The short answer is that the image is that of the fractal basins of attraction for the roots of the transcendental characteristic equation that determines the normal-mode wave numbers of a two-layer acoustic duct with one pressure-release boundary and one rigid boundary. Got that? If not, the long answer is given in what follows.

In this page, I'll give a "layman's" explanation of fractals, in particular, of the fractal shown on my home page. I'll also discuss some of the other ways that nonlinear mathematics, of which fractals are a part, can creep unexpectedly into problems in linear mathematics.

**Fractals**

A *fractal *is a highly irregular geometric object that displays
*self-similarity.
*Self-similarity means that a fractal displays similar shapes, complexity,
and irregularity across all size scales. Fractals are often beautiful and
fascinating to look at.

A fractal can be generated by almost any *nonlinear iterative transformation*.
A *transformation* is a process by which the points in space (in this
case, the two-dimensional space of the image) are shifted from one spot
to another. *Iterative* simply means that the transformation is performed
repeatedly, in a sequence. A transformation is *linear* if
the resulting figure is a simple scaling ('re-sizing') or rotation of
the original figure. Linear iterative transformations, and transformations
that simply shift the whole figure in space, do not result in fractals.
Nonlinear iterative transformations, *i.e.* transformations where
the numbers that represent the locations of points have more complicated
mathematical operations performed on them than simple scaling, usually
result in fractals.

**Newton's Method**

One iterative transformation that finds applications in many branches
of mathematics, physics and engineering, is *Newton's method *of determining
the roots of an equation, which was developed by and named after Sir Isaac
Newton. A *root *of an equation is simply a point where the value
of the equation is zero. The roots of nonlinear equations can be hard to
find by simply playing with algebra. Newton developed an iterative approach
that lets you hunt out the roots and calculate them to an accuracy limited
only by how many times you want to iterate his transformations. All that
the method requires is an initial guess as to where the root might be.
It doesn't even have to be a very good guess.

Newton's method works very well for equations that depend on one variable.
You can use it to answer the question "For what value of *x *does
my mathematical expression equate to zero?" It also works well for equations
that depend on more than one variable, or for equations that depend on
a *complex* variable, but in these cases there are some surprises.
To see what can happen, let's take a simple example. It involves complex
numbers, which are numbers of the form *a+b·i* where *a*
and *b* are regular 'real' numbers and *i* is* *the
so-called imaginary unit, which is √-1.
In this example, we'll know the answer (*i.e.* the roots) beforehand.
The surprise will be what Newton's method comes up with.

Let's suppose we are trying to find the roots of the equation

*y
= x ^{4 }- *1
(1)

Well, it's pretty easy to see that if we plug in *x = *1 or *x
= *-1, we get *y = *0. So, 1 and -1 are roots of the equation.
It happens that *i *and -*i *are also roots of the equation.

Let's see what happens when Newton's method is used to find the roots
of the equation. Figure 1 below shows the *complex plane , *which
is the set of all numbers of the form *x* = *a+b***·***i,
*and
marks the location of the four roots.

Newton's method requires a guess. Denoting the guess as *x _{0}*,
Newton's method allows us to calculate refinements to this guess using
the iterative transformation:

*x*_{n+1}* = x*_{n}*
- y*_{n}*/y'*_{n}
(2)

where *y*_{n} is the expression for *y *(Equation
1)* *evaluated at *x*_{n} and *y*' is the derivative
of *y* with respect to* x*. In our current example, the iterative
transformation works out to be:

*x*_{n+1}* = x*_{n}*
- (x*_{n}^{4 }*- *1)*/ 4x*_{n}^{3}
(3)

It seems reasonable to assume that if we 'guess' that a root is at *x _{0}
=* 0.9, then Newton's method would converge upon the actual root at

It turns out that this naive expectation is wrong. For Equation 1, Newton's
method is a nonlinear iterative transformation. The basins of attraction
of Newton's method for the four roots of Equation 1 are shown in Figure
3:

The borders of the basins of attractions for the roots of Equation 1 form a fractal. The complexity of the interweaving of these borders is infinite. If we were to 'zoom in' on any part of the border, it would look as complicated as Figure 3, no matter how much 'zoom' we applied.

As complex as the fractal in Figure 3 is, it was generated by trying to find the roots of a very simple equation, one for which we could find the roots by simple algebraic manipulation. In many mathematical, scientific, and engineering applications, we require the roots of much more complicated equations, equations for which it would be impossible to find the roots by playing with algebra.

**Characteristic Equations in Acoustics**

One such example comes from acoustics. One way to study the propagation
of sound in a duct is to build up or 'synthesize' the presumably complicated
field of sound waves using simpler sound waves. The simpler waves are called
the
*normal* *modes *of propagation. To find these 'modes', you
must find the numbers that determine the spatial shapes of the sound waves
they characterize. These numbers are called *wave numbers. *The equation
for the wave numbers of an acoustic duct, called the *characteristic
equation*, is generally very complicated, in fact, usually *transcendental*.
A transcendental equation is one that involves non-algebraic functions,
such as trigonometric, exponential, or logarithmic functions. The wave
numbers are given by the roots of the characteristic equation.

For the example used to generate the fractal image on my home page, the duct is a model of propagation in a body of water. The duct has a flat, hard bottom (the bedrock at the bottom of the sea), and a pressure-release top (the sea-air interface). The duct has two layers: the sea water layer, and a layer of sediment. The characteristic equation for the normal mode numbers for this duct is

*y = *tan *(k*_{1}*
h)* tan*(k*_{2}*(-d))*
*+ r*_{2}*k*_{1
}*/ r*_{1}*k*_{2} _{ }
(4)

where

_________________

*k _{1}
= *

and

_________________

*k _{2}
= *

The numbers *k*_{n }are the wave numbers we are
trying to find. They are those numbers that make equation 2 equal zero.
The numbers *c*_{1} and *r*_{1}
are the speed of sound in the sea water and the density of the sea water,
respectively. The numbers and *c*_{2} and*r*_{2}_{
}represent the same quantities for the sediment. The quantities *h*
and *d* are depths of the sea water and sediment, respectively.
The number *f *is the frequency of the sound being transmitted into
the duct. To generate the fractal image on my home page, I used the following
values for the constants:

*
r*_{1 }= 1,000 kg/m^{3 }
*r ^{2}*= 1,900
kg/m

The fractal image shows the basins of attraction of some of the roots
of the characteristic equation 4. In the image on my home page, I had rotated
the fractal so that it looked like a strange horned creature. The unrotated
image is shown in Figure 4 below. The black regions are the parts of the
complex plane for which Newton's method does not produce a root if given
an initial guess in that region. Initial guesses in any of the various
colored regions will cause Newton's method to converge to one of the many
roots of equation 4. The real number axis is the horizontal axis of symmetry
of the figure. The imaginary axis is the far left border. We can see from
the placement of some of the larger basins that there are numerous real
roots.

In the acoustic application that gave us this example, we are interested mainly in the real roots of the characteristic equation, and so we would restrict ourselves to searching the real axis and would not have as much difficulty finding the roots as if we had to search the complex plane. In some wave propagation problems, searching parts of the complex plane is necessary.

**Discussion: Nonlinearity in Linear Problems**

The acoustic wave equation, which is used to calculate the sound field in the duct in our example, is a linear partial differential equation. When a problem is 'linear', most people assume that phenomena of nonlinear mathematics, such as fractals and chaos, cannot rear their ugly (or fascinating, depending on your perspective) heads. The example above shows that chaos and fractals can creep into linear problems via the 'back door'. The back door is opened when nonlinear algorithms are used either to simplify the problem, or to solve some ancillary problem, as is the case with our example, where we used Newton's method to find the normal-mode wave numbers..

As mentioned above, nonlinearity can also enter into linear problems when techniques of nonlinear mathematics are used to simplify the solution of a complex linear problem. In the study of acoustics, it is often necessary to solve the so-called wave equation to study the propagation of sound. Under ordinary conditions, the wave equation equation is a linear, partial differential equation. Its solution is the value of the acoustic pressure at every point in space. The wave equation is also used to study the propagation of other types of waves, including optical and electromagnetic waves.

The solution of partial differential equations, even linear ones, can
be quite difficult to obtain. Until the advent of digital computers, and
modern mathematical techniques that take advantage of them, the wave equation
was only soluble in certain simple cases. Even now, the computer 'solution'
of the wave equation is an approximation (usually very accurate) calculated
using a number of simplifying assumptions. One approach to 'solving' the
wave equation is to treat the problem as if it were not waves that were
propagating, but 'rays', like rays of light. The resulting simplification
of the mathematics is dramatic: the problem becomes that of solving a set
of much simpler *ordinary* differential equations. These ordinary
differential equations are, however, generally *nonlinear*. Depending
on the shape of the boundaries of the space in which the sound propagates,
and the variability of the speed of sound in that space, the resulting
'sound rays' can propagate *chaotically*.

Like fractals, *chaos* is another aspect of nonlinear mathematics
that has no counterpart in the linear world. The word chaos brings to mind
randomness, disorder, or unpredictability. In mathematics, the existence
of chaos in a problem means that there is a practical and inherent limit
to predictability that is built into the equations that describe the problem.
This limit exists even though the problem is completely deterministic.

In acoustics, this unpredictability and disorder take a number of forms. First, a ray that propagates chaotically will trace a very irregular path through space. Second, if two rays leave a sound source, their paths will diverge very quickly, even if the rays started off heading in almost exactly the same direction. The practical implication of this fact is that it quickly becomes impossible to predict where a sound ray emanating from a source will end up. For example, if a ray leaves the source at a 10 degree angle, and a second ray leaves the source at an angle of, say, 10.0000001 degrees (a difference too small to be measured in practice), we would normally expect the sound rays to remain only slightly separated as they progress away from the source. Under chaotic propagation conditions, however, the rays diverge rapidly. Even our ability to predict where a single ray goes is affected by this sensitivity to initial condition. Suppose we use a computer with finite computing precision to predict the path of a ray, and compare the prediction to that made by another computer. Under conditions of chaos, the slightest difference in the ways these computers carry out computations (for example, one computer might have one bit more or less precision than the other) will be magnified into great differences in the predicted ray paths as the rays propagate further from the source.

A third form of disorder affects the number of rays that connect two points in space. We all experience the fact that sound can take multiple paths from a source to a listener. We experience this fact in the form of echo and reverberation. Under conditions of chaos, the number of rays that connect a source and listener increases dramatically as the distance separating the source and listener increases.

A fourth manifestation of disorder can be called 'unlocalizability of
a source'. We can use the differential equations for the ray paths to propagate
a fan of rays from a source. At some distance from the source, we should
be able to take the resulting final locations of the rays and 'back-propagate'
the rays, *i.e.* run them through the ray equations 'in reverse',
and end up back at the source. Under non-chaotic conditions, such a procedure
works, with errors that grow in a manageable way as the distance from the
source increases. Under chaotic conditions, it quickly becomes impossible
to back-propagate the rays to find the source.

It must be emphasized that these manifestations of chaos are not consequences of of computational inadequacy, but are inherent in the structure of the equations.

The above manifestations of chaos are in distinction to what happens when the wave equation is solved instead. Given the same conditions that produce chaos in the rays, the linear partial differential equation that gives the sound pressure at every point in space is very well behaved. Although the resulting sound field may be quite complicated, small changes in the initial location of the source do not radically change the sound field. The sound field can be 'back-propagated' to find the location of the source with reasonable accuracy.

A final remark: The duality between the behavior of sound when considered as being made up of waves, and its behavior when considered as being made up of rays, is directly analogous to the 'wave-particle duality' exhibited by quantum mechanical systems. The probabilities associated with the states of the quantum systems are described by a linear partial differential equation called Schrödinger's equation. The 'short-wavelength', or 'classical' limit of these equations is derived in a manner that is precisely analogous to the way that the ray equations come from the wave equation for sound propagation. The short-wavelength limit of Schrödinger's equation is the set of classical equations of motion for a particle. The quantum or 'wave' description of a physical system does not exhibit chaos, where its classical description as a 'particle' can.

**Links to More Information**

** Fractals**

- The Fractory: Some elementary general information on fractals, along with a gallery and some programs for generating your own. Also has some information on chaos.
- Fractal Explorer: A collection of information, image galleries and Java applets for exploring fractals.
- Fractint Homepage: Fractint is probably the most flexible, versatile and widely used fractal image generation program. It is available on the web for download. It is well documented and there are numerous forums and discussion groups devoted to it.
- Explore Newton's Method: Play with fractals generated by Newton's method.
- sci.fractal FAQ: A FAQ for the Usenet newsgroup sci.fractal that answers many questions about fractals.

- What is chaos?: This site contains "a five-part online course for everyone", and an interactive Java applet for exploring aspects of chaos.
- sci.nonlinear FAQ: A FAQ for the sci.nonlinear Usenet newsgroup. Contains information and links about nonlinear science.
- The Chaos Metalink: Interesting site with links to many sites on chaos and fractals.

- Acoustics FAQ: A FAQ for the Usenet newsgroup alt.sci.physics.acoustics. Useful links.
- Virtilab: The Propagation of sound: An interesting page on sound propagation and ray tracing. Includes a Java applet.
- The KRAKEN Normal Mode Program: There are very few web sites dealing, at an instructional or tutorial level, with underwater acoustics. This site, which gives the theoretical background for a widely used piece of acoustical analysis software, is a pretty good introduction, at the advanced undergraduate, level to the subject of wave propagation and normal modes in ducts.
- Ray tracing for Ocean Acoustic Tomography: Another piece of software background documentation. This has a good development of acoustic ray tracing. In Adobe PDF format.

Back to My Home Page

This page has been accessed 1116 times since May 24, 1999.

You are the 1116th person to access this page.