Gravitational Wave Generator Apparatus

Abstract

An apparatus or structure is proposed for generating high-frequency gravitational waves (HFGWs) between pairs of force-producing elements by means of the simultaneous production of a third time derivative of mass motion of the pair of force-producing elements. The elements are configured as a cylindrical array in the proposed structure and are activated by a radiation wavefront moving along the axis of symmetry of the array. The force-producing elements can be micro-electromechanical systems or MEMS resonators such as film-bulk acoustic resonators or FBARs. A preferred cylindrical array is in the form of a double helix and the activating radiation can be electromagnetic as generated by microwave transmitters such as Magnetrons. As the activating radiation wavefront moves along the axis of the structure it simultaneously activates force elements on opposite sides of the structure and thereby generates a gravitational wave between the pair of force elements. It is also indicated that the Earth is completely transparent to the HFGWs. Thus a sensitive HFGW detector, such as the Li-Baker under development by the Chinese, can sense the generated HFGW at an Earth-diameter distance and could, in theory, be a means for implementing transglobal HFGW communications.

Keywords: gravitational waves, high-frequency gravitational wave generator, HFGW, communication, MEMS, FBARs.

1. Introduction

As will be discussed there exist several sources for high-frequency gravitational waves (HFGWs) or means for their generation. Historically the first generation means, which is the same for gravitational waves (GWs) of all frequencies, is based upon the quadrupole equation first derived by Einstein in 1918. A formulation of the quadrupole (Baker, 2006) that is easily related to the orbital motion of binary stars or black holes, rotating rods, laboratory HFGW generation, etc. is based upon the jerk or shake of mass (time rate of change of acceleration), such as the change in centrifugal force vector with time; for example as masses move around each other on a circular orbit. Figure 1 describes that situation.

Recognize, however, that change in force Δf need NOT be a gravitational force (see Einstein and Infeld quoted by Weber; Grishchuk and Sazhin). Electromagnetic forces are more than 10³⁵ times larger than gravitational forces and should be employed in laboratory GW generation. As Weber points out: “The non-gravitational forces play a decisive role in methods for detection and generation of gravitational waves …”

The quadrupole equation is also termed “quadrupole formalism” and holds in weak gravitational fields (but well over 100 g’s), for speeds of the generator “components” less than the speed of light and for the distance between two masses r less than the GW wavelength. Certainly there would be GW generated for r greater than the GW wavelength, but the quadrupole “formalism” or equation might not apply exactly. For very small time change Δt the GW wavelength, λ_GW = c Δt (where c is about 3 × 10⁸ m per second, the speed of light) is very small and the GW frequency ν_GW is high. The concept is to produce two equal and opposite jerks or force changes at two masses, such as are involved in micro-electromechanical systems (MEMS), for example film-bulk acoustic resonators (FBARs), a distance 2r apart. This situation is completely analogous to binary stars on orbit as shown in Fig. 1.

Fig. 1. Change in centrifugal force of orbiting masses, Δf_cf, that produces GW radiation.

2. Discussion

Next we consider an array of GW sources. Consider a stack of binary star orbit planes, each one involving a pair of masses circling each other on opposite sides of a circular orbit as shown in Fig. 3. Let the planes be stacked one light hour apart (that is, 60 × 60 × 3 × 10⁸ = 1.08 × 10¹² meters apart) and each orbit exactly on top of another (coaxial circles). Let us also suppose that the periods of the orbits were 10 hours. The orbital “frequency” would then be 1/(10 × 60 × 60) = 2.8 × 10⁻⁵ Hz. According to Landau and Lifshitz, on each plane a GW will be generated that radiates from the center of each circular orbit. The details of that generation process are that as the masses orbit a radiation pattern is generated. In simplified terms (from the equations shown in an exercise on page 356 of Landau and Lifshitz) an elliptically shaped polarized arc of radiation is formed on each side of the orbit plane (mirror images). As the two masses orbit each other 180 degrees the arcs sweep out a figure of revolution and the resulting integrated GW radiation is circularly polarized. Together these figures of revolution become shaped like a peanut as shown in Fig. 2. This situation occurs when the orbiting masses move half an orbital period, 180 degrees or 5 hours, on their orbit. Thus the frequency of the GW generated is twice the orbital frequency or 5.6 × 10⁻⁵ Hz.

The general concept of the present HFGW generator is to utilize an array of force-producing elements arranged in pairs in a cylindrical formation. They could be piezo-rods connecting the two masses or individual resonators. In any event they would be analogous to the binary arrays of Fig. 3 in which an imaginary cylinder could be formed or constructed from the collection of circular orbits. As a wavefront of energizing radiation proceeds along the cylindrical axis of symmetry of such a cylindrical array, the force-producing element pairs (such as pairs of FBARs) are energized simultaneously and jerk, that is they exhibit a third time derivative of mass motion, in concert. The jerking generates gravitational waves focused midway between the jerking pairs exactly analogous to centrifugal force jerks of the orbiting binaries.

Fig. 2. Radiation pattern calculated by Landau and Lifshitz, Section 110, page 356.

Fig. 3. GW flux growth analogous to stack of N orbital planes.

2.1. Double helix

A convenient cylindrical array is a double helix exhibited in Fig. 4. In this case the MEMS or FBARs are placed along the opposing ribbons of the helixes. As activating radiation (e.g., magnetron-generated microwaves) moves along the axis of symmetry of the helixes, the opposing FBARs are energized and jerk thereby producing a HFGW. It is important that the activating radiation be phase-coherent.

In order to understand this concept better let us return to the orbit-plane stack of Fig. 3. A GW generated by the first binary (at the base of the stack) should reach the second member of the stack at its center just as the GW arc is formed with the correct polarization and phase to match the upcoming GW. We imagined the polarization plane as the plane of an elliptical arc. Since the orbit planes are one light hour apart the orbiting binaries must be synchronized one hour of motion further along on their orbit from the initial locations, when they were exactly aligned, in order to reinforce the GW moving along the axis of the imaginary orbit-plane cylinder. Analogously the activating radiation of the double-helix cylindrical array must energize each FBAR pair as the GW passes. Thus if the energizing radiation is produced by microwave transmitters along the GW path (axis of symmetry of the helixes) they must be phase coherent. As will be discussed in more detail in the next following sub-section 2.2, the phase coherent HFGW flux or signal increases in proportion to the square of the number of MEMS (e.g., FBARs) HFGW-generation elements, N, according to Dicke and to Scully and Svidzinsky.

Fig. 4. Double-helix HFGW generator FBAR array (patent pending).

2.2. Superradiance

The N-squared build up, termed “Superradiance,” is attributed to two effects: one factor of N from there being N HFGW power sources or generation elements and the other factor of N from the narrowing of the beam so that the HFGW is more concentrated and the flux (in watts per square metre) thereby increased. Utilizing General Relativity, Romero-Borja and Dehnen, and Dehnen and Romero-Borja, computed a superradiance build up of “… needle-like radiation …” HFGWs beam from a closely packed but very long linear array of crystal oscillators. Their oscillators were essentially two vibrating masses that were a distance b apart whereas a pair of vibrating FBAR masses is a distance 2r apart as shown in Fig. 5. However, the FBAR operates in an analogous fashion as piezoelectric crystals. Superradiance also occurs when emitting sources such as atoms “… are close together compared to the wavelength of the radiation …” (Scully and Svidzinsky, p. 1510).

Note that it is not necessary to have the MEMS or FBAR elements perfectly aligned (that is, the FBARs exactly across from each other) since it is only necessary that the energizing wave front (from Magnetrons in the case of the MEMS or FBARs as in Baker, Woods and Li) reaches a couple of nearly opposite FBARs at the same time so that a coherent radiation source or focus is produced between the two FBARs. The energizing transmitters, such as Magnetrons, can be placed along the helixes’ array axes between separate segments of the array or, more efficiently, at the base of the double helixes so that a superradiance microwave beam is projected up the axis of the helixes. The force change, Δf, produced by energizing one off-the-shelf FBAR is 2 newtons according to Woods and Baker.

Fig. 5. Comparison of the Dehnen and Romero-Borja crystal oscillator and the FBAR-pair system.

2.3. Analogy and fabrication technique

In order to clarify the double-helix concept and its fabrication, let us consider a totally different yet analogous situation. It is a storage facility for mattresses. Each mattress is, say, 7 feet by 6 feet and one foot thick (analogous to a gigantic MEMS or FBAR). The storage facility is composed of many coaxial cylindrical structures that are analogous to the cylindrical array of MEMS. The cylindrical structures consist of 7-foot wide compartments between the cylinders’ inside and outside walls and each of these compartments are 6 feet high. Thus one can store one mattress on its side in each compartment.

In order to reach a given compartment, imagine that two escalators are installed on the inside wall of each cylindrical structure. They are in the form of spiral escalator “stairways” and are constructed on exactly opposite sides of each cylindrical storage structure (essentially the ribbons of a double helix of MEMS). As an example, let us consider one of the cylindrical structures that happen to have a diameter of 100 feet. The circumference of the inside wall of the cylinder is about 314 feet so that the foot of the opposite escalator is about 157 feet distant from its opposite. We take the tread of each escalator step as one foot wide (enough room to slide a mattress in or out of its compartment when the escalator is periodically halted). We want to be able to reach each mattress so the escalators must rise 6 feet in 157 feet in the first 6-foot-high floor of the storage structure. Thus the height of each escalator step when it is moving is 6/157 of a foot or about 1/32 of an inch. Two people start up on each escalator simultaneously, which is analogous to a wavefront from a Magnetron moving up a double helix of FBARs. They proceed up from compartment to compartment. At each of the 157 “levels” (N) they reach opposite pairs of mattresses. In the analogous manner the wave front reaches opposite FBARs and excites them and produces a jerk and, therefore, HFGW radiation pattern focused between the FBARs.

But what about the other coaxial cylindrical mattress storage cylinder structures? In order to transport the mattresses the tread width needs to be kept constant, that is, more levels on cylinder structures having inside diameters of more than 100 feet and fewer levels on cylinder structures having diameters less than 100 feet. Thus each level is distinct and every mattress pair is on a uniquely different level (there are N such different levels and, hence, mattress pairs). Also the escalators for each cylinder could be located at different starting points on the circumference of a given cylinder structure. For example, if there were ten structures, then one could place them on different azimuths such as 0, 18, 36, 54, 72, 90, 108, 126, 144 and 162 degrees or at random. Such options may be considered in the fabrication or building process of the imaginary mattress-storage cylinders’ construction or, analogously, the FBAR array fabrication.

In order to develop the double helix winding, a column could be fabricated with the mattresses joined, that is, glue the mattresses back to back one mattress to the next in a long line. This would create a 6-foot by 7-foot cross-section tube or, for the analogous FBARs, a 110 µm by 110 µm thread (or whatever the dimensions of the trimmed FBAR MEMS are). Then place one tube on top of the other after 157 feet. Thus the composite tube exhibits a 7-foot by 12-foot rectangular cross-section. The analogous FBAR construction would be a 110 µm by 220 µm rectangular cross-section thread. The FBAR fabrication would continue by tightly winding the composite threads around a microwave-transparent cylinder or spool, layer after layer in order to form the ribbons of the double helix. Thus the resulting double-helix structure could be inserted in the microwave guide. Returning to the mattress analogy, it is recognized that each escalator passenger may take off at a slightly different time, analogous to a slightly irregular wave front. They all, however, will ascend at the same speed: the speed of light in the structure. Such wavefront irregularities would however be mitigated or eliminated by a properly designed waveguide.

3. Results

As a numerical example or result of a double-helix FBAR array analysis, we will choose the median radius of the overall array as r = 20 cm (convenient laboratory size though usually somewhat greater than λ_GW), Δf = 2 newtons for an off-the-shelf FBAR and Δt = 4 × 10⁻¹⁰ s (equivalent to about a ν_EM = 2.5 GHz frequency or pulse of the jerk or energizing radiation frequency) so that λ_EM = 12 cm and λ_GW = 6 cm (the frequency of the GW is twice that of the frequency of the energizing EM wave) and the power P from the basic GW equation (its derivation can be found in, for example, Baker 2006) is:

Equation 1. P = 1.76 × 10⁻⁵² (2r Δf/Δt)² watts.

For this equation the calculation of the combined Δf of all the pulsating MEMS or FBARs requires more calculation. We will set the length of a double-helix array cylinder as 20 m, but recognize that it can be separated into segments along the same axis with energizing transmitters, e.g., Magnetrons installed on the cylinder axis between the segments. As previously mentioned the transmitters could also be phase coherent and arranged in a line along the double-helix axis at its base. If, for example, there were 1000 one-kilowatt Magnetrons (such as those installed in a conventional microwave oven and feeding in on one hundred 12-cm, λ_EM, wide levels) and each of their beams covered a 10-cm radius circle, then the energizing radiation flux would be 3.2 × 10⁴ watts per square metre. According to superradiance there would result a needle-like microwave radiation directed along the axis of the double helixes amounting to 32 gigawatts per square meter.

In order to create a perfectly planar wave front, with no irregularities, the cylindrically symmetric MEMS array would be contained in a waveguide or possibly a very wide coaxial “cable,” surrounded by a robust one-megawatt heat sink. To increase instantaneous power to the array, bursts of gigawatt power, for example, every millisecond could be employed that would maintain a megawatt average power input.

The walls of the cylindrical array are taken to be 30 cm thick. Thus the volume of the array is π (r1² − r2²) × 20 cubic metres, where r1 is the outside radius = 0.35 m and r2 is the inside radius = 0.05 m. Thus the volume is 7.5 m³. A FBAR (Fig. 6) is a mechanical (acoustic) resonator consisting of a vibrating membrane (typically about 100 µm by 100 µm in plan form, and about 1 µm thickness), fabricated using well-established integrated circuit (IC) micro fabrication technology. A typical off-the-shelf FBAR as shown schematically in Fig. 6 usually has overall dimensions 500 µm by 500 µm by approximately 100 µm thick. For our purposes, in which a high number density is important, we will trim the FBARs to a minimum size. In order to account for fabrication margins we will take the dimensions as 110 µm by 110 µm by 20 µm for an FBAR volume of 2.42 × 10⁻¹³ m³. However, it could be smaller as shown in Fig. 1 of Chan et al. (the MEMS resonator shown there is about 50 µm square by 2 µm thick for a volume of about 10⁻¹⁴ m³).

Fig. 6. Basic FBAR construction (cross-section side view, not to scale).

Thus the total number of FBARs in the double-helix cylindrical array is 3.1 × 10¹³ and the number of pairs is half of that. Thus there will be N = 1.55 × 10¹³ FBAR pairs in the double-helix cylindrical array. Since each FBAR exhibits a jerking force of 2 newtons the combined Δf of all the jerking FBAR pairs is 3.1 × 10¹³ newtons if the jerking pairs (or “orbits”) were collapsed and moved in concert analogous to the orbit plane with the synchronized orbital mass motion.

3.1. Standard approach

As the energizing wave front moves up the axis of the double helix it develops the HFGW wave as if it were a single unified system with one ever accumulating, mass-change effect so that effectively the mass change Δf = 3.1 × 10¹³ newtons. Thus from Eq. (1), with 2r_rms = 2√[(r1² + r2²)/2] = 0.5 m, the total power produced by the double-helix array is P = 1.76 × 10⁻⁵² (0.5 × 3.1 × 10¹³/(4 × 10⁻¹⁰))² = 2.64 × 10⁻⁷ watts. But due to the N levels, each one of which represents an individual GW focus, there exists a “Superradiance” condition in which the HFGW beam becomes very narrow as shown schematically in Fig. B of Scully and Svidzinsky. Thus the HFGW flux, in watts per square metre, becomes much larger at the cap of the peanut shaped radiation pattern. According to the analyses of Baker and Black the area of the half-power cap is given by:

Equation 2. A_cap = A_half-power(N = 1) / N square metres,

where A_half-power(N = 1) = 0.1358 m² for a single level (N = 1) at a distance of 0.282 m (radius of a one square meter area sphere) or (1 m/0.282 m)² × 0.1358 = 1.71 m² at a distance of one meter. Thus Eq. (2) becomes A_cap = 1.71/N square metres (actually one fourth of the HFGW power reaches the cap since half goes to the other side of the peanut-shaped radiation pattern in the minus-z direction in Figs. 2 and 3). Thus the HFGW flux at a one-meter distance from the end of the double-helix cylindrical array is:

Equation 3. S(1) = (P/4)/(1.71/N) = (2.64 × 10⁻⁷/4)/(1.71/1.55 × 10¹³) = 6 × 10⁵ watts per square metre.

From Baker et al., Eq. (6A) of the Appendix, the amplitude of the dimensionless strain in the fabric of spacetime is:

Equation 4. A = 1.28 × 10⁻¹⁸ √S/ν_GW, in metres per metre.

So that at a one-meter distance A = 1.98 × 10⁻²⁵ m/m. From Woods et al. the current estimated sensitivity of the Chinese Li-Baker HFGW Detector is A = 1.0 × 10⁻³⁰ m/m to 1.0 × 10⁻³² m/m with a signal to noise ratio of over 1500 (Woods et al., p. 511) or if we were at a 1.3 × 10⁷ m (diameter of Earth) distance, then S = 3.55 × 10⁻⁹ watts per square metre and the amplitude A of the HFGW is given by A = 1.52 × 10⁻³² m/m. There also could be a HFGW superconductor lens, as described by Woods, that could concentrate very high frequency gravitational waves at the detector or receiver. Thus with the Chinese Li-Baker HFGW detector program successful and the Woods lens practical, the Li-Baker detector will exhibit sufficient sensitivity to receive the generated HFGW signal globally.

3.2. Conservative approach

A more conservative approach would be that there are N = 1.55 × 10¹³ individual GW power sources each with a Δf = 2 newtons. Again from Eq. (1), with 2r_rms = 2√[(r1² + r2²)/2] = 0.5 m, the total power produced by the double-helix array is NP = 1.55 × 10¹³ × 1.76 × 10⁻⁵² (0.5 × 2/(4 × 10⁻¹⁰))² = 1.7 × 10⁻²⁰ watts. But, as already pointed out, due to the N levels, each one of which represents an individual GW focus, there exists a “Superradiance” condition in which the HFGW beam becomes very narrow as shown schematically in Fig. B of Scully and Svidzinsky. Thus the HFGW flux, in watts per square metre, becomes much larger at the cap of the peanut shaped radiation pattern. Again according to Eq. (2), A_cap = 1.71/N square metres. Thus, according to Eq. (3) the HFGW flux at a one-meter distance from the end of the double-helix cylindrical array is S(1) = (NP/4)/(1.71/N) = (1.7 × 10⁻²⁰/4)/(1.71/1.55 × 10¹³) = 3.8 × 10⁻⁸ watts per square metre. From Eq. (4) at a one-meter distance A = 5 × 10⁻³² m/m.

If the FBARs in all of the helix levels are not activated as individual pairs, then the situation changes. For example, let all of the FBARs in a 6-cm wide level (half of λ_EM) be energized in concert. The number of levels would be reduced to N = 20 m/0.06 m = 333. But, because the FBAR pairs in each level act together, Δf = (2 newtons)(1.55 × 10¹³/333). Thus no matter how many activation divisions (N) the changes in Eq. (1) cancel out (the N-squared) and there is no change in HFGW flux. If we were at a 1.3 × 10⁷ m (diameter of Earth) distance, then S = 1.33 × 10⁻²⁰ watts per square metre and the amplitude A of the HFGW is given by A = 3.8 × 10⁻³⁹ m/m.

Although the best theoretical sensitivity of the Li-Baker HFGW detector is on the order of 10⁻³² m/m, its sensitivity can be increased dramatically by introducing superconductor resonance chambers into the interaction volume (which also improves the Standard Quantum Limit; Stephenson) and two others between the interaction volume and the two microwave receivers. Together they provide an increase in sensitivity of five orders of magnitude and result in a theoretical sensitivity of the Li-Baker detector to HFGWs having amplitudes of 10⁻³⁷ m/m. There also could be a HFGW superconductor lens, as described by Woods, that could concentrate very high frequency gravitational waves at the detector or receiver. Thus with the Chinese Li-Baker HFGW detector program successful and the Woods lens practical, so again the Li-Baker detector will exhibit sufficient sensitivity to receive the generated HFGW signal globally.

The HFGW beam is very narrow. From Eq. (4b) of Baker and Black, for N = 1.55 × 10¹³ it would be the inverse sine of 0.737 divided by the square root of 1.55 × 10¹³, that is 1.87 × 10⁻⁷ radians. For N = 333 the angle is 0.0022 radians. This is still narrow, but the double helix configuration certainly reduces the width of the HFGW beam. Additionally multiple HFGW carrier frequencies can be used, so the signal is very difficult to intercept, and is therefore useful as a low-probability-of-intercept (LPI) signal, even with widespread adoption of the HFGW.

3.3. Irregularities

There are at least three irregularities that affect the performance of the double-helix generator design, especially in the standard approach. First is the ability to separate or differentiate the N = 1.55 × 10¹³ FBAR pairs due to irregularities in the fabrication of the helix ribbons. Second is the irregularity in the wave front of the energizing microwave radiation produced by the Magnetrons. Third are irregularities in the delay time between the incidence of the energizing or activating microwave radiation and the FBAR mechanical force change.

At first glance the required positioning accuracy for MEMS, specifically FBARs, of 0.155 picometres would seem to be impossible to achieve using conventional assembly techniques. On the other hand, the tight machine winding of the 110 µm by 220 µm rectangular cross-section FBAR threads in a dust-free environment might have a tolerance of less than a small fraction of a nanometre. It is to be recognized that the simultaneous energizing of two FBARs produces GW radiation at the midpoint of a line exactly between them. If, for example, every ten FBAR pairs are slightly out of alignment and their lines intersect when energized, then the total power of the created GW would effectively be due to a Δf = 2 × 1.55 × 10¹³ × 10 = 3.1 × 10¹⁴ newton force change, but the number of such levels (of 10 common, undifferentiated FBAR pairs) would be N/10 = 1.55 × 10¹². The resulting beam would be much broader and hence the flux would be less. However the power at each GW generation site would be greater. Thus the effect would not be as drastic as one might at first believe. Other scenarios could be imagined in which pairs of FBARs were simultaneously energized at sites not directly across from each other, but hopefully nanotechnology assembly techniques will obviate the problem.

The irregularity in the wave front of the energizing microwave radiation produced by the Magnetrons is a more vexing design problem. If the irregularities in the wave front have cylindrical symmetry, then several superimposed GW beams will be generated in which the total power remains the same, but as in the prior situation, the beam is broadened and the HFGW flux reduced. Proper microwave-guide design of the manifold of multiple Magnetron radiation input will be essential in any event.

There will be a delay between the incidence of the energizing or activating microwave radiation and the FBAR mechanical force change and if the delay is exactly the same for all FBARs, then there is no problem. If the delay has cylindrical symmetry about the axis of the helixes (e.g., due to some thermal effect) then the effect is as previously found, an increase in beam width and a resulting decrease in HFGW flux. Efforts will need to be made to manufacture and assemble the FBARs in a very uniform manner and to carefully control their environment after fabrication in the detector.

4. Conclusions

The overall concept is shown in Fig. 7 in very simplified form. In theory the preferred double-helix array of force-producing FBARs can generate significant superradiant HFGW radiation. A numerical example of a 20-meter long array is presented. Activation-energy radiators or transmitters (such as off-the-shelf Magnetrons) can be utilized to energize MEMS such as off-the-shelf FBARs found in cell phones. Thus point-to-point communication, even at a distance of the diameter of the Earth, might be realized using very sensitive HFGW Chinese detectors or receivers and HFGW lenses to concentrate the HFGW signal at the receivers.

Fig. 7. Simplified concept of the HFGW generator: thousands of Magnetrons produce microwave radiation at 2.45 GHz, which energizes millions of film-bulk acoustic resonator (FBAR) piezoelectric crystals, generating HFGWs at 4.9 GHz. The arrangement is similar to that of Romero and Dehnen (1981).

A HFGW amplitude of the time-varying strain of the fabric of spacetime, A = 3.8 × 10⁻³⁹ m/m, is created at a distance of one Earth diameter from the generator. It is also indicated that the Earth is completely transparent to the HFGWs. Thus with a sensitive HFGW detector, such as the Li-Baker successfully developed by the Chinese, and the Woods lens practical, one could sense the generated HFGW at an Earth-diameter distance and could, in theory, be a means for transglobal communications.

The approach to the laboratory or manmade terrestrial generation of HFGWs is innovative and unique (Baker and Baker). There have been few other advances in the HFGW generation field. The General Relativity crystal oscillator studies by Romero-Borja and Dehnen (1981) and Dehnen and Romero-Borja (2003) are probably the most important up to now, but their reliance on old-style crystals (not modern MEMS technology, discussed in Baker and Baker) and a linear rather than a cylindrically symmetric array resulted in a very inefficient HFGW generator. The methods discussed herein are the most appropriate to the science and engineering of terrestrial HFGW generation. All the relevant literature has been cited that supports the theory and fabrication of the proposed HFGW generator.

Acknowledgements

This research was supported by Transportation Sciences Corporation in the United States and by GravWave LLC internationally. The corrections and suggestions provided by Professor Giorgio Fontana from the University of Trento, Italy, Paul Murad, Gary V. Stephenson of LinQuest Corporation and Dr. Eric Davis of The Institute for Advanced Studies at Austin, are gratefully acknowledged.