Highfrequency gravitational waves or, for short, HFGWs, are superior to lowfrequency gravitational waves or, for short, LFGWs because they carry more information per unit time than LFGWs. With only a few hundred waves or pieces of information received per second or per thousands of seconds, one could not carry on a meaningful conversation at ordinary audio frequencies of 20 to 20,000 Hz or cycles per second (although the range of frequencies individuals hear is greatly influenced by environmental factors). Certainly, television transmission having frequencies of a few to many hundred GHz could never be transmitted by LFGWs. One cannot put HFGWs “on top of” LFGWs, since this is no different from the HFGWs themselves. There is yet another important advantage of HFGWs in that they can be transmitted in narrow beams. The LFGWs will always spread out a lot more due to diffraction.
The concept of diffraction is somewhat more complicated than the simple concept of a wave, but again we need to use our imaginations. Let us consider a harbor with a long breakwater at its entrance. Imagine a series of parallel water waves approaching the breakwater from the sea. There crests and troughs could be parallel. When they reach the breakwater opening some of the waves pass through the opening of the breakwater and then fan out inside the harbor as shown at the top of Figure 71. The angle of this fanning out depends on the distance between the wave crests and toughs, which we know is defined as their wavelength, and all so dependent on the width of the entry opening in the breakwater.
It turns out (after observations and/or theory) that the angle of the wave fanning out decreases in direct proportion to the width of the entry opening in the breakwater. Specifically, the “fan angle” or “diffraction angle” decreases the smaller the wavelength, λ, is relative to the width, w, of the breakwater opening. Diffraction plays an important role in telescope observation. For example, star under observation will have its image blurred due to diffraction and that blurring decreases with the wavelength of the star light (more blurring for longer wavelength redstar light and less blurring for shorter wavelength bluestar light). Also the smaller the telescope lens (smaller lens or smaller mirror diameters) the more diffraction. Telescopes are made with larger lenses or mirrors in order to capture more star light and to reduce diffraction! Specifically, it has been found both experimentally and theoretically that the “angle of diffraction” or spread is proportional to opening width (breakwater width or telescope opening or even diameter of a HFGW beam) divided by the wavelength. Specifically, if θdiff is the angle of widening or the angle of diffraction, then sin (θdiff) = λ/w = c/υw, (71) where λ is the wavelength (= c/ υ), υ is the frequency, w is the width of the entry opening and c is the speed of the wave, in this case the speed of light, c. As in basic trigonometry, “sin” is the sine of an angle. As examples of the use of Equation (71), suppose we have an opening 10 centimeters or 0.1 meters across (w = 0.1 m) and a HFGW frequency of 10 GHz or 1010 cycles per second. In this case, sin (θdiff) = 3×108/10×109 ×0.1 = 0.3 and θdiff = 17.4 degrees. For the larger doublehelix HFGW generator of Chapter 9 the opening at the end is 7 meters across and a frequency of υGW, = 5×109 Hz or higher could be used. From trigonometry, if an angle is small, e.g., θdiff , then the sine of that angle is equal to the angle in radians. Therefore, in radian measure θdiff = c/υw = 3×108/(5×109 ×7) = 8.3 ×103 radians; a narrow beam (remember from Chapter 4 there are about 57 degrees in one radian, that is 3600/2π = 360/(2×3.1416) ≈ 57, so in this case θdiff = 0.4 degrees). Also from basic trigonometry, the sin θ varies from +1.000 to 1.000 as θ varies from +900 to 900. If the sin θ is out of this range, then it must have exceeded its limit and θ >> 900. Thus if we have LFGW at, say 1000 Hz, then sin (θdif) = 30 and θdif >> 90 degrees. That is, a LFGW is totally diffracted away! This is another reason LFGWs are not useful to carry information for communication purposes and one would only utilize HFGWs, which similar to microwaves, could be propagated in narrow beams for communication.
Like all gravitational waves, such as the LFGWs from Einstein’s spinning baton or the pair of neutron stars in PSR1913+16, they would be generated at the center midway between the two baton tips or the masses, m, of the stars as shown in Figure 42 in Chapter 4. The power from this GW focus is not spread out on the surface of a sphere similar to a light bulb or a star (as we previously assumed in Chapter 6 as a first approximation), but spreads as a radiation pattern. Such a pattern is described by Lev Landau and Evgeny Mikhailovich Lifshitz who are a very famous pair of Russian gravitational wave scientists whose book was earlier referenced. Like Einstein’s general relativity, their analyses is extremely complex so we again copy the relationship given in the middle of the page of PROBLEMS to be found on page 356 § 110 of the Landau and Lifshitz book [1975, Low Frequency (LF) GW from Orbiting Objects …]. We utilize their notation and their Equation (110.15) on p. 355 of § 110, This equation is our Equation (72) below, which relates the change in gravitationalwave intensity I to the change in solid angle ∆o.
