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Waves of Water - A Wonder of Non-linear Interaction!


Beep Sounds Related to Problems During Startup of Computers


Fractal Music



Waves of Water - A Wonder of Non-linear Interaction!

I. Introduction

When a pebble is thrown into a pond, at first there is a big wave that resolves into smaller and smaller ripples before disappearing. This is because the wave is made up of a combination of superposition of differing waves of frequency, therefore progress at different speeds. This phenomenon is called the dispersion of waves.  In August 1834, the British engineer John Scott Russell discovered a wave different from this.  In his case, the wave progressed several kilometers without dispersion and neither the form nor the speed changed.  This phenomenon is now referred to as a soliton and is caused by a balance between nonlinear interactions and dispersion effects of a wave. There is mysterious characteristic to a soliton. When solitons collide into each other, they unite momentarily and then separate advancing without changing speed or form just as they were before the collision. This behavior is similar to that of a particle and it is said that non-linear interactions include memory effect.

Note1:  Examples of solitons include a morning tsunami; a tsunami; a wave of the atmosphere; a red spot of Jupiter; vibration inside of an atom; signal transmission of a nerve cell; a tunnel of a magnetic field (whirlpool soliton); soliton of an electromagnetic wave in plasma heating; instanton.

Note2: The morning tsunami of the Amazon River is around 7 meters high and travels 800 kilometers up the river.

Note3: There is also light dispersion.  When sunlight is projected into a prism, it is refracted differently based on the lights wavelength. Therefore the light that comes from the prism disperses into different wavelengths of light into the colors of the rainbow.

II. Linear waves

A soliton is a non-linear wave, but first let us discuss the linear wave. A linear wave is called a micro-amplitude wave. An equation of the wave can be solved as follows.  When the boundary condition of the surface and bottom of the water is fixed, by utilizing the Laplace equation, the speed potential, Φ, can be determined. By substituting Φ into the pressure equation from Bernoulli's principle, an equation for a linear wave is obtained.



The boundary condition of the bottom is y = -h.



The boundary condition of the surface of the water is y = 0.



g = Acceleration of gravity

From the Bernoulli pressure equation, the form η of a wave is,


As for the wave, it is known that in reference to the coordinate, x, and time, t, there is a changing sine wave based on the cosine function.


φundefined function, k: wave number, ωangular frequency

By substituting equation 1.5 in to equation 1.1, with the boundary conditions of the bottom of the water defined by equation 1.2, we get equation 1.6.


Therefore, from the form of the wave, η, in equation 1.4, the following equations can be derived.



When we insert equation 1.3 (the boundary conditions for the surface of the water) into equation 1.5, we can determine angular frequency, ω, wave number, k, phase speed, c (velocity of the wave η), and wavelength, λ.




In the case that η is an arbitrary wave form, η forms from sine waves of countless different waves.  Each sine wave is linear with boundary conditions, so for that purpose they are independently together and follows the equations 1.6 to 1.8.

From equation 1.8, the sine wave’s different wavelengths move with different velocity, and for this reason there is wave dispersion from the occurring wave superposition. 

Note 1:  The coordinates of waves are underwater.

Note 2:  Laplace’s Equation is based on potential velocity, Φ, and the series equation of non-compressible fluids.

  Note 3:  In equation 1.8, when there is a large wavelength, the phase speed, c, involves the calculation of.

III. Non-linear Wave

Non-linear waves are called finite amplitude waves.  For finite amplitude waves, each sinusoidal wave that makes up a wave is not independent but moves in a non-linear interaction. This interaction, when there is a balance of linear dispersion effects, causes the wave change to stop and independent waves called standing waves are formed.

In more than 60 years since 1895, when Russell discovered the soliton, Korteweg and deVries lead the way with their non-linear equation of the soliton (called the KdV Equation).

KdV Equation defines the wave height by the variable u(x,t),


In solving this equation, both scientists discovered the solitary wave solution and the periodic cnoidal wave solution.

The solitary wave solution is,


For nonlinear solitary waves, the speed is dependent on wave length, λ, height,, and width,.


IV. Discovery of the Soliton

In 1965, N.J. Zabusky and M.D. Kruskal both having an interest in the paradox of Fremi-Pasta-Ulam (FPU), they did a computer simulation of a one dimensional nonlinear lattice attached to a spring.  Using numerical integration of the KdV Equation, they confirmed this FPU paradox.  In the initial conditions of a sinusoidal wave, it divided into eight solitary waves, which soon collided.  Before and after the collision, the same wave forms were seen and moved independent of each other.  This solitary wave acted similar to a particle in that it then soon returned to the initial conditions of the sinusoidal wave. 

In line with the photon, electron, and phonon terminology, Zabusky and Kruskal named this solitary wave the soliton.

In 1967 Gardner, Green, Kruskal, and Miura, utilizing the so called inverse scattering transform, elucidated the solution to the results of Zabusky and Kruskal’s computer simulation.

Note 1:  Fremi-Pasta-Ulam (FPU) Paradox: Within a linear criteria, when there is energy exchange and measure of time in reference to nonlinear forms, energy is returned to the initial state.

Note 2:  The Soliton Equation is the equation that solves for a soliton.  Other than the KdV Equation, there are relevant equations, such as Toda Lattice Equation, sine-Gordon Equation, and the nonlinear Schrödinger Equation


V. Linear Waves and Soliton Animation

Soliton Simulation(Click heare) 


VI. References 

Here is a limited list of references on this topic.  Please enjoy learning more about Waves of Water

Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness(Paperback) by John Briggs (Author), F. David Peat (Author)


 Beep Sounds Related to Problems During Startup of Computers


When turning on the PC, it makes a 'beep' sound at least 1 or 2 times . This sound comes from a simple target speaker inside the computer. The beeps are messages sent by the BIOS program to confirm that there is no hardware faults. If the beeps continue, the pattern of the beeps relate to an obstacle of the computer hardware and indicate problems with the corresponding hardware.

 *Note: BIOS (Motherboard Basic Input/Output System) is the control program of a mother board (M/B) of a personal computer. As mother boards may vary between systems, the BIOS program is included in all systems to control the difference of these mother boards to allow Windows to run smoothly.


Meanings of Beep Sounds

BIOS does the self check of the personal computer hardware called POST (power-on self-test) at the time the power supply is turned on. For example, there are the power supply, M/B, CPUs, I/O bus controllers, RAM, keyboard, video card etc. If there is an error, POST displays the error contents in the display.  The message, 'Keyboard error', is an example of when a keyboard is not connected to a computer. Yet, a defect in the main memory and video card may cause the display not to work. The beep sound is an alternative method to determine the presence of the error.  The pattern of the beep sound is not unified between BIOS makers, such as  AMI, Award, Phoenix, Dell, HP, IBM etc. (Award  is merged into Phoenix, but Award BIOS is now maintained by Phoenix).

Beep Sound Patterns (Click here)

*Note: There are some M/B which contain LED that can display an error cord and tell the contents of an error to a user.

*Note: Sometimes while Windows operates there is a beep sound. It is usually due to overheating of the CPU indicating that the power supply should be turned off.


In the case of no beep sound with trouble at the startup of the computer

The problem of CMOS may affect the parameters of the BIOS program stored and is related to no beep sound at all at the startup. By removing the battery attached to the mother board,  the CMOS memory is cleared. (But in the case that a jumper is present, replacing the  jumper  that is attached at the side of the battery is the common way to clear it.)  Clearing the CMOS memory should reset the computer to the default BIOS parameters correcting the errors related to no beeps (i.e. a beep sound should be heard).


Static Electricity

The static electricity is the build up of electricity in a system. The static electricity easily occurs especially when insulators rub against each other because it is difficult for the electricity to escape the insulating material.  For example, glass  when it rubs against silk and silk creates a negative electric charge. The human creates a positive electric charge similarly with glass. The positive charge in the human hand causes the parts, such as M/B and network cards  etc of a personal computer, to break when they are touched (In some cases the electricity is accumulated to the part of the computer, such as a condenser, and discharges the electricity buildup after it is untouched for a while and therefore prevents any damaged to the computer parts). Computer users should touch surrounding metal to discharge electricity in the body before touching the internal parts of a personal computer. There usually is not a problem even if you touch a personal computer, because its internal parts are insulated from its environment.

Note: Electrons move from glass to silk when they are rubbed together.  The protons of the materials do not move between material because they are stable.  In glass, there are only a few numbers of an electrons compared to silk therefore silk retains a negative charge. 


The Future of BIOS

A driver program must be installed during the addition of new hardware on personal computers. At that time, BIOS program sometimes needs to be updated as well. For example, video cards for the latest AGP slot sometimes does not work in the old versions of BIOS so updated versions of the BIOS program need to be downloaded from the homepage of the BIOS maker. The current BIOS update program is easy to operate, because it runs on Windows (Previous versions used to run on Dos).

 As for the present, BIOS loads the boot strap loader* (NTLDR) of Windows to the memory and Windows starts up.  EFI (Extensible Firmware Interface) will do this task in the Longhorn that is the name of Windows next operating system. EFI depends on Platform Innovation Framework for EFI that Intel is advocating. Since the 80's, the appearance of the personal computer has been evolving in many areas.  Everything from the speed of the CPU to the OS structure have been improved and enhanced, but at the same time only BIOS has been left  from progress.  Only recently, the pre-boot environment of BIOS is starting to change with EFI.


*Note:  NTLDR (NTLoader) reads boot.ini and decides which partition the computer should star up from. And it reads a necessary service program at the time of boot-up. Lastly, after reading the kernel of OS (ntosknel. exe), it gives the control to the kernel.

A recommended book for understanding PC hardware

Computer Repair With Diagnostic Flowcharts: Troubleshooting PC Hardware Problems from Boot Failure to Poor Performance  


Fractal Music


Fractal patterns' nature is a self similarity between the whole and the parts that is expanded with a certain scale, which is observed naturally in snow crystals, trees, clouds, rivers, the land and the coastlines. This is called a self correlation in consideration with the time axis. Some examples where fractal patterns are found are the melodies of tunes, heart beats, brain waves, traffic density and the fluctuation of earthquakes and stock prices.

Melodies of Tunes

Fractal music is a melody of tune that has a self correlation. Richard F. Voss was the first to discover that fractal music expresses a beautiful melody, as the fractal pattern that expresses the structure of nature is beautiful. White noise, for example static on the radio that sometimes occurs, has no self correlation. On the other hand, brown noise has the strong nature of self correlation.  For example, some unpredictable phenomenons, such as the random walk of small particles, are brown noise, which relies on the particles past position. The intermediate noise between brown and white noise is called 1/f noise and Voss showed that it makes a good melody for human beings.

Note 1: f is the number of oscillations and 1/f is the terminology of a spectrum theory. A white noise has the spectrum density of 1/f0, brown noise 1/f2, and fractal music 1/f1.

Note 2: Voss showed that the experience of human being and the change of nature are gathered to the circumference of 1/f noise and the pleasure of music might be related with fractal music that has the spectrum density or it might be an imitation of 1/f noise.


The Ways to Make Fractal Music

White Music

White music is made by 2 roulettes. The first roulette is divided into C, D, E, F, G, A, B and the second roulette is divided into a half note, a quarter note and an eighth note. Whenever the roulettes are turned the melody of random white music is produced.

Brown Music

Also, brown music is made by 2 roulettes. The second roulette is the same as in the case of white music. The first roulette is divided into the parts of -3,-2,-1, 0, +1, +2,+3 rather than C, D, E, F, G, A, B.  The initial position is assumed F at the middle of oscillations. It is E at the time of-1、and the next with -2 are C. There is the possibility that has exceeding 1 octave and go on with other than the piano keyboard finally, when this operation is repeated. It needs to make-3,-2, +2, +3 division length small preferably when you divide the roulette to avoid the phenomenon.

 Fractal Music

Fractal music is expressed by scaling Noise* at the center of white music and brown music. There are three roulettes, red, green and blue, that are each divided into six parts.  The sum of all three roullettes makes a maximum value of 18 notes which corresponds to a limited musical scale . It is assumed hear that red is the 3rd bit, green is the 2nd bit, blue (red,green,blue) is the 1st bit while e (000)2 of binary numbers is the initial state of the roulette. When 1 bit, (1)2, is added to the initial state, (000)2, the blue roulette is operated because the first bit is turned, (001)2

(000)2 + (1)2 = (001)2, gives a blue roulette

When you add one more bit to this number, the green and blue roulette are operated because the first bit and second bit are turned.

(001)2 + (1)2 = (010)2, gives a green and blue roulette

When you add one more bit to the number, the blue roulette is operated because the fist bit is turned.

(010)2 + (1)2 = (011)2, gives a  blue roulette

With every additional bit the roulette with the turned bit is operated in the next operation.

For example:

First, all roulettes are turning and the value is calculated as the 8th musical scale when the total is 8 as (1,4,3).

Therefore, the red roulette is 1, the green roulette is 4, and the blue roulette is 3.

Then, 1 bit is added to the initial state binary numbers (000)2 which turns only a blue roulette because it changes into (001)2 and the 11th musical scale is formed from the total of 11 as (1,4,6) with blue's 6 changed.

Next 1 more bit is added to the binary numbers (001)2 making the green AND blue roulette turn because they change into (010)2 and the 7th musical scale is formed from the total of 7 as (1, 3, 3) with green's 3, blue's 3 changed.

This time, 1 bit is added to the binary numbers (010)2 and the blue roulette turns because they change into (011)2 and the 8th musical scale is formed again from the total of 8 as (1, 3, 4) with blue's 4 changed.

Next, 1 bit is again added to the binary numbers (011)2 turning red, green and blue roulettes because it changes into (100)2 and the 7th musical scale is formed from the total of 7 as (1,2,4)2 with red's 1, green's 2 and blue's 4 changed.

This method changes the lowest bits well, while the upper most bits are stable so that it is producing the near situation of 1/f to synthesize only 3 of them.

 *(The sound that the same tone does, even if the tape speed of the sound that we recorded it with the tape is changed.)


Samples of White Music, Brown Music, and Fractal Music

White Music

  Brown Music

  Fractal Music


        Fractal Music, Hypercards and More...: Mathematical Recreations from Scientific American Magazine (by Martin Gardner)

The Science of Fractal Images (Michael F.barnsley, Robert L. Devaney, Benoit B. Mandelbrot, Heiz-Otto Peitgen, Dietmar Saupe, Richard F. Voss)

        Fractal music for healing

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