Wave Articulation Matrix

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.

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

All the cylinders have the same mass, and 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

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

Fig. 1.5:  "Example Dimensions -- 4-Element WAM":https://opendesignengine.net/attachments/download/453/4wam.jpg

Fig. 1.6:  "Photo of 4-Element WAM":https://opendesignengine.net/attachments/download/465/IMG_0061.JPG

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 kind of Logarithmic Spiral.  Its radius multiplies by the Golden Ratio every quarter cycle.  In the picture, 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

A Hyperboloid is also 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.

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

h1.  Wiring

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

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

No matter how many cylinders are in the WAM, let's call the innermost cylinder the Primary Axis, and the outermost cylinder, the Control Ring.  All the cylinders between the Primary Axis and the Control Ring are tied together electrically, and provide the output of the device.

Fig. 3.2:  "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.

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