## On the Existence of Undistorted Progressive Waves (UPWs) of Arbitrary Speeds 0 ≤ v < ∞ in Nature (1997)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Rodrigues97onthe,

author = {Waldyr A. Rodrigues and Jian-yu Lu (b},

title = {On the Existence of Undistorted Progressive Waves (UPWs) of Arbitrary Speeds 0 ≤ v < ∞ in Nature},

year = {1997}

}

### OpenURL

### Abstract

We present the theory, the experimental evidence and fundamental physical consequences concerning the existence of families of undistorted progressive waves (UPWs) of arbitrary speeds 0 ≤ v < ∞, which are solutions of

### Citations

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Citation Context ...presents a unified theory of how to construct UPWs of arbitrary speeds 0 ≤ v < ∞ which are solutions of Maxwell, Dirac and Weyl equations. Our unified theory is based on the Clifford bundle formalism =-=[6,7,8,9,10]-=- where all fields quoted above are represented by objects of the same mathematical nature. We take care of translating all results in the standard mathematical formalisms used by physicists in order f... |

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Citation Context ...ravel with speed less, equal or greater than cs, the so called velocity of sound in water. 21987 Durnin and collaborators rediscovered a subluminal UPW solution of the HWE in cylindrical coordinates =-=[16,17,18]-=- . These are the Bessel beams of section A4 [see eq.(A.41)]. We said rediscovered because these solutions are known at least since 1941, as they are explicitly written down in Stratton’s book [19] . T... |

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Citation Context ... called the even sub-algebra of Cℓp,q. All Clifford algebras Cℓp,q are isomorphic to matrix algebras over the fields IR, IC or IH, respectively the real, complex and quaternion fields. We can find in =-=[73]-=- tables giving the representations of Cℓp,q as matrix algebras. For what follows we need to know the following representations: Complex numbers − Cℓ0,1 ≃ IC Quaternions − Cℓ0,2 ≃ IH Pauli algebra − Cℓ... |

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Citation Context ...presents a unified theory of how to construct UPWs of arbitrary speeds 0 ≤ v < ∞ which are solutions of Maxwell, Dirac and Weyl equations. Our unified theory is based on the Clifford bundle formalism =-=[6,7,8,9,10]-=- where all fields quoted above are represented by objects of the same mathematical nature. We take care of translating all results in the standard mathematical formalisms used by physicists in order f... |

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Citation Context ...n section 3.2 we discuss subluminal UPWs solutions of ME and an unexpected relation between these solutions and the possible existence of purely electromagnetic particles (PEPs) envisaged by Einstein =-=[55]-=- , Poincaré [56] , Ehrenfest [57] and recently discussed by Waite, Barut and Zeni [2,3] . In section 3.3 we discuss in detail the theory of superluminal electromagnetic X-waves (SEXWs) and how to prod... |

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Citation Context ...y launch a superluminal wave in the exterior of a conductor with cylindrical symmetry with appropriate charge density. We discuss more some of Band’s statements in section 4. In 1992 Lu and Greenleaf =-=[43]-=- presented the first superluminal UPW solution of the HWE for acoustic waves which could be launched by a physical device [44] . They discovered the so called X-waves, a name due to their shape (see F... |

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Citation Context ...presents a unified theory of how to construct UPWs of arbitrary speeds 0 ≤ v < ∞ which are solutions of Maxwell, Dirac and Weyl equations. Our unified theory is based on the Clifford bundle formalism =-=[6,7,8,9,10]-=- where all fields quoted above are represented by objects of the same mathematical nature. We take care of translating all results in the standard mathematical formalisms used by physicists in order f... |

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Citation Context ...presents a unified theory of how to construct UPWs of arbitrary speeds 0 ≤ v < ∞ which are solutions of Maxwell, Dirac and Weyl equations. Our unified theory is based on the Clifford bundle formalism =-=[6,7,8,9,10]-=- where all fields quoted above are represented by objects of the same mathematical nature. We take the care of translating all results in the standard mathematical formalisms used by physicists in ord... |

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Citation Context ...ment of the array (about 4mm or 6.67λ in diameter, where λ is 0.6mm in water). This pulse travels with speed cs = 1.5 mm/µs. The distance between the peaks and the surface of the transducer are 104.33=-=(9)-=-mm and 103.70(5)mm for the single-element wave and the Bessel pulse, respectively, at the same instant t of measurement. The results can be seen in the pictures taken from of the experiment in Fig. 2.... |

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Citation Context ...uss more some of Band’s statements in section 4. In 1992 Lu and Greenleaf [43] presented the first superluminal UPW solution of the HWE for acoustic waves which could be launched by a physical device =-=[44]-=- . They discovered the so called X-waves, a name due to their shape (see Fig. 3). In the same year Donnelly and Ziolkowski [45] presented a thoughtfull method for generating UPWs solutions of homogene... |

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Citation Context ...n-dispersive beams. We quote also that Hsu and collaborators [20] realized a FAA to the J0 Bessel beam [eq.(A.41)] with a narrow band PZT ultrasonic transducer of non-uniform poling. Lu and Greenleaf =-=[21]-=- produced the first J0 nondiffracting annular array transducers with PZT ceramic/polymer composite and applied it to medical acoustic imaging and tissue characterization [22,23] . Also Campbell et al ... |

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Citation Context ...GE. They clearly stated also how to use these solutions to obtain through the Hertz potential method (see appendix B, section B3) UPWs solutions of Maxwell 3equations. In 1993 Donnely and Ziolkowski =-=[46]-=- reinterpreted their study of [45] and obtained subluminal, luminal and superluminal UPWs solutions of the HWE and of the KGE. In Appendix A we make use of the methods of this important paper in order... |

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Citation Context ...ed in diverse contexts many times in the literature. It appears, e.g., in the papers of Mackinnon [12] of 1978 and of Gueret and Vigier [13] and more recently in the papers of Barut and collaborators =-=[14,15]-=- . In particular in [14] Barut also shows that the HWE has superluminal solutions. In (∗) In this experiment the waves are sound waves in water and, of course, the meaning of the words subluminal, lum... |

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Citation Context ... the HWE for acoustic waves which could be launched by a physical device [44] . They discovered the so called X-waves, a name due to their shape (see Fig. 3). In the same year Donnelly and Ziolkowski =-=[45]-=- presented a thoughtfull method for generating UPWs solutions of homogeneous partial equations. In particular they studied also UPW solutions for the wave equation in a lossy infinite medium and to th... |

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Citation Context ...us Lorentz group L ↑ + is a symmetry group of g and D. In [62] we maintain that a physical theory τ is characterized by: 32(i) the theory of a certain “species of structure” in the sense of Boubarki =-=[63]-=- ; (ii) its physical interpretation; (iii) its present meaning and present applications. We recall that in the mathematical exposition of a given physical theory τ, the postulates or basic axioms are ... |

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Citation Context ...esults of measurements of the speeds of a (∗) Rodrigues and Vaz are interested in obtaining solutions of Maxwell equations characterized by non-null field invariants, since solutions of this kind are =-=[49,50]-=- necessary in proving a surprising relationship between Maxwell and Dirac equations. (∗∗) A version of [5] was submitted to IEEE Trans. Antennas Propag. in 1991. See reference 40 of [43] . 4FAA to a ... |

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Citation Context ...al UPWs solutions of ME and an unexpected relation between these solutions and the possible existence of purely electromagnetic particles (PEPs) envisaged by Einstein [55] , Poincaré [56] , Ehrenfest =-=[57]-=- and recently discussed by Waite, Barut and Zeni [2,3] . In section 3.3 we discuss in detail the theory of superluminal electromagnetic X-waves (SEXWs) and how to produce these waves by appropriate ph... |

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Citation Context ...omponent in ∧ k (M) of the Clifford field. Besides the vector bundle Cℓ(M) we also need to introduce another vector bundle CℓSpin + (1,3)(M) [Spin +(1, 3) ≃ SL(2, IC)] called the Spin-Clifford bundle =-=[8,81,84]-=- . We can show that CℓSpin + (1,3)(M) ≃ Cℓ(M)/R, i.e. it is a quotient bundle. This means that sections of CℓSpin + (1,3)(M) are equivalence classes of sections of the Clifford bundle, i.e., they are ... |

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Citation Context ... appearing in the literature [2,3,4] of construction of purely electromagnetic particles (PEP) and (ii) a superluminal UPW solution of Maxwell equations called the superluminal electromagnetic X-wave =-=[5]-=- (SEXW). We briefly discuss how to launch a FAA to SEXW. In view of the experimental results presented in section 2 we are confident that such electromagnetic waves will be produced in the next few ye... |

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Citation Context ...cussions it is worth to briefly recall the history of the UPWs of arbitrary speeds 0 ≤ v < ∞, which are solutions of the main equations of theoretical physics. To the best of our knowledge H. Bateman =-=[11]-=- in 1913 was the first person to present a subluminal UPW solution of the HWE. This solution corresponds to what we called the subluminal spherical Bessel beam in Appendix A [see eq.(A.31)]. Apparentl... |

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Citation Context ...we make use of the methods of this important paper in order to obtain some UPWs solutions. Also in 1992 Barut and Chandola [47] found superluminal UPWs solutions of the HWE. In 1995 Rodrigues and Vaz =-=[48]-=- discovered in quite an independent way (∗) subluminal and superluminal UPWs solutions of Maxwell equations and the Weyl equation. At that time Lu and Greenleaf [5] proposed also to launch a superlumi... |

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Citation Context ... + pγ 5 , (B.1) where γ 5 = γ 0 γ 1 γ 2 γ 3 = dx 0 dx 1 dx 2 dx 3 is the volume element and s, vµ, bµν, aµνρ, p ∈ sec ∧ 0 (M) ⊂ sec Cℓ(M). For Ar ∈ sec ∧ r (M) ⊂ sec Cℓ(M), Bs ∈ sec ∧ s (M) we define =-=[9,82]-=- Ar · Bs = 〈ArBs〉|r−s| and Ar ∧ Bs = 〈ArBs〉r+s, where 〈 〉k is the component in ∧ k (M) of the Clifford field. Besides the vector bundle Cℓ(M) we also need to introduce another vector bundle CℓSpin + (... |

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Citation Context ...th the theory. In section 3 we discuss some examples of UPWs solutions of Maxwell equations; (i) subluminal solutions which are interesting concerning some recent attempts appearing in the literature =-=[2,3,4]-=- of construction of purely electromagnetic particles (PEP) and (ii) a superluminal UPW solution of Maxwell equations called the superluminal electromagnetic X-wave [5] (SEXW). We briefly discuss how t... |

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Citation Context ...th the theory. In section 3 we discuss some examples of UPWs solutions of Maxwell equations; (i) subluminal solutions which are interesting concerning some recent attempts appearing in the literature =-=[2,3,4]-=- of construction of purely electromagnetic particles (PEP) and (ii) a superluminal UPW solution of Maxwell equations called the superluminal electromagnetic X-wave [5] (SEXW). We briefly discuss how t... |

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Citation Context ...th the theory. In section 3 we discuss some examples of UPWs solutions of Maxwell equations; (i) subluminal solutions which are interesting concerning some recent attempts appearing in the literature =-=[2,3,4]-=- of construction of purely electromagnetic particles (PEP) and (ii) a superluminal UPW solution of Maxwell equations called the superluminal electromagnetic X-wave [5] (SEXW). We briefly discuss how t... |

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Citation Context ...presents a unified theory of how to construct UPWs of arbitrary speeds 0 ≤ v < ∞ which are solutions of Maxwell, Dirac and Weyl equations. Our unified theory is based on the Clifford bundle formalism =-=[6,7,8,9,10]-=- where all fields quoted above are represented by objects of the same mathematical nature. We take the care of translating all results in the standard mathematical formalisms used by physicists in ord... |

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Citation Context ...q.(A.31)]. Apparently this solution has been rediscovered and used in diverse contexts many times in the literature. It appears, e.g., in the papers of Mackinnon [12] of 1978 and of Gueret and Vigier =-=[13]-=- and more recently in the papers of Barut and collaborators [14,15] . In particular in [14] Barut also shows that the HWE has superluminal solutions. In (∗) In this experiment the waves are sound wave... |

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