2026-08-05
Geometrical origin of the propagating wave
Could a free traveling wave be not an object emitted by an electron, but a new geometric state of the electron itself? This paper examines precisely this possibility: the closed wave structure of the electron opens entirely, changes its frequency-spatial scale, and transforms into a propagating wave, preserving the full energy of the initial state.
Introduction
In the conventional description of radiation, the electron remains a particle, and the electromagnetic field receives only the energy difference between the initial and final states. Here, a fundamentally different process is investigated. It is assumed that a localized electron can completely lose its closed geometry and transition to an unfolded wave state. Therefore, after the transition, two independent objects the remaining electron and the wave separated from it do not exist. There is one and the same physical system, represented before and after the transition by two different geometric states.
The central thesis of the work is as follows:
\[\tag{1} \boxed{ \begin{gathered} \text{the electron does not emit a separate wave, but rather transitions entirely} \\ \text{from a closed state to an unfolded state} \end{gathered} } \] In a closed state, the electron's internal wave cyclically returns to its initial phase and initial spatial position. During a complete geometric transition, this wave opens, its radius increases, and the cyclic phase coordinate unfolds into a transport coordinate. The total energy is not released from the electron, but is conserved as the energy of the same electron in a new wave state.
This transformation requires distinguishing between the internal geometric frequency of the closed electron \(\omega_e\) and the energy, or Compton, frequency \(\omega_C\). In the adopted model, they are related by the fine-structure constant. After uncoupling, it is \(\omega_C\) that becomes the frequency of the unwrapped wave. The simultaneous conservation of energy and the invariant \(r\omega=c\) leads to an increase in the radius from the internal scale \(r_e\) to the reduced Compton scale \(\overline{\lambda}_C\).
This work does not describe the usual atomic transition with photon energy \(E_i-E_f\). A separate geometric hypothesis of the complete electron transformation is considered, for which the energy of the unfolded state is equal to the total initial energy of the electron. The goal of the article is to establish an internally consistent chain between energy conservation, scale change, phase uncoupling, the emergence of the transport coordinate, and the momentum of the traveling wave.
1. One Electron – Several Geometric States
The model is based on a normalized split operator
\[\tag{2} \boxed{ J(a,b)=\j^a(-\j)^b = \ep e^{i\pi b}+\em e^{i\pi a} }. \] Here \(\ep\) and \(\em\) are complementary idempotents:
\[\tag{3} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1, \qquad \j=\ep-\em. \] The parameter \(a\) specifies the internal phase change, and the parameter \(b\) is associated with the external motion of the localized state:
\[\tag{4} a=\varpi t, \qquad \pi\varpi=\omega, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The operator norm is preserved:
\[\tag{5} J\overline J=1, \qquad E_{\mathrm{ext}}+E_{\mathrm{int}}=\mathrm{const}. \] In this concept, different electron states do not require different algebraic operators. The geometric method for reading the continuous exponent \(a(t)\) changes. In the localized state, the phase is read cyclically, while in the expanded state, the total number of completed phase revolutions is stored:
\[\tag{6} \boxed{ \begin{aligned} \text{closed state:} &\qquad \Theta_e=\pi a_e\bmod 2\pi, \\ \text{expanded state:} &\qquad \Theta_{\mathrm w}=\pi(a_{\mathrm w}-a_{\mathrm w0})\in\mathbb R. \end{aligned} } \] Therefore, the traveling wave is introduced neither as a separated part of the electron nor as a result of the action of a new transport operator. It is considered as a complete expanded state of the same electron, described by the same operator \(J\) with a different geometric interpretation of its exponent.
2. The Closed State and Two Electron Frequencies
The velocity invariant is assumed for the internal wave scale of a closed electron.
\[\tag{7} \boxed{r_e\omega_e=c}. \] The quantity \(\omega_e\) here is the internal geometric frequency. The total rest energy of an electron is not directly related to \(\hbar\omega_e\), but to the Compton frequency \(\omega_C\):
\[\tag{8} \boxed{ E_e^{(0)} =m_ec^2 =\hbar\omega_C =\alpha_{\mathrm{fs}}\hbar\omega_e }. \] Hence the frequency ratio
\[\tag{9} \boxed{ \omega_C=\alpha_{\mathrm{fs}}\omega_e }. \] The full spatial period of the internal wave is equal to the length of the closed phase loop:
\[\tag{10} \lambda_e=2\pi r_e. \] The stable closure condition has the form
\[\tag{11} \oint_{\Gamma_e}k_e\,ds=2\pi n, \qquad n\in\mathbb Z. \] For a single phase period, this condition can be represented as the simultaneous return of phase and position:
\[\tag{12} (\varphi,z) \longrightarrow (\varphi+2\pi,z). \] It is the combined return of phase and spatial coordinate that makes the state localized. A possible internal asymmetry of the two branches can disrupt the stability of this configuration, but it is not treated here as the energy of a separate emitted component. Its role is limited to the possible initiation of a complete geometric transition.
3. Complete Geometric Transition and Conservation of Energy
Let's assume that at the moment \(t_0\), the closed wave structure loses the ability to return to its previous spatial center. According to the central hypothesis of the article, the entire electron transitions to the unfolded state. Therefore, the wave energy is equal to the total energy of the original electron:
\[\tag{13} \boxed{ E_{\mathrm w}=E_e^{(0)}=m_ec^2 }. \] If Planck's relation holds for the unfolded state
\[\tag{14} E_{\mathrm w}=\hbar\omega_{\mathrm w}, \] then its frequency is determined by the total conserved energy:
\[\tag{15} \boxed{ \omega_{\mathrm w} =\frac{E_e^{(0)}}{\hbar} =\omega_C =\alpha_{\mathrm{fs}}\omega_e }. \] Thus, conservation of energy does not imply conservation of the intrinsic geometric frequency \(\omega_e\). During the transition, the frequency-geometric representation of the same energy changes: a closed structure with a frequency of \(\omega_e\) transforms into an unfolded structure with an energy frequency of \(\omega_{\mathrm w}=\omega_C\).
After a complete transition, no localized electron remains near the source. Therefore, the equalities \(E_\gamma=E_i-E_f\) and \(\omega_\gamma=(E_i-E_f)/\hbar\), applicable to ordinary radiation, are not included in the scheme under consideration.
4. Changing the radius while maintaining the invariant
Assume that the invariant \(r\omega=c\) connects both geometric states:
\[\tag{16} \boxed{ r_e\omega_e = r_{\mathrm w}\omega_{\mathrm w} =c }. \] Then the radius of the unfolded wave is
\[\tag{17} r_{\mathrm w} =r_e\frac{\omega_e}{\omega_{\mathrm w}}. \] Taking into account expression (15), we obtain the scaling transformation
\[\tag{18} \boxed{ \frac{r_{\mathrm w}}{r_e} = \frac{\omega_e}{\omega_{\mathrm w}} = \frac{1}{\alpha_{\mathrm{fs}}} }. \] Therefore, the inner radius of the electron increases to the reduced Compton radius:
\[\tag{19} \boxed{ r_{\mathrm w} =\frac{r_e}{\alpha_{\mathrm{fs}}} =\frac{c}{\omega_C} =\frac{\hbar}{m_ec} =\overline{\lambda}_C }. \] The total spatial period of the unfolded state is equal to the Compton wavelength:
\[\tag{20} \boxed{ \lambda_{\mathrm w} =2\pi r_{\mathrm w} =\frac{2\pi c}{\omega_C} =\frac{h}{m_ec} =\lambda_C }. \] Thus, the increase in radius is not introduced as an independent postulate. It follows from the simultaneous adoption of the frequency transformation \(\omega_{\mathrm w}=\alpha_{\mathrm{fs}}\omega_e\) and the preservation of the invariant \(r\omega=c\).
5. Scaling of Two-Center Geometry
Let the distance between the centers of two internal wave branches in a closed state be \(d_e\). If, upon a complete transition, the entire electron geometry scales with the same coefficient, then
\[\tag{21} d_{\mathrm w} =d_e\frac{\omega_e}{\omega_{\mathrm w}} =\frac{d_e}{\alpha_{\mathrm{fs}}}. \] For the previously adopted condition \(d_e=r_e\), we obtain
\[\tag{22} \boxed{ d_{\mathrm w}=r_{\mathrm w}, \qquad 2\pi d_{\mathrm w}=\lambda_{\mathrm w} }. \] In this interpretation, it is not the separated wave component that is scaled, but the entire two-center structure of the electron. The ratio between the intercenter distance and the radius is preserved, while the absolute scale increases by a factor of \(1/\alpha_{\mathrm{fs}}\).
6. Disconnection and transformation of phase rotation into translation
Before the transition, one phase rotation returns the wave to the original center. After complete disconnection, the previous closed geometry ceases to exist. The same phase is now reproduced not at the original point, but at the next point in the one-dimensional propagation channel:
\[\tag{23} \boxed{ (\varphi,z) \longrightarrow (\varphi+2\pi,z+\lambda_{\mathrm w}) }. \] One phase rotation turns into one translational spatial step:
\[\tag{24} \boxed{ \Delta\varphi=2\pi \quad\Longleftrightarrow\quad \Delta z=\lambda_{\mathrm w} }. \] This condition corresponds to a traveling wave
\[\tag{25} \Psi(z,t) =A e^{i(k_{\mathrm w}z-\omega_{\mathrm w}t)}, \qquad k_{\mathrm w} =\frac{2\pi}{\lambda_{\mathrm w}} =\frac{\omega_{\mathrm w}}{c}. \] The phase periodicity does not disappear:
\[\tag{26} \Psi(z+\lambda_{\mathrm w},t)=\Psi(z,t). \] The local spatial closure disappears. Therefore, the opening should be understood not as a break in the wave or insufficient time to complete a revolution, but as a topological replacement of return by translation.
7. A Unified Operator and the Expanded Transport Coordinate
A complete transition does not require a separate transfer operator. The wave projection of the same operator \(J\) after rescaling has the form
\[\tag{27} J_{{\mathrm w},\em}(t) =\em e^{i\pi a_{\mathrm w}(t)}. \] This expression is a mathematical projection of the complete state onto the \(\em\) plane, not a separate physical part of the electron. Let the transition begin at time \(t_0\), where \(a_{\mathrm w0}=a_{\mathrm w}(t_0)\). For the unfolded state, we adopt
\[\tag{28} a_{\mathrm w}(t) =a_{\mathrm w0} +\frac{\omega_{\mathrm w}}{\pi}(t-t_0). \] Continuous change of the exponent determines the unwrapped phase
\[\tag{29} \Theta_{\mathrm w}(t) =\pi\bigl[a_{\mathrm w}(t)-a_{\mathrm w0}\bigr] =\omega_{\mathrm w}(t-t_0). \] Multiplying this phase by the new geometric radius yields the transport coordinate:
\[\tag{30} \boxed{ s_{\mathrm w}(t) =r_{\mathrm w}\Theta_{\mathrm w}(t) =\pi r_{\mathrm w} \bigl[a_{\mathrm w}(t)-a_{\mathrm w0}\bigr] =c(t-t_0) }. \] The propagation velocity follows from the exponential expansion and the invariant:
\[\tag{31} \boxed{ \frac{ds_{\mathrm w}}{dt} =\pi r_{\mathrm w}\dot a_{\mathrm w} =r_{\mathrm w}\omega_{\mathrm w} =c }. \] The current value of \(e^{i\pi a_{\mathrm w}}\) does not preserve the number of completed revolutions, so the path traveled cannot be reconstructed from it alone. For the transport description, a continuous expanded value of \(a_{\mathrm w}(t)\) is required.
The transition changes the geometric scale and the way the exponent \(a\) is read, but the operational basis of the model is preserved: the cyclic coordinate \(\pi a\bmod 2\pi\) is transformed into the expanded coordinate \(s=\pi r(a-a_0)\).
8. Phase, Momentum, and Free Propagation
After selecting a one-dimensional channel, the phase of the traveling wave is written as
\[\tag{32} \Phi(z,t)=k_{\mathrm w}z-\omega_{\mathrm w}t. \] Its spatial gradient determines the momentum:
\[\tag{33} \boxed{ p_{\mathrm w} =\hbar\frac{\partial\Phi}{\partial z} =\hbar k_{\mathrm w} }. \] If the unfolded state obeys the massless propagation law \(E=pc\), then
\[\tag{34} \boxed{ p_{\mathrm w} =\frac{E_{\mathrm w}}{c} =\frac{m_ec^2}{c} =m_ec }. \] The temporal and spatial derivatives of a single phase define the energy and momentum:
\[\tag{35} \boxed{ E_{\mathrm w} =-\hbar\frac{\partial\Phi}{\partial t} =\hbar\omega_{\mathrm w}, \qquad p_{\mathrm w} =\hbar\frac{\partial\Phi}{\partial z} =\hbar k_{\mathrm w} }. \] A free wave does not require a constant energy gradient along its path. After the transition is complete, its energy is conserved:
\[\tag{36} \frac{dE_{\mathrm w}}{dz}=0. \] Energy asymmetry can contribute to the breaking of the closure and the choice of direction in the transition region, but further propagation is determined by the unfolded phase coordinate. The condition of constant phase \(d\Phi=0\) yields
\[\tag{37} k_{\mathrm w}dz-\omega_{\mathrm w}dt=0, \qquad \boxed{ \frac{dz}{dt} =\frac{\omega_{\mathrm w}}{k_{\mathrm w}} =c }. \] 9. Compton scale and de Broglie ratio
From expressions (19) and (34) it follows
\[\tag{38} r_{\mathrm w} =\frac{\hbar}{m_ec} =\frac{\hbar}{p_{\mathrm w}}. \] Therefore, the radius of the unfolded state coincides with the reduced de Broglie wavelength for the pulse \(p_{\mathrm w}=m_ec\):
\[\tag{39} \boxed{ r_{\mathrm w} =\overline{\lambda}_{\mathrm{dB},\mathrm w} =\frac{\hbar}{p_{\mathrm w}} }. \] The total phase period is
\[\tag{40} \boxed{ 2\pi r_{\mathrm w} =\frac{h}{p_{\mathrm w}} =\lambda_{\mathrm{dB},\mathrm w} =\lambda_C }. \] In the closed state, theThe quantity \(2\pi r_e\) is the length of the internal phase shift. After scaling and topological transformations, the quantity \(2\pi r_{\mathrm w}\) becomes the longitudinal period of the traveling wave. Thus, the Compton and de Broglie relations receive a unified geometric interpretation in the model.
10. Formation of a Bounded Wave Packet
Harmonic function (25) is infinite in space and describes only the carrier. If the complete geometric transition lasts a finite time \(\Delta t\), the unfolded state must be represented by a wave packet:
\[\tag{41} \Psi(z,t) =A\!\left(t-\frac{z}{c}\right) e^{i(k_{\mathrm w}z-\omega_{\mathrm w}t)}. \] Its longitudinal extent is estimated by the expression
\[\tag{42} \boxed{ L_{\mathrm{packet}}\sim c\Delta t }. \] It is necessary to distinguish between the geometric radius, the carrier period, and the total packet length:
\[\tag{43} \boxed{ r_{\mathrm w}=\frac{c}{\omega_C}, \qquad \lambda_{\mathrm w}=2\pi r_{\mathrm w}, \qquad L_{\mathrm{packet}}\sim c\Delta t }. \] A packet can contain one or more phase periods. The finiteness of the conversion time leads to a finite spectral width:
\[\tag{44} \Delta\omega\,\Delta t\gtrsim1. \] After the transition is complete, the original localized structure is completely absent, and the formed packet continues to propagate as a free solution of the wave equation:
\[\tag{45} \frac{\partial^2\Psi}{\partial t^2} -c^2\frac{\partial^2\Psi}{\partial z^2} =0. \] This transition is presented at the end of this paper.
11. Direction and Limitations of Physical Interpretation
The unfolding of the exponent \(a(t)\) determines the path length and velocity along the selected axis, but does not in itself determine the orientation of this axis in three-dimensional space. Therefore, the direction of the wave vector \(\widehat{\mathbf n}\) must be determined by the dynamics of the complete transition, the geometry of the internal branches, the external field, the boundary conditions, or a probability law:
\[\tag{46} \mathbf k_{\mathrm w}=k_{\mathrm w}\widehat{\mathbf n}, \qquad \mathbf p_{\mathrm w}=\hbar\mathbf k_{\mathrm w}. \] The complete transformation of a stationary electron into a single free wave also requires the fulfillment of the law of conservation of momentum. If the original electron had zero momentum, a directed wave packet with momentum \(m_ec\) cannot arise without a second participant in the process or recoil from the surrounding system.
In addition to energy and momentum, an electron has electric charge, spin, and lepton number. If the unwrapped wave is the same electron, the model must show how these properties are represented in its geometry and how they are observed after the transition. Without such a mechanism, the unwrapped electron cannot be automatically identified with a standard photon, which is electrically neutral and has different quantum numbers.
The proposed scheme is a hypothesis of the complete geometric state of the electron, and not a replacement for the standard theory of atomic radiation. For an electron at rest, it specifies an energy \(m_ec^2\), a frequency \(\omega_C\), and a wavelength \(\lambda_C\); A wave of arbitrary energy \(E_i-E_f\) is not described by this scheme.
12. Sequence of a complete geometric transition
The main relations of the model form the following chain:
\[\tag{47} \boxed{ \begin{aligned} E_e^{(0)} &=m_ec^2 =\alpha_{\mathrm{fs}}\hbar\omega_e =\hbar\omega_C, \\ E_{\mathrm w} &=E_e^{(0)}, \qquad \omega_{\mathrm w} =\omega_C =\alpha_{\mathrm{fs}}\omega_e, \\ r_e\omega_e &=r_{\mathrm w}\omega_{\mathrm w}=c, \qquad r_{\mathrm w} =\frac{r_e}{\alpha_{\mathrm{fs}}} =\overline{\lambda}_C, \\ \lambda_{\mathrm w} &=2\pi r_{\mathrm w}=\lambda_C, \\ (\varphi,z) &\longrightarrow (\varphi+2\pi,z+\lambda_{\mathrm w}), \\ s_{\mathrm w} &=\pi r_{\mathrm w}(a_{\mathrm w}-a_{\mathrm w0}) =c(t-t_0), \\ p_{\mathrm w} &=\hbar k_{\mathrm w} =\frac{E_{\mathrm w}}{c} =m_ec. \end{aligned} } \] This sequence combines three transformations. The frequency transformation replaces the intrinsic geometric frequency \(\omega_e\) with the energy frequency \(\omega_C\). The scale transformation increases the radius by a factor of \(1/\alpha_{\mathrm{fs}}\). The topological transformation replaces the closed phase tour with an unfolded transport coordinate. In all three cases, we are not talking about the separation of part of the energy, but about a complete change in the geometric state of a single electron.
13. What follows from the accepted relationships and what remains a hypothesis
When accepting the equalities \(E_e^{(0)}=\alpha_{\mathrm{fs}}\hbar\omega_e=\hbar\omega_C\), \(E_{\mathrm w}=E_e^{(0)}\), \(E_{\mathrm w}=\hbar\omega_{\mathrm w}\) and the invariant \(r\omega=c\) algebraically follow:
1. Frequency of the unfolded state \(\omega_{\mathrm w}=\omega_C=\alpha_{\mathrm{fs}}\omega_e\).
2. Scale ratio \(r_{\mathrm w}/r_e=1/\alpha_{\mathrm{fs}}\).
3. Radius \(r_{\mathrm w}=\hbar/(m_ec)=\overline{\lambda}_C\).
4. Spatial period \(\lambda_{\mathrm w}=2\pi r_{\mathrm w}=h/(m_ec)=\lambda_C\).
5. With the additional adoption of the law \(E=pc\) — momentum \(p_{\mathrm w}=m_ec\) and the ratio \(r_{\mathrm w}=\hbar/p_{\mathrm w}\).
New provisions, which so far remain physical and geometric hypotheses:
1. An electron is capable of completely transitioning from a localized state to an unfolded wave state.
2. The internal frequency \(\omega_e\) upon opening is transformed into the energy frequency \(\omega_C\).
3. The invariant \(r\omega=c\) is preserved between two different geometric states.
4. Local closure is transformed into the translational phase condition \((\varphi,z)\to(\varphi+2\pi,z+\lambda_{\mathrm w})\).
5. The continuous exponent \(a(t)\) after uncoupling is read as the transport coordinate \(s=\pi r(a-a_0)\).
6. The unfolded state propagates with velocity \(c\) and obeys the law \(E=pc\).
7. The transition dynamics select a one-dimensional channel and its orientation in external three-dimensional space.
8. The geometry of the unfolded state either preserves or consistently transforms the charge, spin, and lepton number of the electron.
9. The complete process, together with the external system, satisfies the laws of conservation of energy, momentum, and all relevant quantum numbers.
The formulas for changing the frequency and radius are consequences of the adopted initial equalities, but the initial equalities themselves and the mechanism for the complete transition require independent physical justification. This separation prevents the internal mathematical consistency of the model from being confused with its experimental validity.
Conclusion
This paper constructs a geometric scheme whose central tenet is the complete transition of an electron from a closed to an expanded state. The electron is not considered as the source of a separate difference wave. Its internal wave structure is completely broken, and the total energy of the original electron becomes the energy of the same object in a traveling wave state:
\[\tag{48} \boxed{ \text{closed electron} \longrightarrow \text{unfolded electron-wave}, \qquad E_{\mathrm w}=E_e^{(0)} }. \] Distinguishing between the internal geometric frequency \(\omega_e\) and the Compton frequency \(\omega_C\) eliminates the apparent contradiction between energy conservation and radius change. Since
\[\tag{49} \omega_{\mathrm w}=\omega_C=\alpha_{\mathrm{fs}}\omega_e, \] preserving the invariant \(r\omega=c\) leads to an increase in the radius:
\[\tag{50} \boxed{ r_e \longrightarrow r_{\mathrm w} =\frac{r_e}{\alpha_{\mathrm{fs}}} =\overline{\lambda}_C }. \] Topologically, the transition is expressed by replacing the spatial return with a translation by one wavelength. The cyclic reading of the exponent \(a\) is unwound, and without introducing a new transport operator, a coordinate is obtained
\[\tag{51} \boxed{ s_{\mathrm w} =\pi r_{\mathrm w}(a_{\mathrm w}-a_{\mathrm w0}) =c(t-t_0) }. \] The momentum of the unwound state is determined by the phase gradient, whereas a constant energy gradient along the free path is not required. When adopting a massless propagation law, the model relates energy, momentum, and the Compton scale:
\[\tag{52} \boxed{ E_{\mathrm w}=m_ec^2, \qquad p_{\mathrm w}=m_ec, \qquad r_{\mathrm w}=\frac{\hbar}{p_{\mathrm w}}, \qquad \lambda_{\mathrm w}=\frac{h}{p_{\mathrm w}}=\lambda_C }. \] The resulting construction internally reconciles complete conservation of energy with changes in frequency, radius, and wave topology. However, it does not yet prove the possibility of such a transition. To complete the model, it is necessary to derive the uncoupling directly from the dynamics of the \(J\) operator, establish a mechanism for selecting the direction, ensure conservation of momentum, and demonstrate a geometric representation of charge, spin, and lepton number.
Thus, the proposed work does not define a mechanism for the usual emission of a photon by an electron, but a more radical hypothesis: a localized electron and a free traveling wave can be two geometric states of a single wave system.
Why the local transmission law leads to the wave equation
Consider a norm-preserving phase mode of the complete state. If the unperturbed state satisfies the condition \(J_0\overline J_0=1\), then such a mode can be represented as
\[\tag{A1} J(z,t)=J_0(z)e^{i\psi(z,t)}. \] The factor \(e^{i\psi}\) changes the phase, but not the absolute value of the state:
\[ J\overline J = J_0\overline J_0 e^{i\psi}e^{-i\psi} =1. \] We introduce two local quantities with the dimension of the inverse length:
\[\tag{A2} q=\frac{1}{c}\frac{\partial\psi}{\partial t}, \qquad p=\frac{\partial\psi}{\partial z}. \] Since \(p\) and \(q\) are derivatives of the same phase function, the coincidence of the mixed derivatives automatically yields
\[ \frac{\partial p}{\partial t} = c\frac{\partial q}{\partial z}. \] To describe the free propagation ofWe introduce an additional local law of the model: the temporal change in \(q\) is determined by the spatial change in \(p\) at the same speed \(c\):
\[\tag{A3} \frac{\partial p}{\partial t} = c\frac{\partial q}{\partial z}, \qquad \frac{\partial q}{\partial t} = c\frac{\partial p}{\partial z}. \] System (A3) has a local conservation law:
\[ \frac{\partial}{\partial t} \left( \frac{p^2+q^2}{2} \right) + \frac{\partial}{\partial z} \left( -cpq \right) =0. \] Therefore, the quantity \((p^2+q^2)/2\) is neither created nor destroyed between adjacent points, and its change within a region is determined by the flow \(-cpq\) through the boundaries.
The system naturally decomposes into two oppositely propagating phase components:
\[ u_+=p+q, \qquad u_-=p-q, \] \[ \frac{\partial u_+}{\partial t} = c\frac{\partial u_+}{\partial z}, \qquad \frac{\partial u_-}{\partial t} = -c\frac{\partial u_-}{\partial z}. \] This shows that system (A3) describes two wave branches propagating in opposite directions with velocity \(c\).
Substituting the definitions of (A2) into the second equation of system (A3) yields
\[\tag{A4} \frac{1}{c} \frac{\partial^2\psi}{\partial t^2} = c\frac{\partial^2\psi}{\partial z^2}, \qquad \boxed{ \frac{\partial^2\psi}{\partial t^2} - c^2\frac{\partial^2\psi}{\partial z^2} =0 }. \] Thus, the wave equation is not introduced separately: it is a consequence of the phase representation and the local transmission system (A3). However, the second equation of this system remains an additional dynamical law. For a complete geometric derivation, it is necessary to show why the two-branch structure \(J\) leads to precisely the symmetric transmission of changes with velocities \(+c\) and \(-c\).