Note:
This composition for four-channel tape can be found here in mp3 format: PLAY. You may either do one of three things. Listen to the composition first, read this article first, or listen while reading. Of these I recommend reading while listening, but your preferences might be to listen to the composition as a piece of music, rather than a programmatic piece of science. Even though the very first listen is a unique experience that will shape future auditions, you can always return to the piece as a piece and disregard the science behind it. Either way, the choice is yours.
Introduction
This 10-minute composition is derived from the methods described in the publication: "Sonification of Particle Systems via de Broglie's Hypothesis". With these algorithms developed, their musical usefulness had to be tested. The ideas are interesting in and of themselves, but what of the final product? Would this system produce music as interesting as itself, even when standing alone from its programmatic nature?
It is interesting to note that even though their is a direct correspondence between the physics and its audification, which makes the music steadfastly programmatic, non-physicists who are aware of this feature will have little understanding of what that correspondence means. It is a very different case when a piece is entitled "Summer on the Danube" and "C. Eligan Genome." Generally titles like the latter will hold little meaning and provide little expectation for an artwork, than will the former title. As in the current example, some titles may be misleading. This music comes form a "simple harmonic system," but that does not refer to the tonal qualities of the work. A physicist would read the title correctly, whereas a musician would not.
Naturally, to test these techniques I composed a 10-minute composition with five 2-minute movements, each utilizing a different aspect of the system. This system of composition with particles is very strange to work with in that common sense takes a hike and expectations are led astray. Before the seven hours worth of computation time to produce one minute of sound, one needs to
be sure that the result will be useful for whatever needs. To circumvent this then a practical system of experimentation needs to be developed to predict and mold the variables--much like composers creating studies to determine what solutions exist. The duration of the final composition will be far exceeded by this time of experimentation and computation. The following section will attempt to explain how I have used the system to create the first composition derived directly from my new methods. Following that is a discourse on how this technique and compositional result identifies with both the experimental and modern experiences of music.
The composition "50 Particles"(Premiered May 24, 1999 at Stanford University; at a very high volume), is the first to use the techniques discussed in this thesis. There are five movements, each with a rigid duration of 120 seconds, constructed using one to several phenomena. There are no pauses between movements, no interruptions; the system begins each movement where it left off. The following sections will elaborate on each movement's composition with graphs and code excerpts where necessary.
Movement 1: Gradual Introduction of 50 Particles into System; Tuning the Harmonic Potential; Adjusting the Observation Apparatus
One at a time the particles come flowing from a source into a three-dimensional harmonic potential. The shape of the potential in which these particles will exist is an integral component of the experiment, if not the most important. This type of potential traps the particles so they don't stray, and is used in many scientific experiments. Initially it is ellipsoidal with the dimension coefficients:
and is described by the formula:
V(x,y,z,t,W)=kx(t,W)x2+ky(t,W)
y2+kz(t,W)z2
where the potential coefficients can depend on time and some set of parameters W--which could be mass, charge, velocity, etc. These coefficient values can thus be changed--altering the shape of the potential and activities of the particles--but the formula remains the same. If any of these constants were to become negative the result could become uncontrollable. Having all three be negative creates a three-dimensional saddle--which is hard to visualize, but easy to hear.
The times at which each particle enters the potential are based on a normal distribution. The code shown below reveals that only one particle can enter the system at a time, and that each time of entrance is based off of the subsequent times.
P_entrance = zeros(1,Num_Particles); \%Initialized Times Array
time_coef = 0.75;
for (i=2:Num_Particles),
P_entrance(i) = P_entrance(i-1)+time_coef * abs(randn(1.0));
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The time_coef value scales the overall distribution to give a time of nearly
120 seconds for all particles, Num_Particles, to enter. Using a normal distribution
gives a nearly linear distribution with enough deviations so that predictability
is avoided. If the particles came in a uniform rate this would be emphasizing
an aspect of form I don't want emphasized, namely predictability.
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Particle entrance times. |
Other initial conditions are decided by random methods within user-defined limits too. The particle charge, the initial velocities, and the entrance positions come from similar code. This uses a uniform distribution rather than the normal one used for the entrance times. Doing this ensures a system with characteristics that don't tend toward particular values. The choice of parameter limits, e.g. the maximum z-velocity is 3, comes from a priori knowledge of the potential. Having tested various particles in the potential, these limits come from preference of what the system should not do in the first movement.
Deciding upon the masses of each particle is an important task since frequency is directly proportional to particle mass. These decisions will restrict future changes to the system because energies (frequencies) might exceed the Nyquist limit. Furthermore, with foresight, the superposition of all the particles at the end of the second movement when the particles are completely at rest will give the restmass spectrum, and is why the mass relationships are carefully constructed. With 50 or more particles the task can be exceedingly time-consuming using preference; but by making an automatic process to create the values is the quickest. The restmass spectrum is shown below, with the Nyquist limit on the right. This distribution comes from a relationship similar to the entrance times.
mrel = 1.1;
munc = 1.0;
m(Np+1) = abs(m(Np)+ munc*randn(1))*mrel
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The mass array is constructed sequentially using the previous entry and a normal distribution, as well as an uncertainty scaling coefficient, \emph munc, and a mass relationship factor, \emph mrel . Many algorithms were tested for this process, and this one gave the most favorable results for what was desired. The smallest mass will have a minimum frequency of 18.5 Hz; and the largest mass a minimum frequency of 1,970 Hz. It is guaranteed then that this piece can span the entire range of human hearing. The area between the Nyquist limit and the spectrum allows a good range of non-aliasing energies that the particles can have.
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Restmass Spectrum of the System with the Nyquist Frequency on the Extreme Right. The "frequency" axis is the number of occurances of a particular mass. |
Over the duration of the first movement, the potential is "tuned." This must be done with care because any changes to the potential drastically change the system by taking or giving energy. The following code is used to change the potential constants.
if (time_current > 40 && time_current <= 45),
B = [10,11,8]*(1+(40-time_current)/5);
elseif (time_current > 45 && time_current < 60),
B = [11,14,9]*(time_current-45)/15;
B2 = B;
elseif (time_current > 70 && time_current < 75),
B = B2*(1+(70-time_current)/20);
end
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In the code B=(kx,ky,kz), which describes the degree to which the harmonic potential is curved. The following graph shows the activity of the potential during the first movement. When all three values become 0 at time_current=45sec the potential is flat, which means there are no forces acting on the particles. This becomes obvious in listening because all frequencies become stationary; essentially energy (potential energy) is taken out of the system. To get back to a harmonic potential the constants are increased, with the drawback of putting energy back into the system.
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"Tuning" the potential. |
Now that the system for the first movement is described satisfactorily, and because computing the 120 second, 4 channel, 44.1 kHz, soundfile takes on the order of 7 hours, a test run at 100 Hz sampling rate is done to make sure the system is working properly, the results will be as predicted, and no frequencies will exceed the Nyquist limit. (Of course, exceeding this limit means nothing traumatic, and could be used as a compositional device with unique affect. As composer, I do not wish to have these effects this early in the experiment.) Though this is not as accurate as the 44.1 kHz sampling rate of the final product, it will give an adequate picture of what could happen.
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The energy dynamics of the system during the first movement. The dashed line at top is the Nyquist frequency. |
Once the system is created decisions need to be made as to how it will be observed. The position of the observer (PS) and the observation range (OR) are as important to this method, as speaker placement and volume are to a stereo. The position can change dynamics and the global spatialization of the system. The observation range restricts what is heard to particles coming in and going out of view, thus complicating the aural experience and keeping interest. In this movement PS remains at the origin; however the OR is manipulated. As in the potential tuning code, the OR modifications happen the same way. The following graph shows the OR during the first movement.
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Changing the range of the observation apparatus. |
It is important to note that only along one-dimension observations will be made. The OR is centered on the x and z of PS, and changes in the y-dimension result in changes of view. If PS is modified in the y-dimension, the OR will then sweep with it and the pitch material will be modified. (This is because what will be observed are particles with lower energy along the y-dimension. As they go up the side of the potential they lose energy, thus frequency. When the OR is centered at the origin what is heard is a glimpse of the particles at a maximum energy.) Changes in the x and z-component of PS will only result in getting closer or further from the origin, thus changes in dynamics and not pitch material.
Movement 2: Adding Viscous Fluid to Reveal the Restmass Spectrum
This movement consists of only one phenomena. By gradually adding a viscous fluid into the potential the particles will slow down until they are suspended motionless at the origin with only their restmass energies. Viscosity usually acts as a damping force in proportion to the velocity of a body within it: as the body's velocity becomes higher the impeding force increases as well. However, the viscosity of the fluid here has the unique property of being dependent on the position of the body. The code below shows that particles further from the observer have a harder time moving fast. A physical analogue might be a fluid in which more viscous elements rise to the "top" of the system.
if (time_current > 130 && time_current < 240),
µ = (3.0 - 3.0*cos(2*PI*(time_current-130)/220))*particle_pos;
end
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The general evolution of this viscosity, µ, at a fixed distance, is shown in the graph below. Since it is proportional to distance, a larger particle_pos means a steeper ascent over time.
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The viscosity coefficient over time. |
The next graph is the result of the experimental phase. It is more than apparent what is happening visually. If all is well, the spectrum at the end of this movement should exactly match the rest-mass spectrum.
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The energy development during the second movement. Notice how the restmass spectrum is revealed by the end of the movement. |
Since the observer is at the origin the mass of sound will accumulate as each particle descends into it. (The volume of any observed particle is inversely proportional to its squared distance to the observer.) By changing the observer's position toward the end a dramatic change in dynamics will result. The next figure shows how the observer rises and descends in the z-dimension, and lasts into movement 3. Here it should be completely apparent that changing the y-dimension in PS would result in having no particles within the OR since they are all at the center.
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The motion of the observer away from system only along the z-dimension. |
The ROO is static, having the values from the end of the first movement.
Movement 3: Sudden Increases in the Coulomb Potential of the Universe; Subsequent Oscillation of Potential Walls
This movement is the most aurally dramatic sections of the piece, which is by design since it holds a special place in the entire piece as the middle movement. It will extend the entire range of human hearing within an instant. (This has quite an effect on the unsuspecting audience.) This movement not only took the longest to compute, but it demanded much time in the experimental stages to remove anomalies and create what was desired. Far from reality, no physicist can accomplish what occurs in this movement; it is pure fantasy where nature's constants are changed at the will of the composer.
At the end of the second movement, we are observing the motionless particles packed neatly together at the origin, and are getting closer to them. They each contain a positive charge, that was randomly issued in the first movement, but because the Coulomb constant has been zero so far, the particles are rather inert, non-interacting. (There are no collisions in this piece.) If the Coulomb potential is suddenly increased, the nicely packed particles will explode into a frenetic system.
There are four large impulses, each having progressively longer durations; and four smaller ones meant to give variations. From the code below, the Coulomb potential is not only changed suddenly, but its value is unique for each particle, depending on the charge, q or the mass, m!
if (time_current > 245 && time_current <=246),
C = 1.9*(time_current - 245)*m;
elseif (time_current > 270 && time_current <= 271.5),
C = 1.2*(time_current - 270)*m;
elseif (time_current > 284 && time_current <= 284.1),
C = 0.5*(time_current - 284)*q;
elseif (time_current > 300 && time_current <= 302.5),
C = 1.5*(time_current - 300)*m;
elseif (time_current > 310 && time_current <= 310.1),
C = 0.5*(time_current - 305)*q;
elseif (time_current > 325 && time_current <= 329),
C = 2.3*(time_current - 325)*m;
elseif (time_current > 355 && time_current <= 355.5),
C = 0.8*(time_current - 355)*q;
end
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This odd dependence was done because with a constant value for all the particles, the larger
masses (higher restmass frequencies) were hardly moving at all (F=m*a), and the small particles were flying past the Nyquist limit uncontrollably. While this may be a nice effect, it is not what I am going for. The accompanying figure shows the ranges of the Coulomb constants for smallest and largest mass in the system. The large ones are functions of mass, while the little blips are functions of charge. The varying effects will be discussed below.
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The Coulomb impulses. Values depended on mass or charge! |
After developing the impulses and running experiments to see the frequency distribution, I had to tailor the action of the viscous fluid so that there would be explosions and returns to restmass frequencies. Gradually over the movement I wanted the particles to become more chaotic, and thus relaxed the action of the viscosity. The viscosity, unlike in the second movement, is only time, not position, dependent.
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The viscous fluid in (discontinuous and un-natural) action. |
At particular times in this movement, I begin to modulate the "walls", which are really volumes, with mass-dependent functions. The code below shows this process. At all other times the potential remains the same as the one the experiment began with.
if (tc > 272 && tc <= 292),
A = [ 0.3,1.35,0.31 ]* (sin(2 * pi * (tc-252)/40)+1)/2;
f = [ 21,47,18 ]* (cos(2 * pi * (tc-272)/40)+1)/2;
kx = A(1) * sin(2 * pi * f(1) * (tc-tco))* m+B(1);
ky = A(2) * sin(2 * pi * f(2) * (tc-tco))* m+B(2);
kz = A(3) * sin(2 * pi * f(3) * (tc-tco))* m+B(3);
elseif (tc > 303 && tc <= 323),
A = [ 0.6,1.65,0.61 ] * (sin(2 * pi * (tc-283)/40)+1)/2;
f = [ 63,160,37 ] * (cos(2 * pi * (tc-303)/40)+1)/2;
kx = A(1) * sin(2 * pi * f(1) * (tc-tco))* m+B(1);
ky = A(2) * sin(2 * pi * f(2) * (tc-tco))* m+B(2);
kz = A(3) * sin(2 * pi * f(3) * (tc-tco))* m+B(3);
elseif (tc > 329 && tc <= 360),
A = [ 1.0,1.9,0.9 ] * (sin(2 * pi * (tc-298)/62)+1)/2;
f = [ 145,320,97 ] * (cos(2 * pi * (tc-329)/62)+1)/2;
kx = A(1) * sin(2 * pi * f(1) * (tc-tco)) * m+B(1);
ky = A(2) * sin(2 * pi * f(2) * (tc-tco)) * m+B(2);
kz = A(3) * sin(2 * pi * f(3) * (tc-tco)) * m+B(3);
else
A = [ 0,0,0 ] ;
f = [ 0,0,0 ] ;
kx = A(1) * sin(2 * pi * f(1) * (tc-tco))+B(1);
ky = A(2) * sin(2 * pi * f(2) * (tc-tco))+B(2);
kz = A(3) * sin(2 * pi * f(3) * (tc-tco))+B(3);
end
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The evolution of the potential can be seen in the graph.
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Potential constants for smallest mass. For larger masses the fluctuations have larger magnitudes. |
The constants are changing rapidly at moments, from 18 - 320 Hz. A small detail of k y is included below so that the function's affect can be seen. For larger masses, the only thing that changes is the magnitude of modulation.
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Detail of potential constant ky modulation. |
After specifying all of these details the experimented is run to check the bandwidth not exceed the Nyquist. The figure shows a beautiful picture of what happens. (The patterns created by the energy versus time graphs can easily be mistaken as waterfall charts--onset times and amplitude envelopes for partials. But this is not what is represented. Wherever there are gaps, white areas, at that time those frequencies are not present. The higher the line, the higher energy/frequency a particle has acquired.) From the first to the last, the explosions become more erratic. The particles begin to wiggle about more and more as potential modulations occur with larger amplitudes. The difference of affect from the Coulomb potential can be seen: where the potential is a function of mass, the effect is major and widespread; where the potential is a function of charge, little disturbance is caused in the larger masses, but the little ones briefly take off.
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Energy versus time. A pretty picture. |
The details of these brief interactions, which become longer as each Coulomb impulse occurs (see rounding off of initial peaks), are quite useful. Below is the first interaction, with a sharp peak and a quick descent.
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Lush wave of sonic information. A real crowd waker. |
This is quite contrasted with the fourth and last impulse, when the particles interact for four times as long. After a rapid ascent the particles have a smooth descent until they catch the fluctuating potential. At one point before 330 seconds some of the frequencies are blurred by frequency modulation. This is what I was attempting to do, but the effect is less noticeable than I would have liked. To create the FM I wanted it is necessary for modulating the potential with a much higher frequency rate. I still need to show my technique to be able to produce FM synthesis.
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The fourth and last impulse. |
The observer's position changes dramatically through this movement creating very effective dynamics. I did not wish for the observer to pass closest at the times of Coulomb impulses. By avoiding this, and since the first explosion happened when PS was close to ground zero, the listener is fooled into expecting another loud crash as s/he passes through the origin. Also for the first time in the piece, the y-position of PS changes. This parameter is not directly noticeable, but I am sure that if this movement were simulated again with the OR fixed on \( y=0 \) there would be much difference.
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Observer's positions during the 3rd movement. |
I also experimented with changing the region of observance, which had only an effect on spatialization. The OR is a sinusoidal function which increases exponentially over time, but since its center position is being modulated by the changing y-position of PS, the graph appears warped. The discontinuity is a mistake, but no audible blip can be heard there.
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Region of Observance. |
This movement by far required the most thought. I spent hours thinking about how to produce the effects I want, and translate that into science, albeit a science fiction. It was hard to pick apart what variables were causing what phenomena, and one by one I had to go through and be positive of their affect. This work before the computation was absolutely necessary since I knew the 120 second movement would take a very long time to simulate. It ended up taking over 50 hours on a 450 MHz machine (Dell R450). Luckily I was happy with the first simulation; my hard efforts resulted in the colorful and dramatic section I was hoping for.
Movement 4:Two Generation Cascading Radioactive Decay
Already having acknowledged the existence of a unique set of particles and heard their characteristic restmass spectrum, the particles will now undergo an \emph irreversible operation of decay. There is no return to the initial system once this occurs. The spectrum would approach zero if the particles were left to decay forever, and would consequently no longer be heard. Over the duration of this movement each particle will split into two and each of those split into two more. By the end of the movement there should be approximately 200 particles in the system.
The times at which the particles decay is (very unphysically) predetermined, since calling a random procedure at each sample step, or integer thereof, is computationally expensive. However the decay times are normally distributed with dependence on the amount of material left, which is accurate to nature. The exponential nature of the decay is seen in the figure of the decay times.
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Radioactive decay times during the 4th movement. |
Normally radioactive decay results in numerous particles with massive energies spilling from the nucleus. The decay velocities are uniformally distributed between -10 to 10, plus the initial velocity of the decayed particle. This will obviously result in energies higher than the Nyquist limit, but that is not of concern for me here. (In several of my first examples of radioactive decay, particles would blast apart with such force that soon every particle was aliasing with a such a high frequency that a strange pitched noise would result, sincerely adding to the effect. Indeed, here I am encouraging this.)
During this movement I alter the viscous fluid so that particles would not blast out of the system. With such a high viscosity, nothing moves very far before being stopped. The result is a nice pitched pop. Toward the end I completely take away the viscosity and create a mysterious excitation force before the beginning of the last movement. This is an attempt, though unsuccessful, to
get things moving again.
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Viscosity and excitation force. |
Like in the first movement I have a similar evolution of the OR. With this shape activity should become less and less. I was hoping for effects similar to my first experiments will small ORs, but what resulted is not close. (This is because the viscosity was too high; and the excitation force too small.)
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Region of observation for the 4th movement. |
Again similar to the previous movement, I oscillate the potential walls. This had little effect on the system because of the high viscosity.
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Oscillating the potential walls. |
The most dramatic effect comes from oscillating the observer's position very rapidly.
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Oscillating the observer's position. |
It becomes so fast that the entire system is amplitude modulated and the auditory scene becoming gritty--a nice contrast to the purity of the previous 6 minutes. (It is speculated that the grit comes from the 200 particles appearing as 600 from the amplitude modulation creating band-specific noise.) To create this effect in real life, oscillate head to and fro over 12 inches at 115 times a second.
As the observer slows down, the system seamlessly modulates back into the sound of pure particles. A detail of this is found in the next graph.
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Detail of PSy over 10 seconds. |
At the end of this movement there are 201 particles in the system, and the tonal material has degraded. The spectrum, or bandwidth, extends much lower than the first set of particles. The new restmass spectrum is shown in the following graph.
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The new restmass spectrum after radioactive decay. |
Movement 5: Reduction of the System Via Least Energies
At randomly predetermined times the system is scanned and the particle having the least kinetic energy is removed. It is as if it fell out of the potential through an expanding gap. This is similar to the introductory process, but instead of sequentially adding heavier particles one at a time, the least energetic particles are systematically removed. The times at which these events happen are decided upon exactly the same as in the first movement.
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Particle exit times. |
Since through the fourth movement the sonic material has significantly degraded to many particles with small masses concentrated at the minimum potential, a subtle means of moving the system to higher energies has to be devised. Modulating the potential is a good way to guarantee energy will be put into the system, but because many of the particles are resting at the origin they would be unaffected by this.
A general forcing function as added as impulses, that will move the particles with some force independent of their mass. It is like a sideways acting gravity because all particles no matter what the mass move at the same rate. These impulses over the movement are shown in the following figure.
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The "sideways gravity." |
This effect is peculiar to visualize and just as strange to hear. It is almost an unnatural effect, an unscientific audible scene.
Once the particles are moving about they will respond to a fluctuating potential. The following graphs are the changes to each potential constant over time. After each excitation force and potential modulation I want the general trend of energies to increase, to approach the limit of human hearing.
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The potential constants during the 5th movement. |
It took much time to find a good combination of all the parameters to allow for this. I found that complicating some while keeping others simple gives the best results.
The viscosity of the fluid was rather simple. In the beginning it acts to keep the particles from getting too energetic too soon. Half-way into the movement, its value is relaxed to allow for higher energies. The sharp edges, discontinuities, provide an interesting counterpart to the overall continuous flow of the piece.
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The Viscosity during the 5th movement. |
Having completed the construction of the potential it is necessary to test for adequate results as usual. This took much more simulation time since the movement begins with 201 particles. The general trend of particle extraction is quite clear. There is certainly aliasing, but I am willing to leave it be. (Every vertical line in the graph is where a particle leaves. It really should just disappear, but its array position is replaced by another particle still in the potential. This is interpreted by MATLAB as a continuous line.)
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Inverse evaporative reduction during the 5th movement. |
After the design is finished the observation parameters must be created. Like in the third movement with a sinusoidally varying PS, here I have a similarly varying OR. I had to be careful not to restrict too much vision at times because little might be heard.
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The sinusoidal region of observance during the 5th movement. |
The last parameter included is the observer position. The only thing I require is a movement away from the system toward the end of the experiment. Only the x and z parameters are changed because the OR should stay focused on the origin.
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The observer's position, and final decrescendo, during the 5th movement. |
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