuous signal (music), I would expect the initial conditions to be anything but invariant.