Wave Articulation Matrix

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h1. Wave Articulation Matrix

The Wave Articulation Matrix is composed of concentric steel cylinders.  The simplest design uses four cylinders.  More sophisticated devices may have 6, 8, or any even number of cylinders.  Each cylinder, except the outermost, is wrapped with a thin layer of dielectric material over a portion of its length.  See the 'Assembly' section below for exact details.

Fig. 1.1:  "Wave Articulation Matrix, 8-Element":https://opendesignengine.net/attachments/download/446/6frw2.jpg

All the cylinders have the same mass, and the same surface area.

The length of the innermost cylinder is equal to the circumference of the outermost cylinder.  The length of the outermost cylinder is equal to the circumference of the innermost.

Fig. 1.2:  "Physical Dimensions -- Side View -- 8-Element WAM":https://opendesignengine.net/attachments/download/439/side_dimensions.jpg

Fig. 1.3:  "Physical Dimensions -- Top View -- 8-Element WAM":https://opendesignengine.net/attachments/download/455/top-view.gif

Fig. 1.4:  "Perspective Views -- 8-Element WAM":https://opendesignengine.net/attachments/download/451/primary-axis2.jpg

The WAM can be described as a Log Periodic Nested Waveguide.  It is Log Periodic, in that, the radius of each element (counting from the innermost element, outwards) is equal to that of the previous element, multiplied by a constant.  Likewise, the length of each element (counting from the outermost element, inwards) is equal to that of the previous, multiplied by that constant.  The constant we use (our base of logarithms) is the Golden Ratio.

In this document, this particular configuration will be referred to as The Fountain.

A Log Periodic Dipole Array is Log Periodic in one dimension, so a curve traced through the tips of its elements is a logarithmic curve.  The Fountain is Log Periodic in two dimensions simultaneously (radius, and length), so a curve traced through the bottom edges of its cylinders is a Hyperbola.

Fig. 1.5:  "The Fountain is a Log Periodic Nested Waveguide":https://opendesignengine.net/attachments/download/477/log-periodic2.jpg

If we want to build a 4-element Fountain around a 0.75 inch diameter acetel rod, using 0.002 inch thick steel shim stock, then Lambda = 10.008 inch.

Fig. 1.6:  "Example Dimensions -- 4-Element Fountain":https://opendesignengine.net/attachments/download/453/4wam.jpg

Fig. 1.7:  "Photo of 4-Element Fountain":https://opendesignengine.net/attachments/download/465/IMG_0061.JPG

h1.  Two Waves, Whose Frequencies are Consecutive Integer Powers of the Golden Ratio, Heterodyne in a Unique Way

Fig. 9.1:  "Powers of the Golden Ratio":https://opendesignengine.net/attachments/download/450/golden-ratio-powers.gif

Fig. 9.2:  "...Heterodyne to Produce More Terms of the Series":https://opendesignengine.net/attachments/download/441/heterodyne.gif

h1.  Assembly Instructions

_Note:  We recommend using a 100 watt soldering iron to solder the steel cylinders._

A 4-element prototype can be constructed, using 0.002 inch thick steel shim stock to form the cylinders.

Fig. 8.0:  "Physical Dimensions of a 4-element Fountain":https://opendesignengine.net/attachments/download/453/4wam.jpg

Begin by choosing a value for Lambda (how big you want your Fountain to be).  All physical dimensions are derived from this value.  You may wish to consider what diameters are available for the rod material which you will build your structure around.  Then choose a lambda based on that.  For example, we decided to use a 0.75 inch dia. Acetel rod, so our Lambda came out to be 10.008 inches.

Steel shim stock, 0.002 inch thick, is reasonable to work with.

Fig. 8.1:  "0.002 inch thich Steel Shim Stock":https://opendesignengine.net/attachments/download/490/as_steel-shim.jpg

The shim stock can be measured, then cut with a utility knife.

Fig. 8.2:  "Cut Steel Shim Stock with a Utility Knife":https://opendesignengine.net/attachments/download/485/as_cut-shim.jpg

You will cut out 4 rectangles of steel to roll up into cylinders.  The "Circumference" of the cylinder, in Fig. 8.0, is the width of the rectangle.  The "Length" is the length.

Remember that the steel has a grain to it.  Cut your rectangles such that the grain runs in the direction which they roll up to form cylinders.  

The meeting edges of the cylinder should be sanded with fine emery cloth to remove oxide layer.  Then soldered together using high-acid flux and lead free silver solder.

Fig. 8.3:  "Steel can be Soldered with a High Acid Flux":https://opendesignengine.net/attachments/download/488/as_silver-solder.jpg

You will need wooden spacer rings to hold the metal cylinders apart.  We had ours laser-cut from good quality plywood.  The "OD" and "ID" dimensions in Fig. 8.0 refer to the dimensions of these wooden rings.

Fig. 8.4:  "Wooden Spacer Rings, Laser-Cut":https://opendesignengine.net/attachments/download/489/as_spacer-rings.jpg

Wrap the first steel rectangle around a dowel or rod and solder.  This is cylinder number zero.  We used a 12 inch length, 0.75 inch diameter Acetel (Delrin) rod for our prototype.

Next, wrap a thin layer of dielectric material (4 mil plastic sheeting) around the metal cylinder.  The dielectric layer should be as long as the next metal cylinder, and flush with the top of the cylinders.  (see Fig. 8.4.1, and Fig. 8.6).

Fig. 8.4.1:  "Dielectric Layers":https://opendesignengine.net/attachments/download/497/dielectric-layers.jpg

Now, assemble the wooden spacer rings around the first cylinder, on top of the dielectric.  You may wish to use cyanoacrylate glue (superglue) to keep everything in place and flush at the top.  Water-based glues may not be ideal, due to their material characteristics at microwave frequencies.

Wrap the next steel rectangle around the wooden rings and solder.  This is cylinder number one.  This cylinder, and the next one, each get a wire soldered to them, which provides the output of the device.  When cylinder one is all soldered, it will look like Fig. 8.5.  Wrap a layer of dielectric around it, and it will look like Fig. 8.6.

Fig. 8.5:  "The First Two Elements":https://opendesignengine.net/attachments/download/483/as_2-rings.jpg

Fig. 8.6:  "The First Two Elements, with Dielectric":https://opendesignengine.net/attachments/download/486/as_dielectric.jpg

Now assemble the next set of wooden spacer rings on top of the structure you have created thus far.  It will look like this:

Fig. 8.7:  "Two Completed Cylinders, with Spacer Rings Ready for Third":https://opendesignengine.net/attachments/download/491/as_wooden-rings.jpg

Wrap the next steel layer around the wooden rings and solder.  Also solder to this cylinder a wire for output.  Wrap dielectric around it.  Put the next set of wooden rings around that.  It will now look like this:

Fig. 8.8:  "Three completed Cylinders, with Spacer Rings Ready for Fourth":https://opendesignengine.net/attachments/download/492/as_wooden-rings2.jpg

Wrap the last steel layer on and solder.  Now you are done.  The last cylinder does not get a layer of dielectric.  Here is your completed Fountain:

Fig. 8.9:  "Four Completed Cylinders.  You're Done!":https://opendesignengine.net/attachments/download/484/as_complete2.jpg

Notice that during the assembly, cylinders 1 and 2 had wires soldered on before the dielectric was wrapped around.  These provide the output of the device.

h1.  From a Golden Spiral to Gabriel's Horn

I would like to show you how the geometry of the Fountain can be derived from a Golden Spiral.

Let's take a look at a Golden Spiral.

Fig. 2.1:  "A Golden Spiral":https://opendesignengine.net/attachments/download/447/golden-spiral3.gif

A Golden Spiral is a Logarithmic Spiral.  Its radius multiplies by the Golden Ratio every quarter cycle.  In Fig. 2.1, the Golden Ratio is written as the Greek letter Phi.

Just to make things simpler, let's have our Golden Spiral grow by a factor of Phi every complete cycle, instead of every quarter cycle.  Let's also look at our spiral sideways, and allow it to exist in the dimension of time.  Now what does it look like?

Fig. 2.2:  "A Golden Spiral in Time":https://opendesignengine.net/attachments/download/448/golden-spiral-2.gif

The envelope of the wave on the right is an exponential curve; the amplitude of the wave is growing exponentially.  

Can our wave grow in any other way?  Yes.  Its frequency can grow as well as its amplitude.  Let's make a wave where each time it completes one cycle, its amplitude has multiplied by the Golden Ratio, and its period has been divided by the Golden Ratio.  Now what does our wave look like?

Fig. 2.3:   "A Golden Spiral with a Hyperbolic Envelope":https://opendesignengine.net/attachments/download/444/hyperbolic-wave.gif

Fig. 2.3.1: "Explanation of the Logarithm Used in the Equation":https://opendesignengine.net/attachments/download/449/logarithm.gif

This wave has a hyperbolic envelope, not an exponential one, as before.

We can also flip our wave around.  This is perhaps the more general form of the equation.

Fig. 2.4:   "A Golden Spiral with a Hyperbolic Envelope -- a Chirp":https://opendesignengine.net/attachments/download/458/wave.gif

Fig. 2.4.1: "The Constant K Determines How Quickly the Wave Collapses":https://opendesignengine.net/attachments/download/457/enter-time1.jpg

If we rotate the envelope of this wave around the Z-axis, we create a Hyperboloid.

Fig. 2.5:  "Gabriel's Horn":https://opendesignengine.net/attachments/download/460/below.jpg

This Hyperboloid is known as Gabriel's Horn, because it looks like the trumpet blown by Archangel Gabriel on the Last Day.  It has finite volume, yet infinite surface area.  This is the correct shape of a vortex in water.

The cylinders of the Fountain are formed by taking slices of Gabriel's Horn.

Fig. 2.6: "Wave Articulation Matrix and Gabriel's Horn":https://opendesignengine.net/attachments/download/443/hyperboloid.gif

Fig. 2.7: "Wave Articulation Matrix -- Perspective View":https://opendesignengine.net/attachments/download/438/beautiful.jpg

Gabriel's Horn is pertinent to Hyperbolic Geometry, and the work of N. I. Lobachevsky (Lobachevskii).

h1.  Wiring Diagrams

The circuit used to excite the WAM is very similar to a RADAR Modulator.

Fig. 3.0:  "RADAR Modulator is very Similar to WAM Exciter":https://opendesignengine.net/attachments/download/479/RADAR23.jpg

Fig. 3.1:  "Simple Wiring Diagram":https://opendesignengine.net/attachments/download/475/wiring-simple.jpg

Fig. 3.2:  "Complete Wiring Diagram":https://opendesignengine.net/attachments/download/495/wiring-complete2.jpg

Fig. 3.3:  "Wiring Diagram -- Perspective View":https://opendesignengine.net/attachments/download/496/wiring-complete-3.jpg

Let's call the innermost cylinder the Primary Axis, and the outermost cylinder, the Control Ring.  Regardless of the number of cylinders in the Fountain, all the cylinders between the Primary Axis and the Control Ring are tied together electrically; these provide the output of the device.

The Control Ring is tied to the low side of the switch.  It is held at a positive low voltage (12v).

Fig. 3.4:  "Output Circuit":https://opendesignengine.net/attachments/download/456/wiring-top.gif

We recommend using a Hydrogen Thyratron, Spark Gap, or Gas Discharge Tube as the switching mechanism.  This is because these devices switch from "OFF" to "ON" very quickly -- on the order of tens of pico seconds.  Some fast IGBTs can switch in 20 - 50 nano seconds, but this is still 1000 times slower than a Thyratron.  Why do we need such a fast switch?  The switching time is directly proportional to the diameter of the first cylinder surrounding the Primary Axis.  The Modes which this cylindrical waveguide can support are around 11 GHz for the 4-element WAM shown above.  Larger structures would tolerate slower switching times.

h1.  Not a Steady State Device

The Fountain is not a steady state device.  It is not excited by RF alternating currents.  Rather, it is excited by Transients.

Fig. 4.1:  "Transients vs. Steady State AC":https://opendesignengine.net/attachments/download/478/fig8.jpg

For an excellent introduction on Transients, please see:

"Steinmetz, Charles Proteus, Elementary lectures on electric discharges, waves and impulses, and other transients":https://archive.org/details/elementarylectur00stei

h1.  Pulsed DC, not AC

The Fountain is excited with pulsed DC, not AC.  In a pulsed DC circuit, the magnetic field always spins the same direction.  This is in contradistinction to the magnetic field in an AC circuit, which reverses direction repeatedly.

Fig 5.1:  "The Magnetic Field in a Pulsed DC circuit Always Spins the Same Direction":https://opendesignengine.net/attachments/download/480/ac-dc.gif

h1.  Discharge to Low Voltage, Not to Ground

In order to create the phenomenon, the following must be done:

1.  One plate of a capacitor is connected to Earth Ground.

2.  The other plate is charged to a high positive voltage (300-600V).

3.  The capacitor is rapidly discharged (through a spark gap, or thyratron) to a low positive voltage (12V).

4.  The capacitor is charged up again, and the process repeats.

Fig. 6.1:  "The Low Side of the Switch is Biased at +12v":https://opendesignengine.net/attachments/download/481/bias.gif

One terminal of the spark gap is connected to a conductor, which is connected to the high voltage plate of the capacitor.  The other terminal of the spark gap is connected to the positive terminal of a 12v battery.  The negative terminal of the battery is connected to Earth Ground.

h1.  Transformation Between Extensive Space, and Gegenraum

Consider the concept of "Duality" in Projective Geometry,  applied not to a particular solid, but to space itself.

Consider the transformation between the Infinite Plane (Euclidean Space), and the Point at Infinite Distance (Gegenraum).  This transformation is represented by Gabriel's Horn.

See the Projective Geometry developed by George Adams and Rudolph Steiner.

Fig. 7.1:  "Growth Measure from Gegenraum to Extensive Space":https://opendesignengine.net/attachments/download/454/gegenraum.jpg

h1.  Recursive Process

We believe that a recursive process iterates once, each time the Fountain is charged and discharged.

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