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We discuss the electromagnetic wave equation and dispersion relation for moving media as an example of the challenge of integrating vectors and dyadics and their covariant counterparts into the teaching of special relativity. We discuss two methods of deriving the dispersion relation and its associated wave equation. One is a direct approach for which the starting point is Maxwell's equations combined with the Minkowski constitutive relations in the laboratory frame [GAMMA] in which the medium is moving uniformly. The second approach, starting in the medium's rest frame [GAMMA]', establishes the invariance properties of the dispersion relation, and then proceeds to derive its counterpart in [GAMMA]. Application to the calculation of the group velocity in each frame is also discussed. [DOI: 10.1119/1.2772281]

A theory of steady state nonlinear wave propagation is developed using a Volterra series formalism. Under the term 'steady state wave propagation' we imply a state in which wave parameters remain constant in the time- and space-domain. Detailed consideration is given to a quadratic homogeneous medium Conditions for the existence of such a state in static and dynamic meanings are investigated. It is shown that the main condition is a smallness of the total wave energy. Other necessary factors are medium dispersive properties and a correlation between nonlinearity and dispersion values. An estimate for high harmonics' amplitudes ratio in a steady state is obtained for a case with a negligible small dispersion. But in such a case this state turns out to be unachievable in evolutionary processes. Sufficient closed form conditions for dispersive properties providing the dynamic achievability of a steady state are obtained.

The propagation attenuation of Doppler signals through lossy material and moving scattering media is studied. The investigation discovered two distinct processes at work in such media which are not seen in lossless materials. One is the case wherein complex Doppler effects manifest themselves. This holds true when the scatterer shifts in uniform. Another phenomena is when the scatterer is oscillating about a set point. Numerical simulations performed to demonstrate the theoretical discussion are also shown.

Identifying invariance properties helps in simplifying calculations and consolidating concepts. Presently the Special Relativistic invariance of dispersion relations and their associated scalar wave operators is investigated for general dispersive homogeneous linear media. Invariance properties of the four-dimensional Fourier-transform integrals is demonstrated, from which the invariance of the scalar Green-function is inferred. Dispersion relations and the associated group velocities feature in Hamiltonian ray tracing theory. The derivation of group velocities for moving media from the dispersion relation for these media at rest is discussed. It is verified that the group velocity concept satisfies the relativistic velocity-addition formula. In this respect it is considered to be ‘real’, i.e., substantial, physically measurable, and not merely a mathematical artifact. Conversely, if we assume the group velocity to be substantial, it follows that the dispersion relation must be a relativistic invariant. [ABSTRACT FROM AUTHOR]

Deals with a study which proposed a model for Doppler effects involving translatory and moving surfaces on which certain boundary or source conditions are prescribed. Discussion on integral representations and wave fields; Limitations of spatiotemporal analysis; Origin of consistent and inconsistent approximations.

The generalized Doppler effect spectrum produced by particles moving on arbitrary trajectories is analyzed. Scattering by a single particle moving in the farfield of a transmitter and a receiver is considered. It is shown that the Doppler spectrum exciting the particle is due to two factors: The longitudinal (or radial) component of the motion produces a spectrum similar to the conventional Doppler effect, by affecting the phase of the excitation wave. In addition, spectral effects are produced by the particle moving transversely (or angularly) through the spatially modulated radiation pattern of the transmitter. By virtue of the reciprocity properties of transmitters and receivers, each spectral component of the excitation signal again gives rise to spectra induced by the longitudinal and the transversal components of the motion relative to the receiver. The combined Doppler spectrum observed at the receiving transducer or antenna output is a convolution of all four spectra. The behavior of an ensemble of scattering particles is analyzed. The statistics are found by defining the particle and/or the trajectory parameters as random variables. Presently, it is assumed that the particles are identical, their positions are uncorrelated, and multiple scattering is ignored. It is shown that the interference of scattered waves from various particles, manifested in the coherent radiation, gives rise to a spectrum that might be different from that of a single particle. In special cases, the combined spectrum degenerates into a single frequency, and when this is the transmitter's frequency, the Doppler effect completely disappears. This explains the fact that when we have moving media, but the boundary surfaces are at rest, there is no Doppler effect. The results of the present analysis contribute to our understanding of Doppler velocimetry methods using ultrasound or laser radiation in medical and industrial instrumentation. [ABSTRACT FROM AUTHOR]

The Doppler effect approach is a powerful tool used in various areas of physics to analyze motion in a wave field, by observing the frequency shifts of scattered waves. In acoustics, Doppler effect analysis is used for diagnostics in biomedical systems and for industrial applications, e.g., in cases where scatterers are present in moving fluids. The theoretical models used to analyze such systems usually ignore the fact that the moving ambient medium might produce effects of the same order of magnitude as the scatterers themselves. It has even been argued that in certain circumstances such effects should completely cancel the Doppler frequency shifts. A model is developed here that contributes to our understanding of the scattering in the presence of moving objects and space- and time-dependent moving media. The model is restricted to irrotational flows, neglects velocity effects except of the first order in the Mach number v/c, and assumes slow variations in the ambient medium. These restrictions facilitate the analytical discussion of specific canonical problems. The present study indicates that Doppler effects can be produced by moving scatterers in a medium at rest, scatterers at rest in a moving medium, and in configurations in which combined motion of scatterers and media take place. The Doppler effects are of the same order of magnitude in all cases. This vindicates the simple models used for research and applications, which assume that the moving objects produce Doppler effects but neglect the flow of the surrounding medium. [ABSTRACT FROM AUTHOR]

Transient scattering as a tool for identification of objects and their physical properties through their natural (resonance) complex frequencies is now under intensive investigation by mathematicians and engineers. This promising method of inverse scattering is pertinent to mechanical as well as electromagnetic waves. Presently this problem is considered in the context of acoustics for time varying objects, such as pulsating cylinders and spheres. Together with a choice of a reasonably simple physical model and boundary conditions, this provides a framework for discussing a few canonical problems. Mathematically, the method employed here is essentially a perturbation technique. The representation in terms of compact symbolic differential operators which are manipulated algebraically helps to avoid a lot of cumbersome detail. The results display the creation of new spectral components and the associated new poles (i.e., new resonance frequencies) due to the time variation of the objects. The additional new features, when compared to the presumably known signature of the original object at rest, facilitate the identification of the details of the motion. [ABSTRACT FROM AUTHOR]