User:LightYear

This be me and my page, and I welcome you. The handles LightYear and LiteYear are, in many cases, Heath Raftery, presently of sunny Mayfield in the land Down Under. There are reports of time spent at Glen Innes High School, and further activities at University of Newcastle. Newcastle is now my home and I'm happy to settle here for a few years.

Background
My educational background is in Mathematics and Computer Engineering, and it is from those disciplines that most of my mental passions and pursuits arise. Computer Engineering, despite common impression, is not generally about using computers. It is much closer to Electrical Engineering than to Information Technology for example. In fact, it differs from Electrical Engineering only in a little borrowing from Software Engineering, and an exploration into microprocessors and embedded design. In the interest of equal time, my Mathematics degree also concerned themes different to common impression. To the surprise of many, I have not been trained to be particularly good at multiplying multiple-digit numbers in my head. In fact, very little of my degree concerned arithmetic - I rely on a calculator for all but the most basic operations. On the other hand, I have at my disposal very powerful ways of visualising problems, and the knowledge of many tools to solve problems that not many people would associate with mathematics.

My current areas of passion are embedded systems that change people's lives, social software that does the same, the Web2.0 phenomena and entrepreneurialism. My professional pursuits have steered towards heavy industrial product development.

I use and support the Mac, C, Objective-C, and open source.

Outside the theoretical realm I enjoy riding, both mountain and motor. Soccer is my favourite sport, but I'm more of a player than a watcher. I practice martial arts and rock climbing, although not at the same time. I love a beer or ten, BBQs, socialising, live music and solving the world's problems.

I rant and rave on my blog. I conduct business under the name HRSoftWorks.

Wikipedia Interests
My contributions around these parts are generally in the field of science and engineering, but occasionally other topics catch my interest. Recently, I've authored or contributed to the following pages


 * Dependent and independent variables
 * Ultrasound
 * Barkhausen effect
 * Joule
 * Tower of Terror (roller coaster)
 * Versus
 * Chart
 * Line chart
 * Drop (unit)
 * Drop (liquid)
 * Good To Great
 * Stockdale paradox
 * Ultraviolet

Nonlinear acoustics
Working on this at the moment, but not ready to publish yet. LightYear 06:19, 15 January 2007 (UTC)

Traditionally, the propagation of an acoustic wave is modeled using an [[LTI_system_theory|LTI (Linear Time-Invariant)] system. One important characteristic of such a system is that if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, but always with the same frequency.

If our acoustic wave consists of a signal frequency, than it may be represented as $$x(t) = A \sin (2 \pi f t)$$ where $$A$$ is the amplitude, $$f$$ is the frequency and $$t$$ is the time variable. Under an LTI system model, we can represent the wave's medium by $$H(\bullet)$$. Then,


 * $$H(x(t)) = B \sin (2 \pi f t + \phi)$$

where $$B$$ is a new amplitude and $$\phi$$ is a phase shift or more simply, a time delay. The important thing to note is that the output of this LTI system consists of a single sinusoid at the same frequency as the input.

In general, most real-life mediums for the passage of acoustic waves are not linear and cannot be accurately modeled by an LTI system. To accurately model such a medium, a non-linear model must be used. Consider first, a simplified derivation of the one-dimensional linear model.

Imagine the medium in the model consists of particles separated by springs. If we consider the motion of one of those particles as the wave energy impinges on it, then from Newton's Laws of Motion, we know that


 * $$F = m \ddot{x} ,$$

where $$F$$ is the resultant force on the particle, $$m$$ is its mass and $$a = \frac{d^2 x}{dt^2} = \ddot{x}$$ is its acceleration. In a linear model, we know from Hooke's Law that the resultant force on the particle is


 * $$F = -k x ,$$

where $$k$$ is a contant that relates to the density of the material. Combining the two equations, we find


 * $$m \ddot{x} + kx = 0$$
 * $$\ddot{x} + \frac{k}{m}x = 0 .$$

Solving the linear differential equation and simplifying, we arrive at a solution for $$x(t)$$, the motion of the particle.


 * $$x(t) = \alpha \cos ( \sqrt{k/m} t ) + \beta \sin ( \sqrt{k/m} t )$$
 * $$x(t) = A \sin ( \omega t + \phi ) ,$$

where $$\alpha$$, $$\beta$$, $$A$$, $$\omega$$ and $$\phi$$ are all constants. The result represents the simple harmonic motion that the particle undergoes in the linear system. This gives the system the properties found in the LTI_system_model.

Now consider the non-linear system and in particular, the increasing density of the medium at the peaks of the incident acoustic wave. We may wish to model the tendancy of a wave to travel faster during these periods of higher density by considering a non-linear spring. Therefore, instead of assuming the particle is governed by Hooke's Law, which only applies to linear systems, we may instead model the force on the particular as a non-linear function of its displacement from its resting position. That is, the resultant force on the particle is


 * $$F = f(x) ,$$

for some non-linear function $$f$$. Continuing the derivation using the new resultant force, we find


 * $$m \ddot{x} + f(x) = 0$$
 * $$\ddot{x} + \frac{f(x)}{m} = 0 .$$

Solutions to the resulting non-linear differential equation are not trivial.

In practice, generating a non-linear model is difficult, and there are no generic properties one can assume the model will exhibit. Finding an accurate, non-linear model of various mediums under acoustic wave propagation is a popular research topic. Many solutions can only be evaluated numerically...
 * While the summary of an LTI is useful, perhaps it could be a bit more brief and refer the reader to the full article if they are interested in a more complete description.Dudecon 19:54, 15 January 2007 (UTC)
 * Agree we don't want to concern ourselves with the linear model in this non-linear article much, but that's a pretty short summary and included a treatment not found in LTI_system_theory, which is also linked. LightYear 06:51, 16 January 2007 (UTC)