1. Introduction

The known physics can be essentially reduced and divided in two branches: one associated with particle phenomena and the other involving wave phenomena. The wave phenomena permeate many areas of physics such as mechanics, thermodynamics, electromagnetism, relativity, and quantum mechanics. One important theme associated with wave phenomena is propagation [^{1},^{2}]. Essentially, wave propagation can be affected by the changing of the physical medium properties (for example, in electrodynamics the electrical permittivity ^{[3]} Understanding how a wave propagates in a given system is of fundamental importance to understand the physics it represents.

Every propagation analysis initiates obtaining what it is called the wave vector

One typical problem involving wave propagation is the case of waves propagating in opposite directions in the same physical medium. Such problem is graphically shown in Figure 1, in which snapshots in times ^{4},^{5}] to manipulate objects in the microscale, ^{[6]} and in fiber optics for telecommunication. ^{[7]} Theoretical researches are associated with the breaking of symmetry in nonlinear resonators, ^{[8]} wave attraction in resonant systems, ^{[9]} and its dynamics in forced systems.^{[10]}

The problem involving counterpropagation of waves is always in focus of mechanical and electromagnetic books as a part of the propagation studies of wave phenomena. However, the approach limits itself to obtain the effective amplitude of the resultant wave as a function of characteristics of the individual waves. Reference ^{[11]} goes a little bit further and obtain the resultant wave in the medium but still just perform an analysis of the wave amplitude. Many books treat this problem still in a more restrict context, in the sense that one of the waves is a result of reflection of the other, since this configuration permeates many mechanical and electromagnetic systems, such as waves in strings of musical instruments, ^{[12]} waves in energy and communication lines, ^{[13]} to name a few examples. In this situation, dimensionless quantities like standing wave ratio (SWR) and the reflection coefficient acquire importance, which are associated with the amplitude of the individual waves. [^{14}
^{15}-^{16}] However, aspects related to the propagation of the resultant wave -- which are associated to the phase of the resultant wave -- are not taken in to account.

Qualitatively, it is well-known that when the amplitudes of the counterpropagating waves are equal, the resultant wave presents a stationary pattern. It is merely an oscillation. It is a standing wave. But when the amplitudes of the counterpropagating waves differ, the resultant wave still presents a travelling behavior. These situations are well explored in the books in the perspective of the amplitude of the resultant wave. In the context of these systems, the SWR and the magnitude of the reflection coefficient respectively change from infinity to some finite value and from unity to some smaller value when the resultant pattern migrates from stationary to travelling. [^{14}
^{15}-^{16}] Every literature in the field of wave phenomena stops the analysis of counterpropagating waves exactly at this point. No further discussions about the subject is presented. And from here the present work starts.

Waves are not just composed by amplitudes but also of phases. Opportunely, the existence of a phase is the unique restriction imposed by the wave equation ^{[17]} in its solution. To be a solution of the wave equation, a given arbitrary function

In the same manner as mentioned above for the amplitude, the problem of counterpropagating waves can be explored qualitatively with a view on the phase of the resultant wave, by the means of its phase velocity. When the amplitudes of the individual waves are equal, the stationary situation, the phase velocity of the resultant wave is zero. However, when the amplitude of the individual waves differ, the phase velocity of the resultant wave is not zero. That is, the phase velocity of the resultant wave has changed between both situations, to say from zero to some finite value. Moreover, since both situations are controlled by the individual amplitudes of the counterpropagating waves, so it is the phase velocity as well. In this way, some fundamental and conceptual questions arise in this subject: what is exactly the phase velocity of the resultant wave in this system? Does the phase velocity really depend of the amplitudes of the individual waves as the simple above observation qualitatively induces? If it depends, which is the expression for the phase velocity in this system? The absence of literatures that discuss this subject and answer the questions formulated right above motivates us to develop the present work.

In this way, the purpose of the present work is to obtain the resultant wave ^{[2]}, the theory associated with the phase velocity is shortly discussed. In section ^{[3]}, the resultant wave in the medium is analytically determined, being its phase velocity calculated. In section ^{[4]}, the analytical results are confronted with numerical simulations. In section ^{[5]}, the conclusions are presented. Finally, in section ^{[6]}, the future works, which are consequences of all shown in this paper, are presented.

2. The theory

Textually, the phase velocity of a given wave is defined as the velocity an observer should develop for a given reference phase of the wave seems to be static, without any relative motion between the observer and the reference point of the wave. Mathematically, defining a plane wave of amplitude ^{18},^{19}]

and supposing a reference phase

which implies that

The rate of equation (3) within the interval of time

For equation (4) to be valid in any instant of time

formally resulting in [^{18},^{19}]

or

in which

The right above equations (6) and (7) mathematically define which must be the phase velocity

3. Analytical results

The present problem consists of two counterpropagating plane waves. The plane waves considered are also uniform, which imply that its amplitudes, although might be different, are constants. One of the waves is represented by

Assuming parallel polarization between waves,

It is time to represent mathematically the individual waves

in which

and the other consists of flipping the signal of the temporal term of the phase

In equations (11) and (12),

In this way, the resultant wave

if the wave

if the wave

In order to determine the phase velocity

3.1 Spatial approach

For expressing the resultant wave

By the use of Euler and De Moivre relation

Looking for matching the equation (1) with the expression right above of equation (16), it is obtained respectively the amplitude

and the phase

being the resultant wave

Note that the resultant amplitude

The phase velocity

in which

It can be also observed that the phase velocity is indeed function of the individual wave amplitudes

The Figure 2 graphically illustrates the analytical results for the resultant wave

A final analysis of the analytical results of this section must confront the behaviour of the phase velocity plotted in Figure 2(c) with the propagation of the resultant wave plotted in Figure Figure 2(a). For that, consider the spatial coordinates comprised in the interval

The next subsection will treat of the second approach, which is associated with the representation

3.2 Temporal approach

Instead of what has been performed in the previous section 3.1, evidencing the term

which becomes

when the Euler and De Moivre relation is applied to the temporal terms

From the matching of equation (1) with the expression between the squared brackets of equation (23), it is obtained respectively the amplitude

and the phase

being the resultant wave

Note that the resultant amplitude

The phase velocity

One can directly verify that the phase velocity

in which

In the same manner as performed in the previous subsection, it can be also observed that the phase velocity is indeed function of the individual wave amplitudes

As performed in the previous section, the Figure 3 graphically illustrates the analytical results for the resultant wave

Similarly with what has been done in the previous section, to conclude the analysis of the analytical results of this section, one must confront the behaviour of the phase velocity plotted in Figure 3(c) with the propagation of the resultant wave plotted in Figure 3(a). For that, consider the interval of time *t*. Again, this behaviour of

3.3. The equivalence between both approaches

The spatial and temporal approaches presented before produced different expressions for the phase and amplitude of the resultant wave. However, it is expected that the analytical results for the wave **r** or the time *t*. This is the case of the phase velocity of the resultant wave, which is an expression derived from just the phase of the resultant wave. It is then necessary a detailed inspection of these expressions to connect the results produced by both approaches.

For evaluating compatibility between the phase velocities

since *n* is an integer number. Inserting this reference phase in equation (18) results in

while inserting in equation (25) results in

Considering that

Essentially, the equation (32) establishes, for any given time *t* in which *t* to follow the same reference phase in both spatial and temporal approaches employed to represent the resultant wave. In this situation, since the reference phase adopted is the same in the approaches, it is expected that the phase velocities

The figure 4 shows the absolute value of the phase velocity *r* by the means of the equation (32). For each spatial coordinate *t* is obtained through the equation (32) and then *t* together with the phase velocity *t*, which is the corresponding spatial position *r* through the equation (32). It can be observed again a nice agreement. In figure 4 and 5, it is adopted

The analytical expressions of both approaches satisfied the expected results in the limit situations explored above in each subsection. Also, it has been observed that both approaches are equivalent when the same reference phase of the resultant wave is considered. However, for full validation, the analytical expressions obtained for the phase velocity through both approaches must be confronted with the results obtained from the direct numerical simulation of equation (9) with equation (10) and equation (11) or equation (12). In this way, in the next section, a direct comparison between the analytical and numerical results for the resultant wave and its phase velocity will occur.

4. Comparison with numerical simulations

The analytical results presented in the previous section 3 are confronted here with numerical simulations of the counterpropagating waves. In Figure 6 and Figure 7, as the first comparison, it is shown the analytical and numerical results for the resultant wave *t* and of spatial coordinates

The phase velocity *r* or along the time *t*. Since wave propagates, the reference phase *t* as the spatial coordinates are followed. For each interval of time *r* as time *t* evolves. For each displacement

in which the subscript *n* stands for numeric. To follow spatial or temporal coordinates will be suitable when the comparison of the numerical results for the phase velocity occurs respectively with the analytical results provided by the spatial and temporal approaches. Since the resultant wave *r* or the time *t*. This numeric procedure will be applied right below when the analytical results from both approaches will be confronted with the numerical results for the phase velocity.

Figure 8 and Figure 9 presents the results along the spatial coordinates *r* for respectively *r*, once the amplitude of the resultant wave also depends of the spatial coordinates *r*, and it can induces to errors in following the reference phase point *r* for each one of these times *r* in each one of the times

Similarly, the Figure 11 and Figure 12 presents the results for respectively *t* in distinct spatial positions. This is the only difference between both Figures. For both, *t*, once the amplitude of the resultant wave also depends of the time *t* in this approach, and it can induces to errors in following the reference phase point *t* for each one of these spatial coordinates *r _{n}* is cumulatively plotted in the panels (a) of Figures 11 and 12. In these Figures, each color represents the resultant wave picture along time

*t*in each one of the spatial coordinates

*r*. The total number of curves plotted in the panels (a) of Figures 11 and 12 are 100, which shows to be adequate to describe the phase velocity in the interval of time

_{n}*r*occurs in a time

_{n}*t*and is detached by a '+' sign. The interval of time

_{n}5. Conclusions

The superposition of plane waves in a physical medium can produce a resultant wave with a phase velocity distinct from the former plane wave components. Such is the case of a wave incident obliquely over an infinite and perfect conducting surface, in which the phase velocity of the resultant wave in the incident medium becomes also -- although it remais constant -- a function of the incidence angle.

In the present situation of counterpropagating waves, it was demonstrated that the superposition of plane waves can also produce a resultant wave with phase velocity that depends of the spatial coordinate *r* or the time *t*, depending of the approach adopted. As shown, both approaches are equivalent, since they provide results that are identical to each other when the same reference phase is considered.

Considering the temporal approach, mathematically, the phase velocity of the resultant wave could be expressed as *f* accounts the propagative effects -- resultant of the superposition of counterpropagating waves -- in the present case, which is a function of the amplitudes of the individual waves, *t*. It must be also observed that, although *t*,

One interesting feature identified in this work is associated with the fact of the phase velocity of the resultant wave to depend of the individual wave amplitudes. This interesting feature can be explored in electromagnetics as will be right now discussed. In charged particle accelerators, resonant interaction between charged particles and the generated electromagnetic waves is intended to cause particle acceleration. If the resultant wave of the system studied in this work is electromagnetic and propagates in a plasma, the resonant interaction of the resultant wave with the charged particles of the medium can be controlled by the amplitudes of the individual waves. More, by counterpropagation, superluminal waves can lower down its phase velocities to interact resonantly with charged particles of the medium.

This mechanism identified in this work is relevant to develop new charged particle acceleration concepts and structures. The controlling feature of the phase velocity of the resultant wave through the individual amplitudes of the plane wave components is practical from the implementation point of view. Although the amplitude control can occur through two distinct generators, it is more suitable to establish counterpropagation through reflection. In this way, instead of controlling individually the amplitude of the counterpropagating waves, one can just control the reflection coefficient, and then so the phase velocity of the resultant wave for accelerating purposes.

By the end, the analytical expressions in both representations predict the well-know results of the limit situations involving the amplitudes of the individual waves. Perfect agreement of both representations with the numerical simulations is also found.

6. Future works

Future works will explore this amplitude-dependant characteristic of the phase velocity of the resultant wave in the wave-particle interaction phenomena in a electromagnetic system. For that, counterpropagating waves will be considered to evolve in a plasma, and the resonant interaction between the resultant wave and the charged particles of the medium will be analysed. An analytical description of how the individual counterpropagating wave amplitudes control the resonant interaction is expected to be obtained.