^{ }

**Generation of
High-Frequency Gravitational Waves (HFGW) by Means of an Array of Micro- and Nano-devices**

(Paper HFGW-03-117, MITRE Gravitational-Wave
Conference,

by

Robert M. L. Baker,
Jr.^{†}

The
process by which High-Frequency Gravitational Waves (HFGW) are generated by
means of the time rate of change of the acceleration of a mass or masses,
termed a “jerk” or a “shake,” is developed. Arrays of micro- and nano-devices,
termed energizing and energizable elements, are utilized to generate a train of
coherent gravitational waves. As the waves progress along such devices they are
reinforced by the energizable elements, under the control of a computer logic
system in order to be modulated for applications such as communication. Starting
with a theoretical non-rotating, but ratcheting or jerking rim, linear devices,
such as a stack of ratcheting rims, evolve. Two specific devices are described
to illustrate the concept: the first such device involves a barrel or stacks of
concentric rims whose surface or edges are covered with an array of ultra-small
micromagnets (energizable elements) surrounded by a sheath of ultra-small
microcoils (energizing elements); it generates approximately 380 kW. The second
is much smaller, 18 mm in length, and through use of a superconducting lens
delivers a flux of 6.3x10^{-7} watts per square meter at a 7 km distant
receiver. Other HFGW generators
utilizing nanowire lattices for a high-frequency nanomechanical resonators and
parallel current-carrying plates are also investigated and the question of null
GW generation for symmetrical systems is considered.

The problem, which all of the devices
discussed in this paper solve, is to cause a system of masses, which could be
mini-magnets, micro-devices, nano-devices, individual molecules, submicroscopic
particles, or individual electrons (as in a superconductor), to move in concert with a jerk in order to *build
up* (generate) HFGW with either planar or cylindrical wave propagation. Such
jerking masses produce a very long sequence of HFGW pulses having significant
average power and ability to carry information without generating
incapacitating heat, causing disruptive g loads, or producing overpowering EM
radiation.

A area

a semi-major axis of a two-body orbit

*a *acceleration

B magnetic flux density

c speed of light or, alternatively, the electron mobility speed

__________________________________________________________________

†
Senior Consultant, GRAVWAVE® LLC and Transportation Sciences Corporation,

Copyright
2003 by R. M. L. Baker, Jr. Published by The MITRE Corporation with permission.

D_{a}_{b} quadrupole
moment-of-inertia tensor

d diameter or distance between conductors

E energy

e eccentricity of a two-body orbit

F force per unit length

f force

**f _{cf}** centrifugal-force vector

G universal gravitational constant

h strain
in the space-time continuum , Δ*l/l *caused by GW passage

I moment of inertia

i current

*l *length

M mean anomaly for a two-body orbit

m mass of an object on orbit in characteristic units or of any object in kilograms

*m *sum
of the masses of a pair of binary stars or mass of a rod in kilograms

N noise or index of GW refraction or number of pulses in an activation time interval

n mean motion for a two-body orbit or number of objects or elements or number of rim or ring sections in a stack

*n* number of coil turns or number of concentric rims or rings

P the magnitude of the power of a gravitational-radiation source

p parameter
or semilatus rectum = a(1-e^{2} )

q charge or periastron distance, a(1-e)

R resistance or range

r radial distance to an object on orbit; alternately, the effective radius of gyration

*r *radius
of a magnetic core, piston or barrel or stack of rims

S GW flux

*s *distance
or displacement

t time

t` spinning-
rod time

V volume or speed

v true anomaly of a two-body orbit

*v* velocity
or frequency

x axis of orthogonal coordinate system

y axis of orthogonal coordinate system

z axis of rotation orthogonal to x and y axis

a attenuation or diffraction angle

D small increment

Df_{cfx }incremental x component of
centrifugal force

Df_{cfy} incremental y component of centrifugal
force

Dt time increment

δ fraction of a linear-motor, GW generator’s barrel radius that is an energizing-element sheath

and/or energizable-element core

d*l* thickness
of a given rim or ring

dm differential mass

dt differential time or activation time interval

θ the central angle of a rotating rod

k_{I3dot} coefficient (constant or function) of
the kernel in the d^{3}I/dt^{3}
formulation of the quadrupole

l wavelength

m = m_{1} +m_{2} =
sum of masses on a two-body orbit in characteristic units

m_{0} permeability of free space

n frequency

s absorption cross section

t characteristic time; for
heliocentric unit systems = 5.022x10^{5} seconds

w angular rotational rate

1 refers to mass one

2 refers to mass two

a current in one wire

b current in adjacent wire

cf centrifugal

d diffraction

GW gravitational wave

*l *longitudinal

p phase

r radial

t tangential

x x component

y y component

1. INTRODUCTION

The general concept of the
devices discussed in this paper is to simulate scientifically acceptable
generation of gravitational waves (GW) like those that are produced by
energizable celestial systems such as rotating binary stars, star-black-hole
collisions, star explosions, star collapse, binary black holes, spinning black
holes, etc. through the use of smaller macro- and micro-, terrestrial or
laboratory energizable systems. Such terrestrial systems generate well over 40
orders of magnitude more force intensity by virtue of their use of
non-gravitational forces (nuclear or electromagnetic compared to gravitational)
than a typical celestial system and well over 12 orders of magnitude greater
frequency (THz or QHz {10^{15} Hertz; the term Quadrahertz, QHz, is
preferred over Petahertz or PHz} and higher compared to KHz or very small
fractions of a Hz) than a typical celestial system. Terrestrial energizable
systems produce significant and useful GW according to the various designs of
the devices to be described, even though they are orders of magnitude smaller
and less massive than the extraterrestrial celestial systems. In the various designs of these devices,
large numbers of small energizable elements are energized in sequence or in
concert, by energizing or stimulating elements, to emulate the motion of a much
larger and extended body in order to enhance the generation of GW.

The specific
concept, which will be expanded upon, requires applying a long series of rapid
“jerks” or “shakes” or third-time-derivative motion to a mass or series of
masses, using relatively strong magnetic, electric, or nuclear forces. The
devices described in the present paper will be shown to generate significant *High-Frequency Gravitational Waves* or **HFGW **without disruptive g loads. The
effect will be measurable in the laboratory since it affects or warps the *spacetime* geodesic over very small
distances (due to high frequency and short GW wavelength) and thereby will
produce HFGW detectable by utilizing detectors described in this Conference. If
the energizable elements are uncharged, then there may be little or no
attendant electromagnetic (EM) radiation.

In order to
illustrate the concept, a circular rim, which does not rotate, but ratchets or
jerks, is described. This rim is then evolved into a practical gravitational-wave
generator. The system of masses described in this paper (and subject to jerks)
can be small or mini magnets micro- or nano-devices, molecules, sub-microscopic
particles, electrons (e.g., in a superconductor; about 10^{20} per
cubic centimeter), etc. The misconception that the laboratory generation of GW
is not feasible is fed by the example of a spinning rod given in most
introductory textbooks. Such a rod utilizes the change in the centrifugal force
vector to generate GW and is torn apart well before any significant GW is
generated. The devices discussed herein are completely different and utilize
electromagnetic forces and reciprocating, not rotational, motion in order to
generate GW.

2. JERK FORMULATION OF THE QUADRUPOLE EQUATION

There is no new
Physics here, simply a different approach or formulation of the conventional equations
utilized to estimate GW power in order to render engineering applications more
apparent. I will employ the standard quadrupole equation, which was originally
formulated by Einstein in 1918, to compute the HFGW power. I will formulate
that basic quadrupole approximation in terms of a change in force, Δf,
over a short time interval, Δt, which is defined as a “** jerk.**”
The derivation of this basic jerk equation will be accomplished by two separate
analysis paths: one starting with the third derivative of the moment of inertia
formulation of the quadrupole equation and the other starting with the spinning
rod (or binary orbit) formulation of the quadrupole equation. The resulting
jerk equation will be numerically checked against the known result for the
binary star pair PSR 1913 + 16.

2.1* Derivation from Third Time Derivative of
the Moment of Inertia*

As is well known and noted
specifically in a letter (dated *National*
*Security Agency*: “Because of symmetry, the quadrupole moment can be
related to a principal moment of inertia, I, of a three-dimensional tensor of
the system and … can be approximated by

-dE/dt » -G/5c^{5}
(d^{3}I/dt^{3})^{2}
= - 5.5x10^{-54} (d^{3}I/dt^{3})^{2}.” (1A)

or from Eq. (110.16), p. 355 of Landau and
Lifshitz [5] :

P
=│- dE/dt │ =κ (G/45c^{5})(d^{3}D_{a}_{b}/dt^{3})^{2} [watts] (1B)

or

P = 1.76x10^{-52} (d^{3}I/dt^{3})^{2} [watts].
(1C)

This is Einstein’s *quadrupole* *equation**.*

In
Eq. (1A), k in Burdge’s notation is G (not, however, the Einstein tensor) and
the units in Eq. (1C) are in the MKS
system [watts] not the cgs. In order to introduce the jerk concept let us
consider the hypothetical example of a rim that, like the ratchet wheel of a
mechanical watch, need not be uniformly rotating or, in fact, not rotating at
all. In this case, for a collection of
masses, which are small permanent magnets, along the rim,

I = dm r^{2}
[kg-m^{2}],
(2)

where

dm = mass of an individual magnetic sites around the
rim [kg], and

r = the distance from a pivot out to any single dm on
the rim [m] (or more exactly, the *radius
of gyration* of the rim). Thus

d^{3}I/dt^{3}=dmd^{3 }(r^{2})/dt^{3}= 2rdm(d^{3}r/dt^{3} )+…^{ }(3A)

Approximately, by delta differentiation,

2r{δm(d^{3}r/dt^{3})}
≈ 2r{δmΔ(d^{2}/dt^{2})/Δt} (3B)

^{ }

and, by noting that by

f_{r}
= δm( d^{2}r/dt^{2}), (4A)

we have, again by delta differentiation,

δmΔ(d^{2}r/dt^{2})
= Δf_{r} (4B)

where f_{r} = radial force on dm and Δf_{r }is the rapid increase in
f_{r} over time Δt (**the
jerk**). The third derivative of I is, therefore, approximated by

d^{3}I/dt^{3} @ 2r Df_{r}/Dt , (5)

in which Df_{r} is the nearly instantaneous **increase** in the force on magnetic (or
other energizable element) sites, dm, caused by the magnetic field of
current-carrying coils (or other energizing elements) when they are turned on
and off or pulsed by transistors or ultra-fast switches resulting in a jerk.

Let
us now visualize a stack of such rims; each one composed of a circle of small
permanent magnets that are surrounded by a close-by ring of coils (please see
FIG. (3B)). In this regard, the coils adjacent
to the periphery of each rim are sequenced (at the local GW speed, say the
speed of light) along the stack of rims from one rim to the next in order to
generate or build up the train of coherent HFGW as they move through the stack
of rims (energizable magnetic sites). In
order **not** to build up acceleration
the jerks are reciprocating; but (arguably) due to the square in the kernel of the quadrupole equation, the GW radiates in
both directions along the axis of the circular rims (through their centers) no
matter which direction the peripheral magnetic masses are jerked. In summary,
by substituting Eq. (5) into Eq. (1C),

P = 1.76x10^{-52} (2rDf_{r}/Dt)^{2} [watts], (6)

which is the **jerk
formulation of the quadrupole equation.**

** **2.2 *Derivation
from a Spinning Rod*

An alternative derivation of Eq. (6)
is as follows: From Eq. (1), p. 90 of
Joseph Weber [1] one has for Einstein's formulation of the gravitational-wave
(GW) radiated power of a rod spinning about an axis through its midpoint having
a moment of inertia, I [kg-m^{2}], and an angular rate, w [radians/s] (also please see, for example, pp. 979
and 980 of Misner, Thorne, and Wheeler [2], in which I in the kernel of the
quadrupole equation also takes on its classical-physics meaning of an ordinary
moment of inertia):

P
= 32GI^{2 }w^{6 }/5c^{5} = G(Iw^{3})^{2}/5(c/2)^{5}
[watts] (7)

or , with I = r^{2}m (r being the radius
of gyration of the rod)

P = 1.76x10^{-52}(Iw^{3})^{2}
=

1.76x10^{-52}(r{r*m*w^{2}}w)^{2} [watts] (8)

where {r*m*w^{2}}
can be associated with the magnitude of the rod’s centrifugal-force vector, **f _{cf}**. Equation (8) is the

The
**f _{cf}** vector reverses every
half period at

f_{cf}
Δ** **f_{cf }= f_{cfx}Δ f_{cfx} + f_{cfy}Δ
f_{cfy} (9A)

and when one associates the components Δf_{cfx,y }with f_{cfx,y} Δθ
and, after dividing by Δt` (t` being spinning-rod time), and noting that
Δθ/Δt` =ω,

f_{cf}
Δ** **f_{cf }/ Δt` = (f_{cfx}^{2}
+ f_{cfy}^{2})ω
. (9B)

Thus
Δ** **f_{cf }/ Δt` = f_{cf}ω ; but
Δt` = ½Δt since the period of the GW is half the period of the rod,
so that

2
Δ** **f_{cf }/ Δt = f_{cf}ω, (9C)

but f_{cf}= {rmω^{2}} so

2
Δf_{cf}/Δt = {rmω^{2}}ω (9D)

and substituting Eq. (9D) into Eq. (8) yields

P
= 1.76x10^{-52 }(2rDf_{cf } /Dt)^{2}, (10)

where (2rDf_{cf }/Dt)^{2} is the kernel of the quadrupole
approximation equation and Df_{cf } /Dt is, again, the **jerk.** Equation (10) is
identical to Eq. (6), but arrived at by a different analysis path.

Equation
(6), like Eqs. (1), (7), (8) and (10), are approximations for GW power and may **only hold accurately **for r << l_{GW }and _{ }for speeds of the
GW generator components far less than the speed of light, c. Please see, for
example, Pais [3], p. 280 and Thorne [4], p. 357. (On the other hand, Leonid P.
Grishchuk at this Conference suggested that the requirement that r << l_{GW } may not be a stringent one.)

* *2.3*.
Validation Based on Orbit of PSR 1913+16*

As
a numerical validation of Eq. (10), that is a validation of the use of a jerk
to estimate gravitational-wave power, let us utilize the approach for computing
the gravitational-radiation power of the pulsar **PSR 1913+16** observed by Hulse and Taylor [6] to demonstrate the
existence of GW.

2.3.1 *Orbital Elements*

Since the observation of the binary pulsar PSR 1913+16 (identifies right
ascension of 19 degrees 13 minutes and declination of 16 degrees North)
represents the only *experimental
confirmation* of gravitational waves, insight into the jerk approach can be
found in the analyses of such a double-star system. Thus please bear with the
following rather laborious arithmetic.

The pair of PSR 1913+16 stars will coalesce in 3.5x10^{8} years
due to GW radiation and produce a rather continuous GW until that time. It is
the pair’s coalescing that **exactly **agrees
with GW-generation theory (utilizing orbital mechanics) that indirectly
confirms the existence of GW. According to J. H. Taylor, Jr. [6], the period of
their mutual rotation is 7.75 hours (or **2.79x10 ^{4}
[s]**), periastron is 1.1 solar radii (one solar radius is 6.965x10

2.3.2 *Gravitational-Wave Power—the Quadrupole
Equation*

In the case of a
binary star pair such as PSR 1913+16, the magnitude of the GW power, P, is
computed from the quadrupole equation, which for two masses on orbit about one
another is given, for example, by an equation on p. 356 of L. D. Landau and E.
M. Lifshitz [5]^{ }or Peters and Mathews [7]. The time-constant factor
in the equation for P is

8G^{4}m_{1}^{2}m_{2}^{2}μ/(15c^{5}). (11)

The *time-variable factor* in P is a function of the true anomaly, v, and
orbital eccentricity, e, as given in [5] :

(1+ecosv)^{4}([1+{e/12}cosv]^{2}+e^{2}sin^{2}v)/(a[1-e^{2}])^{5}. (12)

In conventional astrodynamic/celestial-mechanics notation (see Samuel Herrick [8]) this factor (i.e., Eq. (12)) is

p/r^{6}+(dr/dt)^{2}/12mr^{4
}, (13)

where p is the orbital
“parameter” or semilatus rectum (= a{1 – e^{2}}) in [AU], r is the
radial distance between the two masses [AU], t is the characteristic
time measured in k_{s}days or in units of 5.022x10^{6 }[s] for
a heliocentric-unit system (utilized by Taylor [6] and others for PSR 1913+16),
m
is the sum of the two masses, i. e., μ = m_{1 }+ m_{2}
[solar masses], and as usual G = 6.67423x10^{-11} [m^{3}/kg-s],
and c is the speed of light = 3x10^{8} [m/s]. Note that one AU
(astronomical unit) = 1.496x10^{11} [m].

The GW power
radiated, P, which causes a perturbation in the semi-major axis, a, (and an
attendant secular decrease in the orbital period) is obtained by integrating
the time-variable factor, Eq. (13), over the orbital period using the mean
anomaly, M, as independent variable, which is directly proportional to the time
(that is, M = n [t-T], where n is the mean motion {w in Landau and Lifshitz’s
[5] notation, p. 357} n= 2π/Period = 2π/2.79x10^{4} = **2.25x10 ^{-4} [1/s]**, and T is
the time of periastron passage).

2.3.3 *Accuracy of the Results*

The value of the average GW power, P, is computed from observations that define the eccentricity (based primarily upon Doppler-shift determination of the range rate at periastron and apastron), semi-major axis, and orbital orientation angles of PSR 1913+16. The error in the computed value of P is related to the observational error leading to the determination of the orbital elements as well as the determination of the masses of the pair of neutron stars. For example, a 0.1 percent change in the measurement of range rate at periastron results in a 0.28 percent change in GW power, P, and a 0.1 percent change in the mass of the stars results in a 0.33 percent change in GW power.

2.3.4 *Centrifugal Force and Acceleration*

The x and y
average delta **centrifugal force **component(s),
Df_{cfx,y} (which will later be utilized to validate the
fundamental jerk equation numerically) are both

*m*an^{2}= (5.56x10^{30})(2.05x10^{9})(2.25x10^{-4})^{2}
= **5.77x10 ^{32} [N]** (14)

divided by *m *yields the average centrifugal acceleration = 103.78 [m/s^{2}]
= 10.6 [g’s]. At periastron, r = q =
a(1-e) = (2.05x10^{9})(1-0.641) = 7.36x10^{8} [m] (with e =
0.641), the centrifugal acceleration is q(dv/dt)^{2} where dv/dt = Ö(mp)/r^{2}
(please see Baker [9] , p. 13). In this latter case m = 2.8 [solar masses], a =
2.95 [solar radii] = (2.95)(6.965x10^{8} [m/solar radii])/1.496x10^{11}
[m/AU] = 0.01373 [AU], p = a{1-e^{2}} = 0.01373{1- 0.4109} = 0.00809
[AU], and q = r = 7.36x10^{8} [m]/1.496x10^{11}[m/AU] = 0.00495
[AU]. After inserting these numbers I have dv/dt = (Ö[2.8x0.00809]/[0.00495]^{2})/5.022x10^{6
}[s/k_{s}day] = 1. 223x10^{-3} [radians/s]. Thus the
centrifugal **acceleration **at
periastron of the star pair is q(dv/dt)^{2} = (7.36x10^{8} [m]
)(1.223x10^{-3} [radians/s])^{2} = 1.101x10^{3} [m/s^{2}]
= **112** **[g’s]** – apparently *still
within the weak-field approximation of Einstein’s GW equations*.

2.3.5 *Comparison of Results*

From
Eq. (14) I computed that each of the components of force change, Df_{cfx,y} = 5.77x10^{32} [N]
(multiplied by two since the centrifugal force reverses its direction each half
period) and Dt = (1/2)(7.75hrx60minx60sec) = 1.395x10^{4} [s] for the half
period. Thus using the jerk approach:

P = 1.76x10^{-52}{(2rDf_{cfx}/Dt)^{2} + (2rDf_{cfy}/Dt)^{2}} =

1.76x10^{-52}(2x2.05x10^{9}x5.77x10^{32}/1.395x10^{4})^{2}x2

= **10.1x10 ^{24}**
[watts] (15)

versus the result of **9.296x10 ^{24}** [watts] using Landau and Lifshitz’s more
exact two-body-orbit formulation given by Eqs. (1.1) and (1.2) of Baker [10]
integrated using the mean anomaly not

3. COMPUTATION OF HFGW POWER

There
are some very sophisticated and exact computer simulations of the generation of
gravitational waves (please see, for example, S. F. Ashby, *et al* [11]). The quadrupole approximation utilized herein by me
and, for example, by Romero and Dehnen [12] and others at this Conference is
probably less exact. On the other hand,
the computer simulations are less relevant to the devices involved in the
generation and detection of HFGW in the laboratory. These computer simulations
describe GW generation by strong-field astrophysical phenomena (e.g., neutron
stars, black holes, etc.), coupled spacetime and general relativistic hydrodynamic
equations, and are usually restricted to gravitational forces ; not
non-gravitational forces involved in laboratory HFGW generation. I will first
discuss the meaning of the term quadrupole

* *3.1* Meaning
of Quadrupole*

The
basic physical process for generating a gravitational wave is the third (or
higher) time derivative of the motion of a mass, termed a "jerk" or
“shake” or Δf/Δt, that is, Δf is an increase in force, f, on the
mass carried out over a small time interval, Δt. As noted in Baker [10], that physical process
produces a gravitational wave with a ** power**
given by, for example, the quadrupole approximation (as originally derived by
Einstein) or it could be determined directly from the special and general
relativity equations (using a computer-implemented numerical integration as,
for example, discussed in. Ashby,

Other
algorithms, often most complicated, can define other GW properties such as
direction, polarization, constructive/ destructive interference, etc. This
situation is similar to

The two-body
approximation itself is ** not**
the physical law at all, but only one means of describing the resultant motion
– a “lowest-order solution.” In the case of a nuclear-reaction-generated
gravitational wave, in which a nuclear particle is ejected from a nucleus, it
is like a small rocket, or in the case of electrons shaken in a resonance
cavity, plasma beam, superconductor, etc., there is a third time derivative of
the motion of the nucleus in the first case or electron mass in the second
case, or a jerk, which produces gravitational waves whose power can be
estimated, for example, by the quadrupole approximation. Thus when I mention a “quadrupole-produced
gravitational wave” I’m really implying the fundamental physical concept of the
jerk and not the computational means for establishing the power of the
gravitational wave.

* *3.2* Harmonic Motion.*

As far as a harmonic motion of a mass or a pair of masses is concerned (harmonic oscillator), gravitational waves are generated. Just as in the case of a pendulum, the usual descriptor of harmonic motion, there exists a third time derivative of the pendulum bob. It is the jerk of that bob that produces the gravitational wave, which can be estimated using a quadrupole approximation or computed exactly by means of a rather complicated solution of the equations of special and general relativity.

4. LABORATORY MICRO-
AND NANO-SCALE HFGW GENERATION DEVICES

** **In this
section I will describe an Individual Independently Programmable Coil System or
** IIPCS
** (U. S. Patent No. 6,160,336),
miniature integrated circuits, which provide for the emulation of a device that
is much more extensive than the individual energizable elements (e.g., the
small permanent magnets), and I will sumarize what all of the HFGW-generation
devices accomplish.

4.1 *Individual Independently Programmable
Coil System*

I
will now discuss HFGW generation devices that utilize, for example, microchip
and nanotechnology in order to generate HFGW in the laboratory. For the very
large number of ultra-small, sub-millimeter coil elements utilized in some of
the devices discussed, a miniaturized integrated circuit can be utilized (please
see, for example, the coil turn of *Al*
utilized by Y. Acremann, *et al *[13]).
They will be embedded in or imprinted on a silicon chip, organic material, or
in connection with polymer-based or superconductor devices. They will consist
of multiple layers with appropriate sequencing time delays to ensure near
simultaneity of the magnetic fields interaction as the direct-current train of
approximately one-picosecond or shorter pulses simultaneously traverse each
coil set on the chip levels. The timing sequence could be integrated in the
chip with the ultra-fast switches or transistors or through other
semi-conductor devices. The myriad of
these small coils in a three-dimensional array (please see FIG. (1)) act on the
field of a small magnet to produce the jerk.

* Since the jerk is generated by
an electromagnetic process, there could be significant EM radiation generated
that could reduce the efficiency of the device.* It should, however, be emphasized that **it is not the magnetic field that
generates the HFGW**, but rather the mass of the magnets (or other
energizable elements) that are jerked that generates the GW. The magnetic
material exhibits magnetic sites (perhaps on a molecular level or ferromagnetic
atoms) that, of course, include electrons; but in this case (as opposed to a
superconductor HFGW generator) it is not the electron mass being jerked that
produce the GW, but rather the actual magnet’s mass.

4.2
*Miniaturized Integrated Circuits*

* *A
preferred design (U. S. Patent No. 6,417,597) utilizes conventional computer chips
or wafers of a computer logic system, containing IIPCS circuit elements (U. S.
Patent No. 6,160,336). These circuits are about 18 micrometers or less apart and
include a synchronizing clock, input/output
ports, and sub-millimeter coils on 50 to 100 micrometer centers. The chips are about 6 mm to 9 mm square and
are obtained from silicon wafers. These
chips are sewn into a circuit-board roll with an approximately
25-micrometer-diameter gold thread.
Several layers of this roll (for example, 25) are connected in a fixed
location or band adjacent to the moving or non-moving (jerking or non-jerking) spindle’s
rim and they form the IIPCS in the spindle rim’s magnetic field. (The rolls of
chips just mentioned are routinely fabricated by French-owned *Oberthur Card Systems* {a plant exists at Rancho *Gemplus,
Schlumberger* {*Frost & Sullivan*.)

4.2.1 *Coil Sets*

In the proposed miniaturized integrated circuit devices, as exhibited in
FIG. (1), there will be a very large number of ultra-small, sub-millimeter or
microscopic coil sets or elements, 56, embedded in or imprinted upon a silicon
chip, 57, in multiple layers. Ultra-fast micro-switches or transistors of the
IIPCS, 58, will launch a long a series of current pulses, 59, of approximately
nanosecond to picosecond or less duration moving at the electron’s mobility
speed (approximately light speed, c) that will be timed to reach the individual
coil sets or elements almost simultaneously (with the same rise time as
discussed in Y. Acremann *et al *[13]^{
}). These pulses can travel along several individual conductors, as in
FIG. (1), or along one single conductor per line, as in FIG. (2), and thereby
interact with the magnetic field of a nearby magnet on the rim, 60, in
concert.

This interaction will result in a third-time-derivative motion or jerk of
the uncharged magnetic masses on the rim to generate a train of gravitational
waves. **The effect is exactly the same as
a rotating or ratcheting rim with the change in centrifugal force (jerks) replaced
by the reciprocating jerks of the magnets attached to the rim.** The
ultra-fast switches are preferably semiconductor-based, such as the
semiconductor optical amplifier (SOA), a semiconductor nonlinear interferometer
such as a nonlinear *Sagnac *interferometer
on a phosphide semiconductor chip, etc. (please see, for example, D. Cotter, *et al* [14], pp. 1523-1528).

4.2.2 *Pulse Duration*

The pulse duration will be such as to completely energize any given coil set as it passes through it in order to produce a magnetic field interaction. As has been emphasized, the interaction will result in a third-time-derivative circumferential motion or jerk of a cylindrical stack of rims, 63, shown in FIG. (3A) (figure 2 of US Patent 6,417,597) and generate a long GW train of successive GW pulses having axis, 29. This stack or barrel is surrounded by and immediately adjacent to a sheath of IIPCS-controlled coil sets, 64. The cross–section of the barrel or an individual rim of the stack is shown in FIG. (3B) (figure 8A of US Patent 6,160,336). The coils (myriads of them represented by a single coil) 26, interact with the tiny rim magnets, 24; produce jerks along axes, 27, which emulate a ratcheting rim, 15. In the case of a design with the current-pulse train on a single conductor interconnecting a line of coil sets, FIG. (2), there will be a build up of impulses to full value as the current-impulse train progresses down the line of coil sets. Use of a single conductor wire for each line of coil sets reduces the resistive power loss. In each line of coils set in series 61 there will be time delays, 62, between coil sets to ensure simultaneity of the current pulses reaching any given coil set. The myriad of miniature coil sets (incased in layers of chips) will energize (jerk) each tiny rim magnet.

4.2.3 *Parallel-Conductor Stacks*

In FIG. (4), ultra-fast switches or transistors of the IIPCS, 58, will launch a long series of direct-current pulses acting in either direction, 59, of approximately nanosecond or picosecond or less duration moving at the electron’s mobility speed along individual conductors or single interconnecting computer wires in order to produce current pulses, 59, acting in concert to generate modulated jerks and resulting HFGW (GHz to THz and higher frequencies) with axis, 29 (or perpendicular to that axis). The current pulses will be timed to reach parallel-plate conductors, 66, which may have different masses or may have ballast, 67, attached and/or carry different current, and/or have different modulus of elasticity and/or are constructed differently in their mountings for the purpose of exhibiting asymmetrical mass displacements, jerks or “hammer blows.” The asymmetry is required in order to avoid the null situation to be discussed in Section 9. (Such a concept of utilizing the force between parallel current-carrying conductors is similar to the nanowire or nanoplate devices to be described in Section 8.)

4.3 *Emulation of a Much More Extensive Body*

As a GW front
passes by the energizable, e. g., in the case of parallel-plate, elements
(schematically shown in FIGS. (5A) and (5B) as 80, 84, 86, and 88) or
individual members of a stack of (jerking) rims as shown in FIG. (3A), they are
energized in sequence thereby increasing the wave’s amplitude. In FIG. (5B)
such an effect is schematically illustrated as GW 83 ,85, 87, and 89 build up
to accumulate the GW, 82, wave front shown also in FIG. (5A). **Thus a linear device having a much longer
effective length (or radius of gyration), r, or a cylindrical stack of rims
having a much larger mass than any single rim is emulated.** Again, this is
all subject to experimental verification. It is to be emphasized that **any unwanted EM radiation can be screened
out.**

In FIG. (6),
ultra-fast switches or transistors of the IIPCS, 58, will launch a long series
of current pulses, of approximately picosecond duration along individual
conductors or single, interconnecting conductor wires that will be timed to
reach individual, sub-millimeter micro- or nano-electromechanical elements, or
piezoelectric crystals, etc. 56. This will be in sequence to reinforce and
cause a *build up* of the amplitude of
a **coherent **GW beam (as in FIG. (5B))
having axis, 29. The ensemble of electromechanical elements (including other
kinds of energizable elements such as nanomachines) will also be embedded in or
imprinted on a silicon chip in multiple layers. FIGS. (7A) and (7B) (figures 7A
and 7B of US Patent 6,160,336) exhibit the ultra-fast, micro-switches (1.1d,u
to 1.4 d,u and 1.5 *l*,r to 1.8 *l*,r) set so no current flows through the
coils and ultra-fast, micro-switches (2.1 d,u to 2.4 d,u and 2.5 *l*,r to 2.8 *l*,r) set so that the current flows through the coils right to left.
Other switch setting can reverse this current direction.

** **4.4** ***Summary*

The problem, which all of the devices
discussed in this paper solve, is to cause a system of masses, which could be
mini-magnets, micro-devices (e.g., small plates), nano-devices (e.g.,
nanowires), individual molecules, submicroscopic particles, or individual
electrons (as in a superconductor) to move in concert with a jerk in order to *build
up* (generate) HFGW with either planar or cylindrical wave propagation. Such
jerking masses produce a very long sequence of HFGW pulses having significant
average power without generating incapacitating heat, causing disruptive g
loads, or producing overpowering EM radiation.

As I have emphasized, the problem is solved in several alternative ways by utilizing an array of energizable elements (e. g., rim-magnets, coils, parallel plates, piezoelectric crystals, dielectrics, capacitors, nanomachines, high-temperature superconductors, electrons, nuclear particles, laser beams, etc.) to be activated by energizing elements (e.g., coils, submicroscopic particles, laser beams, etc.) under computer-logic-system control.

As already noted, these energizable elements
are activated or energized in the correct sequence with correct timing by the
IIPCS computer (computer-controlled logic system) to accumulate a GW (moving at
local GW speed in the energizable mass, which may or may not be near to the
vacuum light speed) as the GW front moves in the mass or collection of masses.
Essentially, the IIPCS causes the entire mass or collection of masses, or rims,
or molecules to jerk effectively in unison or in step with the GW wave front
and generate coherent HFGW. That is, the jerk will progress in step with the GW
front and build the GW amplitude up – somewhat similar to a cyclotron pulsing a
charged particle as it circles around in its magnetic field, or, possibly, like
a traveling-wave amplifier and similar to the coherent GW generation suggested
by Romero and Dehnen [12]. Energizable elements (that jerk when energized) are
energized in sequence as the GW front passes. As has been seen, these elements
taken together emulate a much larger, more extensive mass. **That is, the entire mass “appears” to the GW (as it passes) to be a
single larger mass (e.g., a solid massive cylindrical flywheel) being jerked
cohesively.** Experiments suggested at this Conference would not only shed
light on such HFGW characteristics, but also, as suggested by Y. Acremann, *et al* [13] in their discussion of the
processional motion of the magnetization vector “… forms the basis for
realistic models of magnetization dynamics in a largely unexplored but
technologically increasingly relevant (picosecond) time scale.”

5. MAGNETIC FIELD BUILD UP AND HEAT LOSS

** **I will commence the analysis using the
theoretical example of the ratcheting or jerking rim and then evolve the device
into both a stack of rims and into a linear form. It should be recognized again
that a rotating rim could generate GW, but in order to generate significant and
continuous GW its rotational rate would need to be so large that the rim would
be torn apart! Thus a rotating rim has been replaced by a ratcheting or jerking
rim and that rim will be replaced by a stack of rims composed of individual
jerking rim elements and finally by a linear motor.

* *5.1*
Magnetic Field Build Up*

Although of little concern in most applications,
the length of time to "build-up" the magnetic field of the coils is
important here as it is in the experimental work of Y. Acremann [13]^{.} The electrons must complete sufficient coil
turns (moving at the electron’s mobility speed – about 2.3x10^{8 }[m/s]
or nearly light speed) in approximately a picosecond to "launch" most
of the magnetic field that produces the impulsive force (like a “hammer blow”)
or jerk when it interacts with the static magnetic field of permanent or
electromagnets carried around by the ratcheting rim. Thus, they must be __very__ tightly wound
with each coil "set" having a total length of less than 0.3 mm
(0.0003[m] or 300 micrometers). If each of the ultra-small, sub-millimeter coil
sets consist of two coils or turns, as exhibited in FIGS. (7A) and (7B), then
their diameters are on the order of d = 0.3/2 p = 0.05[mm] = 50 [μm] or less. (Note that
the single-turn coil of *Al, *utilized
by Y. Acremann, *et al *[13] was about
6 [μm] in diameter.) The coil wire could be made of gold having about a 0.015-mm
or 15-micrometer diameter. The resistance for such wire at room temperature is
about 135 [ohms/m] -- high-temperature superconducting (HTSC) material would be
useful here. As will be seen from Section 5.4.2, in the spin-up jerk mode the
IIPCS will need to build up **0.26 [Tesla]**
flux density, at the appropriate polarity and interval, e. g., every 0.044 [m]
for magnets of that rim spacing (the requirements for the spin-down jerk mode
are essentially the same, but reversed). The magnetic flux density, B, is given
by

B = m_{o}*n*i/*l *[Tesla]
(16)

where m_{o} = 4px10^{-7} (permeability of free space), *n* is the number of coil turns, i is the
current through the coils [amps], and *l*
is the length of the coil conductors [m]. The double coil sets will be placed
on 50 to 100 [micrometer] centers, so that there will be about 2x100x100 = 2x10^{4}
coil turns on each square-centimeter level of the stack of 25 coil levels or
layers. With *l *= 0.044 [m], *n* =
25x2x10^{4 }= 5x10^{5}, i = 9.1x10^{3}/5x10^{5}
= 0.018 [amps] or 18 milliamperes, , *n*i
= 9.1x10^{3} [amp turns], so that Eq. (6) yields **B = 0.26 [Tesla].**

5.2 *Heat Loss*

The total length of 15-micrometer- diameter
gold wire across any given layer or level is 100(rows) x100 (coil and
jumper/time-delays)x(600 micrometers) = 6 [m]. For the 25 layers or levels
there will 150 [m] of wire with a resistance of 150[m] x135[ohms/m] = 2.025x10^{4}
[ohms]. Since on average every other pulse interval across a conductor wire
will carry no current (resulting of course, in a lower average GW power), the
heat loss per centimeter of chip stack or semi-conductor layers is

(1/2)i^{2}R
= **3.28 [watts]**. (17)

This heat loss can be reduced by 32% by using
25-micrometer-diameter wires for the time-delay jumpers, but high-temperature
superconductors (HTSCs) for this purpose are contemplated. In addition
there may be some energy loss or resistance occasioned by EM radiation
generated during the GW-generation process **–
reduced or eliminated since the jerked masses are uncharged**. Such an EM energy
loss can be reduced by the design of the energizing coil elements and
controlling the direction of current pulses by the IIPCS. Concerns of the influence of magneto resistance
(MR) of both the conductors and the semiconductor circuits and the dynamics of
the impulsive magnetic field buildup should be addressed during experiments as
would be the aforementioned EM radiation, which could significantly reduce the
efficiency of the HFGW generator. Note that alternating currents are **not **utilized (only direct-current,
positive pulses) in order not to drive the electrons to the conductor’s skin
and thereby increase resistance. This is probably not a problem if
superconductors are utilized and their utilization is contemplated in most of
the HFGW devices described in this paper.

5.3
*Direction of the Generated HFGW*

The HFGW is expected to progress both ways
along the axis of the rims (or cylindrical stack of independent rims) since
there is a square associated with the kernel of the quadrupole equation—so
there is no preferred direction along the axis of the cylinder. Such a concept
will be subject to experimental verification and the propagation direction may
be dependent upon whether or not there is circular polarization or cross “+”
polarization or a combination. Between the rims HTSC lenses can be inserted to
concentrate the HFGW along the axis of the rings. As has been emphasized a
large number of ultra-fast switches, preferably semiconductor based (exhibited
in FIGS. (7A) and 7(B)), would be activated simultaneously along the
progressing gravitational wave front in one of the two directions by the
coil-control computer, with communication lines of nearly equal length to all
switches. The coil-control computer logic system of the IIPCS could also
activate coil sets inside and outside the rim, not just outside as shown in
FIGS.(3A) and (3B) and concentric “layers” of rims and coils could be utilized.

The coils, which are closely adjacent
to magnets or magnetic sites in the rim or rims, are to be sequentially
activated in order to generate coherent HFGW in one direction (non-coherent
HFGW will propagate in the other direction). Depending upon the proximity of
the coils and the duration of the current pulses, there may be currents induced
in one coil juxtaposed to another. *Any
induced currents will produce deterministic reverberations for which the
computer logic system of the IIPCS that can be programmed to account for.*
In any event, the reverberations would subside as the current-produced magnetic
pulses either collapse or clear the ensemble of chip layers at light speed.

* *5.4* Linear-Motor,
Linear-Jerk GW*

* *The foregoing discussion of a ratcheting rim is
included first in order to bridge the gap between the rather conventional
celestial GW mechanisms involving circular motion and the terrestrial
laboratory cylindrical stack of “rotating” or jerking rims. Another laboratory
device is similar to that proposed by Romero and Dehnen [12] and involves a
linear motor device* *involving linear
motion. (Yet another variant of the ratcheting rim is a ratcheting rod shown in
FIG. (8), which could be oriented at various angles.) In the following
subsections I discuss the linear-motor concept, estimate the coil-magnet force
(utilized for all the devices discussed in this paper), and by means of a
numerical example, determine the material acceleration.

** **5.4.1*
Concept*

The linear-motor design of the
HFGW-generation device, sometimes referred to as a *linear induction motor* or LIM, is visualized to involve a single
sector of the ratcheting rim with the impulsive forces being longitudinal, Df_{l}_{,}
rather than radial. If the rim magnets and adjacent coils were peeled off from
the rim and laid out flat, then the result would be a linear motor. Please see
FIGS. (9A) – (9D) for a schematic of the progression of such a “peeling.” In a very
hypothetical case , a 2000-meter-radius exemplar device once peeled would be
2πr = 2πx2000 = 6283 [m] in length and, since for this linear mass
distribution I = (1/3)mr^{2} and d^{3}I/dt^{3} @ (2/3)r Df* _{l}
*/Dt , the effective radius or radius of gyration is 6283/3 = 2094 [m]
≈ 2000 [m] (a measure of the mass distribution) and I assume that it is a
tube 3 [m] in diameter. One-centimeter-wide chip rolls would be placed
longitudinally along the sides of central, cylindrical, permanent- (or
electro-) magnetic tube, core, piston, or barrel, 63, consisting of an array of
magnetic energizable element sites, 57, as shown in FIG. (3A) if one replaces
the stack of jerking rims by a solid cylinder of magnets surrounded by a sheath
of coils. The thin (approximately one cm thick) band of Alnico 5 permanent
magnets could be replaced by far stronger electromagnets that face outward as
in FIG. (3B).

5.4.2* Estimate of Force*

In general, permanent magnets** **exhibit irregular magnetic fields and
associated forces. As a rule of thumb a band of juxtaposed 1.75-inch-long (0.044[m])
magnets will produce in excess of 30 pounds per 1.75 inches (or 206 pounds per
foot) of tangential force. Each 1.75-inch permanent magnet has a flux density,
B, of about 2,600 gauss or **0.26 [Tesla]**
developed every 4.4 [cm]. This matches the magnetic flux density of the
juxtaposed coils from Eq. (16). Thus each meter-long, square-centimeter segment
of the roll would produce about {(206 pounds)/(2.2 pounds per kilogram)} {3.28
feet per meter} = 307 [kg/m] x{ 9.8 Newtons per kilogram weight}or F ~ **3000 [N/m] **of longitudinal force, f* _{l}*, per meter and all together
they form a sheath of sub-millimeter coils (energizing elements) surrounding
this central magnetic core, tube, piston, or barrel. I will be extrapolating
these numbers to micro- or nano-magnets so it is important to establish a
specific magnetic force per unit volume, which of course would be much larger
if electromagnets replace the permanent magnets and HTSCs were introduced. The
impulsive force per unit volume for the meter long square-centimeter cross-section
magnet and closely adjacent coil combination, is Δf

5.4.3 *Numerical Example*

As a numerical example for the
linear-motor design, there would be about one roll or 25-layer strip of chips
spaced around and adjacent to the cylindrical barrel of the linear motor (64,
shown in FIG. (3A)) in a longitudinal direction (parallel to the barrel axis)
every two centimeters forming the sheath. There would be px3[m]x100[cm/m]/2[cm] = 471 strips around the
barrel’s circumference, each one having a length of 2πx1000 = 6283 [m] so
that

Df* _{l}*=(471)(6283[m])(3000[N/m])
=

and with k_{mr3dot} = 32 (the theoretical quadrupole-approximation value to be established
experimentally since r may not be less than l_{GW}),

P= 1.76x10^{-52} (⅔x6283x9x10^{9}/10^{-12})^{2}
= 0.25 [watts]. (19)

Thus, with the reference area being the two 3 [m]
diameter “barrels” or “pipes” or “tubes” or cylindrical ends (GW propagating in
both directions so the area is doubled) with a thickness of one centimeter,
area = 2 (3π) (0.01) = 0.19 [m^{2}], the generated HFGW flux is
about 0.25/0.19 = 1.3 [watts/m^{2}] near the hypothetical device. The
average HFGW flux or signal would be about 1 [watt/m^{2}].** **As a point of reference I again
compare our terrestrial HFGW generator to celestial Low-Frequency Gravitational
Wave or LFGW generation.** **Thus, for
the sake of argument (although admittedly, it is like comparing “apples to oranges”)
the 1 [watt/m^{2}] is compared to **4x10 ^{-16}
[watts/m^{2}]** maximum signal from a 500 mega parsec [Mpc] distant,
1000 black-hole (BH) radius semimajor-axis binary black hole (BBH) osculating
orbit and

For
cylindrical GW, in case the barrel magnets participated in harmonic oscillation
(each end’s uncharged magnetic sites moved in and out harmonically relative to
the other), the reference area would be (6283)(3p) = 6x10^{4}
[m^{2}] and the GW flux would be 0.25/6x10^{4} = 4x10^{-6}
[watts/m^{2}]. Although the jerked masses are uncharged, the
high-frequency electromagnetic fields may generate significant EM radiation
that will be studied in any experimental effort.

5.4.4
*Material Accelerations*

The acceleration, *a*, caused by the jerk is obtained by
multiplying the activation time, δt, by the time rate of change of
acceleration. The time rate of change of acceleration is d*a*/dt ≈ Δ*a*/Δt
and by *a* so Δ*a* = Δf/m and

d*a*/dt
≈Df/mDt. (20)

From Eq. (18) Δf = 9x10^{9} [N], m
(mass) per meter for the cylinder of magnets is equal to (3.8[kg/m] of magnet
strips)(471 strips per meter of cylinder)(6283 [m] cylinder length) = 1.12 x 10^{7}
[kg], and Δt = 10^{-12} [s]. Thus

d*a*/dt = 9x10^{9}/(1.12x10^{7})(10^{-12})
= 8x10^{14} [m/s^{3}] (21)

Let us suppose that the total activation time for
jerking a magnet in one direction, δt, is three picosecond pulse lengths
or 3x10^{-12} [s]. Therefore, a* *=
(d*a*/dt)δt = (8x10^{14})(3x10^{-12})
≈ 2.4x10^{3} [m/s^{2}] or 245 g’s and far less than, say,
the acceleration experienced by a bullet in the barrel of a gun. Thus the
magnet would not disintegrate. The acceleration is also probably within the
weak-field limit of Einstein’s equations since for PSR 1913 + 16 the
acceleration at periastron is over 100 g’s (Section 2.3.4) and probably much
grater acceleration is encountered for a spinning neutron star for which the
Einstein equations presumably hold.

* *In the *extreme
case* of 100 picoseconds of continuous jerk (in the same direction), δt
= 10^{-10} [s], the speed would build up to

d*s*/dt = (d*a*/dt)dt^{2}/2 = (8x10^{16})x10^{-20}/2
= 0.0004 [m/s] (22)

and the displacement of the magnetic mass
(composed of many magnetic surface sites, 57, of the linear motor, piston, or
barrel shown in FIG. (3A)) is

*s* = (d*a*/dt)dt^{3}/6 = (8x10^{16}/6)x10^{-30}
= 1.3x10^{-16} [m]. (23)

Again there could be considerable “motion” of the
magnetic mass, but even in the most unlikely case of an extremely long series
of jerks in the same direction (100 pulses), it goes a very small distance
before the IIPCS reverses the built-up acceleration, speed, and displacement
and *the stresses in the material of the
device would be minimal. *

In
general, for permanent-magnet and coil combinations of all of the devices
discussed in this paper, which exhibit an N-pulse long activation time, the acceleration is

d*a*/dt = {(Δf per meter)/(mass per
meter)}N (24)

with δt = N Δt. We have calculated that
Δf per meter = 3000 [N/m}, that mass per meter = 3.8 [kg/m], and with N =
3 (the Δt’s cancel out), d*a*/dt =
(3000/3.8)x3 = 2368 [m/s^{2}] or 245 g’s as before. **Since the Δt (and, therefore, the
frequency) cancel out, the material (magnet) acceleration is only dependent
upon the ratio of force to mass of the magnets and the number of current
pulses, N, in the total time that the magnet is activated (jerked) in one
direction before the IIPCS reverses the jerk.**

6. HIGH-INTENSITY HFGW GENERATOR

A high-intensity HFGW-generation would necessarily involve
a much shorter pulse duration, e.g., ten attoseconds or 10^{-17} [s] (100
QHz frequencies). In this regard, ^{-18} [s]....” I will
configure the generator as before as a barrel composed of a stack of individual
and separate rims whose edges are again covered with an array of ultra-small
micromagnets (energizable elements) surrounded by a sheath of ultra-small
microcoils (energizing elements). The device is again represented by the
schematic drawn in FIG. (3A) whose cross section or individual separate rims
are shown in FIG. (3B) except that there are multiple concentric rims or rings
around the cylinder axis. If the HFGW spreads out from one rim to the next,
then a thin HTSC lens (please see paper HFGW-03-120) can be inserted between
the rims in order to concentrate the HFGW down the axis of the stack as its
intensity is built up. At the end of the stack of rims or rings there is a
final HTSC lens that concentrates the HFGW on a focal plane.

The power of the device is given by a variant of Eq.
(6). Each rim or ring has magnets on its periphery that are energized by an
adjacent shell of coils (using the IIPCS for timing). As already noted there
are many concentric rims or rings at, say, two-centimeter intervals along the
axis of the rim or ring stacks to allow for a one-centimeter thick (maximum
thickness) HTSC lens to be sandwiched in between the one-centimeter thick,
δ*l, *ring sets*.* I will set the overall length of the
generator to be *l* = 500 [m] so that
there are n = (*l*/δ*l*)/2 = 25,000 rims (or sets of
concentric rings) along the axis of the cylinder (FIG. (3A)). I utilize the
force per meter that has been calculated for the (rather weak) permanent
magnets of F = 3000 [N/m] and set the radius of the outermost rim to be r_{0}
= 10 [m], so that the rims or rings in any cross section number *n = *(r_{0}/δ*l*)/2 = 500. The power of the HFGW
generator is given by (half goes the opposite way)

^{k=500} ^{}

P =½x1.76x10^{-52}∑ {2r_{k}
Fn*n*(2πr_{k} )/Δt}^{2}
[watts]^{}

_{k =1, step2}^{}

(25)

where r_{k} = 2k_{ }δ*l* [m] and Δt = 10^{-17}
[s]. The numerical result is about **380
kW**.

At
the focus of the last or end lens, the size of the diffraction pattern on the
focal plane defines the maximum HFGW flux. The diameter of the diffraction
pattern is 1.22λ_{GW}, where λ_{GW} = c Δt and
for the ten-attosecond (100Qz), λ_{GW} = 3x10^{-9} [m].
Ideally all of the HFGW power is concentrated in this diffraction pattern,
which has an area of π(1.22x3x10^{-9}/2)^{2} = 1.05x10^{-17}
[m^{2}]. So the flux is **3.7x10 ^{22}
[watts/m^{2}]**. Such a hypothetical and ideal flux density compares
favorably with the ultra-intense laser pulses of 10

7. MINITURIZED HFGW GENERATOR

For
the purpose of having a specific numerical example of a miniaturized HFGW generator,
suppose that the dimensions of the transmitter or miniaturized device involve
an energizable-element stack of tiny rims (e.g., rings or circles of
microscopic magnets) that is 6 [mm] thick surrounding a 3 [mm] radius energizing-element
core (e.g., microscopic coils). The diameter of the device is 2(3+6) = 18 [mm].
The radius of gyration would be 6 [mm]. Let us also suppose that the device is
18 [mm] in length. At the receiver, which I assume to be 7 [km] away, I will
introduce a d = 18 [mm] diameter superconducting lens to gather and focus the
HFGW in order to concentrate or amplify the signal at the receiver. I will
consider that Δf* _{l} */ΔV
can be increased ten fold by increased magnetic efficiencies due to the use of
superconducting electromagnets (rather than rather weak permanent magnets) to
3x10

P
= ½x 1.76x10^{-52}{(2)(0.006)(1.22x10^{3})/10^{-16}}^{2} = **1.89x10 ^{-18}**
[watts]. (26)

This
power from the forward, “coherent-radiation” end is distributed over an area
defined by the diffraction pattern at a distance of 7 [km] or range, R = 7x10^{3}
[m]. The diffraction angle, α* _{d}*
, at the apex of a cone of HF GW is given by (please see Conference paper
HFGW-03-120)α

The area of the conical spread of the HF GW is

α
= π(α* _{d}* R/2)

The
18 [mm] diameter lens, which concentrates the HF GW at the receiver, has a grasp
or GW gathering power, or amplification of
(d/λ_{GW})^{2} =
{(0.018)/(3x10^{8})(10^{-16})}^{2} = **3.6x10 ^{9}** . Putting it all
together the signal at the receiver is {(1.89x10

Note
that *the HFGW signal at the receiver is
inversely proportional to the sixth
power of the system’s pulse length, Δt,* (including the lens at the
receiver). The foregoing is a bit of a simplification since, like the
discussion of the high-intensity design in Section 6, one would turn to a
concentric, cylindrical-layer construction – not to a simple sheath and single
rim configuration. Thus the energizing elements (e.g., coils) and energizable
elements (e.g., magnetic sites on the rims) would be close enough for the GW
waves (of wavelength cΔt = (3x10

8. NANOMECHANICAL RESONATOR

** ** A recent paper by Melosh, *et al *[17] raises the possibility of
utilizing high-frequency nanomechanical resonators for generating HFGW. They
indicate some 10^{11} junctions (presumably where resonators could be
located) per square centimeter of wafer or chip. The resonance frequency of the
wires is “... expected to range from tens of MHz to GHz.” At the GHz frequency,
Δt ≈ 10^{-9} [s]. The Δf is more difficult to estimate.
Let us assume that the force applied to the resonators is like the force
between two parallel current-carrying conductors mentioned in Section 4.2.3 and
exhibited schematically in FIG. (4). In this case the classical equation for
the force (impulse) is

Δf = μ_{0} *l*i_{a}i_{b}/2πd, (28)

where
μ_{0} = 4πx10^{-7} , *l* is the length of the wire section, d is the distance between
wires, and i_{a} and i_{b} are the currents flowing through any
adjacent pair of wires. The wires can be as small as 20 [nm] in diameter for a
wire cross section of about 3x10^{-16}[m^{2}].
If one were able to conduct a TAmp/m^{2} or 10^{12} Amps/m^{2}
(a most difficult problem especially if superconductors are contemplated), then
the current in the two wires would be three tenths of a milliamp or 3x10^{-4}
[A]. I will assume that at the resonator the wire section, *l*, subject to the force and the distance, d, between wires are
about equal. Thus Eq. (28) yields

Δf
= (4πx10^{-7}) (3x10^{-4})^{2}/(2π) = 3.6x10^{-21}[N] (29)

and
if I had about a three-meter stack of these (assumed 1 [mm] thick) wafers or chips
(so that the radius of gyration for a computer-logic system control to build up
a coherent GW along the stack, would be r = 1 [m]), and with 10^{11} resonators per
wafer, Eq. (6) yields

P=
3x10^{3}x1.76x10^{-52}x{2(3.6x10^{-21})(10^{11})/10^{-9}}^{2}

^{ }

^{ }= 6.8x10^{-50} [watts]. (30)

Clearly
this is **not** a viable HFGW generator.
Even if one could impress a QHz frequency on the nanowires one could only
expect a 10^{12 }improvement – not enough.

A better way to implement a
system like this would be to utilize an array of parallel plates that are, for
example, one centimeter on a side and one millimeter thick. A two-meter square
plate would hold about ten thousand of them plus associated circuitry and,
again, there could be a three-meter stack for a one-meter radius of gyration
for the build up of a coherent beam. In this case the area of the “wire” would
be (0.01) (0.1) = 10^{-3} [m^{2}]. Assuming only 10^{5 }[A/m^{2}],
a 100 [A] current would pass through each pair (probably superconducting to
avoid heat). Let us suppose that I can separate the plates by 100 [nm] or 10^{-7}
[m] and that for the centimeter plates *l*
= 0.01 [m]. Thus evaluation of Eq. (28) yields

Δf
= (4πx10^{-7})(0.01)(100)^{2}/2πx10^{-7}=200
[N] (31)

with
QHz current pulses Δt = 10^{-15} [s]. There are 10^{4} of
the energizable elements on each plate as the GW wave passes by and if the
plates are one-half centimeter apart (therefore 3[m]/0.05 = 600 of them in the
stack) , so that Eq. (6) yields

P=
600x1.76x10^{-52}{2x10^{4}x200/10^{-15}}^{2} =
1.7x10^{-6} [watts]. (32)

All of this is very hypothetical, but much closer to a realistic HFGW generator.

9.
SYMMETRY AND NULL GW GENERATION

** **There
exist some situations in which a jerk exists in a system of masses, but there
is no attendant GW generation. One often refers to these as *symmetrical situations*. A situation in
which a system is so symmetrical that one can think of the GW as canceling out
and becoming null. Of course there are reactive jerks when a star collides with
a black hole resulting in some acceleration of both bodies, or one neutron star
orbits another with action and reaction on both bodies and in both cases GW is
generated. Also GW would be expected to be generated with the reactive jerks of
coils and magnets, motion of resonate cavity walls, Cooper pairs in a
superconductor, electrons in a dielectric, etc. But there are situations, such
as an isotropic explosion of a star, in which there is a jerk due to an
expanding shell of gas and **no** GW is
generated.

This
situation warrants some attention. Consider FIG. (10A) in which there is shown
the cross section of an exploding shell of gas having diameter, d. I consider
two opposite small incremental masses of the shell, Δm_{1} and
Δm_{2, }which are jerked to the top and bottom of the figure. Let
us suppose that they are jerked so that each mass generates two oppositely
moving GWs, normal to the direction of the jerk in a plane perpendicular to the
plane of the figure. I show as λ_{GW}. The usual assumption for
the efficacy of the quadrupole approximation for estimating GW power is that
λ_{GW} »d, which is the case for most celestial LFGW generators.
Thus I have the situation shown in FIG. (10B) and the GW cancel out and become
null.

10.
CONCLUSIONS

** **The
jerk formulation of the quadrupole approximation was derived in two alternative
ways for the HFGW power of a laboratory HFGW generation device. The formulation
was numerically validated by analysis of a well-known GW-generating binary
pulsar, PSR 1913+16. By means of numerical examples, it was shown that the
resistive heat loss and device component acceleration are well within tolerable
limits.

Micro- and nano-scale HFGW
generator components have been described in connection with a computer logic
system to facilitate the generation of coherent HFGW. A device consisting of a cylindrical
stack or jerking rims was also described in connection with two specific
designs: a high-intensity HFGW generator and a miniaturized HFGW generator.
Moreover, the outputs of HFGW flux from these devices range from 6.3x10^{-7}
[watts/m^{2}] to 3.7x10^{22} [watts/m^{2}]. In either
case the fluxes are greatly increased by increasing the frequency of the jerks
to the QHz or higher and by including a HTSC lens to concentrate the HFGW.
Linear motor HFGW-generator designs were configured and studied. The situation
in which component symmetry prohibits the generation of GW due to destructive
GW interference was examined. A nanomechanical resonator concept is analyzed,
but not found to be especially efficient. And finally, the major conclusion is
that **laboratory HFGW generation devices
are feasible, practical, and** **warrant
experimental investigation**.

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