# Conditions for Scalar and Electromagnetic Wave Pulses to Be “Strange” or Not

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## Abstract

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## 1. Introduction

## 2. Methods

- Every component of the electric (and magnetic) field is a scalar-valued field that obeys the wave equation. Hence, in order to judge for a chosen wavefunction $\psi \left(\overrightarrow{r},t\right)$ whether the corresponding EM pulse is strange or not, it is sufficient to evaluate the integral$${S}_{\psi}\left(\overrightarrow{r}\right)=\underset{-\infty}{\overset{\infty}{{\displaystyle \int}}}\psi (\overrightarrow{r},t)dt.$$
- If EM field vectors are derived by the standard procedure of constructing the magnetic vector potential or the Hertz vector from $\psi \left(\overrightarrow{r},t\right)$, even simple expressions of $\psi \left(\overrightarrow{r},t\right)$ may result in too cumbersome ones for the EM field vectors, and the integral of Equation (1) may be difficult to evaluate. Moreover, as the procedure involves taking derivatives with respect to time and/or spatial coordinates, a strange $\psi \left(\overrightarrow{r},t\right)$, i.e., one with property ${S}_{\psi}\left(\overrightarrow{r}\right)\ne 0$, generally results in a usual EM field, i.e., ${\overrightarrow{S}}_{E}\left(\overrightarrow{r}\right)=0$.
- The notion of strangeness also applies to wave fields that are scalar valued by their physical nature, e.g., sound waves.

#### 2.1. Evaluation of the Wave Pulse Energy

#### 2.2. Time-Domain Representation

## 3. Results

#### 3.1. General Conditions for a Pulse to Be “Usual”

#### 3.1.1. Sufficient Conditions

#### 3.1.2. Necessary and Sufficient Conditions

#### 3.2. Spherically Symmetric Pulses: Some Examples

#### 3.2.1. Even and Odd Lorentzians

- (a)
- If $f\left(s\right)$ is an even function, $\psi \left(\overrightarrow{r},t\right)$ is odd with respect to time (hence, automatically not strange) and $\overrightarrow{E}(\overrightarrow{r},t$) is even, but nevertheless not strange. The magnetic field is odd and, hence, not strange.
- (b)
- If $f\left(s\right)$ is odd, $\psi \left(\overrightarrow{r},t\right)$ is even with respect to time (but nevertheless not strange) and $\overrightarrow{E}(\overrightarrow{r},t$) is odd, i.e., automatically is not strange. The magnetic field is even, but still not strange.

#### 3.2.2. Error Function

#### 3.3. Propagation-Invariant Pulses: Some Examples

#### 3.3.1. Superluminal X-Waves

#### 3.3.2. Subluminal Arctan-Wave

#### 3.3.3. Luminal Localized Wave

#### 3.4. Strange Fields Generated by Sources

#### 3.4.1. Bonnor Fields

#### 3.4.2. Single-Cycle Dipole Electromagnetic Fields

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 1.**Dependencies of the real part (red) and imaginary part (blue) of the wavefunction in Equation (12) on time (for specificity—in the optical femtosecond domain) along the propagation axis (where $\rho =0$). Ordinate scales are normalized to $Re\psi \left(0,0\right)$ (see (

**a**)), but notice the change of the scale in (

**b**–

**d**). Spatial locations: (

**a**) $z=0\mathsf{\mu}\mathrm{m}$; (

**b**) $z=0.4\mathsf{\mu}\mathrm{m}$; (

**c**) $z=0.8\mathsf{\mu}\mathrm{m}$; (

**d**) $z=-0.8\mathsf{\mu}\mathrm{m}$. Values of the parameters: ${a}_{1}=0.1\mathsf{\mu}\mathrm{m}$, ${a}_{2}=0.2\mathsf{\mu}\mathrm{m}$, $b=0.1\mathsf{\mu}\mathrm{m}$.

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**MDPI and ACS Style**

Saari, P.; Besieris, I.M.
Conditions for Scalar and Electromagnetic Wave Pulses to Be “Strange” or Not. *Foundations* **2022**, *2*, 199-208.
https://doi.org/10.3390/foundations2010012

**AMA Style**

Saari P, Besieris IM.
Conditions for Scalar and Electromagnetic Wave Pulses to Be “Strange” or Not. *Foundations*. 2022; 2(1):199-208.
https://doi.org/10.3390/foundations2010012

**Chicago/Turabian Style**

Saari, Peeter, and Ioannis M. Besieris.
2022. "Conditions for Scalar and Electromagnetic Wave Pulses to Be “Strange” or Not" *Foundations* 2, no. 1: 199-208.
https://doi.org/10.3390/foundations2010012