Red shift, velocity shift, and physically non-observable values

1. Seismology

The spectrum shift of the seismic wavelet towards the low frequencies when increasing the distance between the source and the receiver in random media is established in seismology [1]. Such spectra "red shift" depends on the velocity heterogeneity of elastic medium without absorption.

One of the main characteristics measured in seismology is the wavelet propagation velocity and for the random media the velocity shift is also established [2]. The physical explanation of this velocity shift between the random media and homogeneous media is that the wave preferably propagates in high-velocity zones in Fresnel volume.

Seismology application of the spectral-temporal concepts based on the Fourier transforms detects the problem of physically non-observable values [3, 4, 5]. Shots and earthquakes excite the simple and on a relative time scale short wavelets, i.e. seismic wavelets have the least time of coherence. There is a nonstrict definition of a seismic wavelet: cycle and half and not much more. The Rayleigh, Berlage, Ricker wavelets are used for solving theoretical problems of seismic wave propagation, which outwardly look like the physically observable wavelets. But the effort of answering the simple question if the displacement seismic wavelet can be the unidirectional pulse in the elastic medium shows the paradox, which can be denominated as the infinitely low frequencies paradox. It is known that the absolute value of Fourier spectrum of any unidirectional pulse contains low and infinitely low frequencies, which amplitude increases when decreasing the frequency, and the maximum value of the spectrum amplitude is at zero frequency. The unidirectional pulse has the finite time duration and extension but the wavelength of the pulse dominating frequency is infinite in space independently of wave propagation velocity. Fresnel volume, or the wave transmission volume, is also equal to infinity for the unidirectional pulse.

Spectra of the time function can be defined with the help of Fourier transforms or with the help of the physical spectral analyzer. It's known that for physical measuring of the frequency or of the harmonic component period in the pulse spectrum we need the time not less than one period of the measured harmonic component. The finite unidirectional pulse duration and the finite measurement time do not allow determining the harmonic component frequency with the period more than the time of observation. For example, it is impossible to determine the amplitude and the frequency of harmonic component with 10 sec period during the 1 sec measurement time. Thus, there are physically non-observable values in Fourier spectra.

The main characteristic of the spectral analyzer is its resolution. For instance, physical spectral analyzer can consist of the set of the electrical resonators with Df bandpass and f0 resonance frequency. The resolution will be the same along the frequency axis if the physical spectral analyzer uses constant Q-factor resonators, i.e. f0/Df=const. The resolution will vary along the frequency axis if the physical spectral analyzer uses varying Q-factor resonators, or Df=const. These two types of the spectral analyzers give two different spectra for the same time function. Fourier spectral analysis is equivalent to the physical spectral analyzer using varying Q-factor resonators [5]. Increase of the amplitude in the spectra of unidirectional pulses when f®0 Hz comes out from the decrease of the Fourier spectral analysis resolution when f®0 Hz. Fourier spectral analysis is convenient for the mathematical description of the signal characteristics but Fourier spectra contain physically unobservable values.

The modified Fourier spectral analysis is equivalent to the physical spectral analyzer using constant Q-factor resonators [5]. Modification of the Fourier spectra excludes physically non-observable values from the spectral-temporal concepts and as a result the quantitative values of Fourier spectra width change. The infinitely short rectangular pulse, or Dirac pulse, has got the spectrum only in the area of infinitely high frequencies. Dirac pulse cannot contain physically observable harmonic oscillations in its spectrum because Dirac pulse is physically unobservable value. Heaviside had introduced delta-function [6] just as a derivative of the unit step function, but Dirac [7] suggested it again in a modern form. Dirac delta function is a filter operator with the infinitely wide frequency bandwidth, i.e. such filter does not operate in any way.

The Heaviside unit step function has an infinitely wide modified Fourier spectrum equal to 1 along the whole range of the frequencies. Physical spectral analyzer put together from the set of constant Q-factor resonators confirms such conclusion. The Heaviside unit step function has got an infinitely high frequency in its spectrum because the function has got an instantaneous jump that means the infinite velocity of energy propagation. The Heaviside pulse has got the infinite-low frequencies in its spectrum because the pulse duration is equal to infinity that means the infinite time of measurement of the frequency and amplitude of harmonic components with the infinite oscillation period. Physically observable signals cannot contain in their spectrum the infinitely high and the infinitely low frequencies.

Modification of the Fourier spectra opens new and unexplored properties of the uncertainty relationship. Velocity of propagation of unidirectional pulse, or shock waves, depends on the relative width of the pulse spectra. Oscillating pulse propagation velocity is constant in given nondispersive medium. The elementary wavelet characteristics have been determined in the transition zone from unidirectional to oscillating pulses. The relative spectrum width of the elementary wavelet is equal to the relative spectrum width of Planck's radiation law.

Thus, in order to answer the question: what is wave energy motion? we need to answer the question: what is non-wave energy motion? It is established in quantum physics that the particles have some wave properties but we can't explain the phenomena of the macrocosm by the laws of microcosm. The uncertainty relationship came to the spectral-temporal concepts from the quantum physics. I prove in the paper [5] the uncertainty relationship is the law of classical physics.

References

1. Shapiro S.A. and Kneib G., 1993. Seismic attenuation by scattering: theory and numerical results, Geophys. J. Int., 114, 373-391.

2. Mueller G., Rott M., and Korn M.,1994. Seismic-wave traveltimes in random media, Geophys. J. Int., 110, 29-41.

3. Bliznetsov M.T., 2001. Elementary wavelet, Geofizika, #1, 52-60 (in Russian).

4. Bliznetsov M., 2002.Uncertainty principle and elementary wavelet, 27th EGS General Assambly, Nice, France, Abstr. 02-A-01137.

5. Bliznetsov M.T., 2005. Spectral analysis resolution and study of the uncertainty relationship, Physics Essays, vol. 18, #1 (in press).

6. Heaviside O., 1925. Electrical papers, Copley Publishers, Boston.

7. Dirac P.A.M., 1947. The principles of quantum mechanics, Oxford University Press, New York.