So What Is That 'Fractal' Image All About?

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 = x4 - 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.
 

Figure 1


Newton's method requires a guess. Denoting the guess as x0, Newton's method allows us to calculate refinements to this guess using the iterative transformation:

       xn+1 = xn - yn/y'n                     (2)

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

       xn+1 = xn - (xn4 - 1)/ 4xn3        (3)

It seems reasonable to assume that if we 'guess' that a root is at x0 = 0.9, then Newton's method would converge upon the actual root at x = 1. In fact, using this guess, we find that x1 = 1.0179,  x2 = 1.0005,  x3 = 1.000000329,...  and we see that Newton's method converges very rapidly to the root at x = 1. Similarly, if we 'guess' that a root is at x = 0.9·i, we would expect Newton's method to rapidly take us to the actual root at x = i. Naively, then, we might divide the complex plane into four symmetric regions as shown in Figure 2. We might expect that if any point in a given region is used as an initial guess for Newton's method,  then the method would converge to the root in that region. Thus, any guess in the blue region would be expected to cause Newton's method converge to the root x = 1, because all points in the blue region are closer to the root at  x = 1 than to any other root.. A region for which all the points in that region end up converging to a fixed point when acted on by an iterative transformation is called a basin of attraction. Thus, we might expect that applying Newton's method to Equation 1 results in the four basins of attraction shown in Figure 2 below.
 
 

Figure 2


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:
 
 

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 (k1 h) tan(k2(-d)) r2k1 / r1k2       (4)

where
                _____________
      k1 = [(2pf/c1)2 -kn2]

and
                _____________
      k2 = [(2pf/c2)2 -kn2].

The numbers kn are the wave numbers we are trying to find. They are those numbers that make equation 2 equal zero. The numbers c1 and r1 are the speed of sound in the sea water and the density of the sea water, respectively. The numbers and c2 andr2 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:

           r1 = 1,000 kg/m                  r2= 1,900 kg/m3
            c1 = 1,500 m/s                        c2 = 1,600 m/s
           h  = 100 m                             d  = 50 m
           f  = 35 Hz

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.
 
 

Figure 4


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.

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 Fractals

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