Wave pulses with unusual asymptotical behavior at infinity

Abstract

The behavior of wave signals in the far zone is not only of theoretical interest but also of paramount practical importance in communications and other fields of applications of optical, electromagnetic or acoustic waves. Long time ago T. T. Wu introduced models of 'electromagnetic missiles' whose decay could be made arbitrarily slower than the usual inverse distance by an appropriate choice of the high frequency portion of the source spectrum. Very recent work by Plachenov and Kiselev introduced a finite-energy scalar wave solution, different from Wu's, decaying slower than inversely proportional with the distance. A physical explanation for the unusual asymptotic behavior of the latter will be given in this article. Furthermore, two additional examples of scalar wave pulses characterized by abnormal slow decay in the far zone will be given and their asymptotic behavior will be discussed. A proof of feasibility of acoustic and electromagnetic fields with the abnormal asymptotics will be described.

Figures (7)

• Figure 1: The real and imaginary parts of $U\left( \rho,z,t\right)$ at the instants $ct=0$ and $ct=4$ ($c\equiv1$). As the plot is axisymmetric, the axis $x$ represents any axis transverse to the propagation axis $z$ and, in distinction from the radial coordinate $\rho,$ also takes negative values. The sign of the imaginary part has been reversed. The pulse width parameters are chosen as follows: $ct_{s}=0.3$ and $z_{s}=0.1$.
• Figure 2: "Top view" of the 3D plot of the real part of $U\left( \rho,z,t\right)$ at four instants $t$. See caption of Fig. 1.
• Figure 3: The real and imaginary parts of $f\left( \rho,z,t\right)$ at $ct=0$ and $ct=10$. At positive times, the plot has mirror symmetry $z\rightarrow-z$ The pulse width parameters are $a_{1}=1$ and $a_{2}=2$. See also caption of Fig. 1.
• Figure 4: Radial dependence of the imaginary parts of the expanding and converging waves defined in Eq. \ref{['splexp']} and Eq. \ref{['splcon']} at instants $ct=30$, $ct=170$, and $ct=-100$. All curves have been multiplied by $-R$ and the third curve additionally by 50. The pulsewidth parameter $ct_{s}=2$.
• Figure 5: Radial dependencies of the real parts of the waves defined in Eq.s \ref{['Psiiexp']}, \ref{['Psiicon']}, and \ref{['Psii2']} at instants $ct=30$ and $ct=160$. All curves have been multiplied by $R$. The pulse width parameter is $ct_{s}=2$. For comparison, a logarithmically diverging dependence is shown by the dotted curve, whereas the factor $1/2$ has been inserted in accordance with the presence of the same factor in Eqs. \ref{['Psiiexp']} and \ref{['Psiicon']}.