Abstract
In this paper we study a classical and theoretical system which consists of an elastic medium carrying transverse waves and one pointlike high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this system. Afterwards we always consider the case where the concretion is not the wave source any longer. Then the concretion obeys a general and covariant guidance formula, which leads in lowvelocity approximation to an equivalent de BroglieBohm guidance formula. The concretion moves then as if exists an equivalent quantum potential. A strictly equivalent free Schrödinger equation is retrieved, as well as the quantum stationary states in a linear or spherical cavity. We compute the energy (and momentum) of the concretion, naturally defined from the energy (and momentum) density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation is fulfilled, it strikingly appears that the energy and momentum of the concretion not only are written in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas.
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Notes
Let, for instance, the mass density expressed by a 3D Gaussian function, \(\rho (\mathbf {r},t)\!=\! \frac{m_0}{(\sigma \sqrt{2\pi })^3}\exp \left[ \frac{(\mathbf {r}\mathbf {\xi })^2}{2\,\sigma ^2}\right] \), where \(\sigma \rightarrow 0\). Hence, Eq. (6) becomes \(\left( \frac{\gamma _m}{c_m^2}\frac{\partial \frac{1}{\gamma _m}}{\partial t}\frac{\partial \varphi }{\partial t} + \left[ \!\frac{\mathbf {v}}{c_m^2} \, \frac{\partial \varphi }{\partial t}(\mathbf {\xi },t) + \mathbf {\nabla }\varphi (\mathbf {\xi },t)\right] \cdot \frac{\mathbf {r}\mathbf {\xi }}{\sigma ^2}\!\right) \rho (\mathbf {r},t) = 0.\) This leads, when \(\mathbf {r}=\mathbf {\xi }\), to \(\frac{\partial }{\partial t} (\frac{1}{\gamma _m})=0\).
Eq. (7) becomes in the lowvelocity approximation, \(\mathbf {v}\,\psi (\mathbf {\xi },t) = \frac{c_m^2}{\mathrm {i}\,\Omega _m}\mathbf {\nabla }\psi (\mathbf {\xi },t)\), whose real part yields \(\mathbf {\nabla }{F}(\mathbf {\xi },t)=\mathbf {0}\). Put into the real part of Eq (7), we get \(\frac{\partial F}{\partial t}(\mathbf {\xi },t)=0\).
In any reference frame where the concretion has the velocity \(\mathbf {v}\), the particle derivative is written as \(\frac{\mathrm {d}}{\mathrm {d}t}= \frac{\partial }{\partial t} + \mathbf {v}\cdot \mathbf {\nabla }\).
The rest mass energy has no importance in quantum mechanics à la Schrödinger.
Let \(\omega _{\mathrm {circ}, v} = \frac{v}{r_c\,\sin \theta _c}\) the angular velocity of the concretion. The velocity of the concretion derived from the guidance formula yields \(\omega _{\mathrm {circ}, v} = \frac{v^2}{c_m^2}\frac{\Omega _m}{m}\). This leads in the lowvelocity approximation (note \(v = \omega _{\mathrm {circ}, v} = 0\) when \(m= 0\)) to \(\omega _{\mathrm {circ}, v}\ll \Omega _m\).
The angular momentum density derives from the stressenergy tensor. Spatial integration around the location of the concretion and using timeaveraged values (in the same manner as for \(W_\mathrm {conc}\) and \(\mathbf {p}_\mathrm {conc}\)) lead to \(\mathbf {L}_\mathrm {conc}\).
\(\langle \partial _\mu \varphi \partial ^\mu \varphi \rangle = \left( \frac{(\Omega _m + \omega )^2}{c_m^2}\frac{v^2}{c_m^4}(\Omega _m+\omega )^2\right) \,\langle \varphi ^2\rangle \), by using the guidance formula (7) and \(\frac{\partial F}{\partial t}=0\) at the location of the concretion. Next, the lowvelocity approximation (related to \(\omega \ll \Omega _m\)), Eq. (19) and the condition (14) are used.
A paper including an external potential is in preparation.
Note that this difference vanishes when the concretion is in symbiosis with the wave (cf. Section 2.3).
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Acknowledgements
We thank Hervé Mohrbach and Alain Bérard for help and discussions. We would like to thank the Reviewer for his constructive suggestions and insightful comments on the paper.
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Appendix
Appendix
Lagrangian Density of a System with a Bead Versus One with an Elastic Medium Concretion and Some Consequences
The system presented in [6] consists of (i) a bead oscillator, i.e., a punctual mass with an “internal clock”, by which the mass tends to oscillate at a proper pulsation \(\Omega _0\) through a quadratic potential, and (ii) the same (homogeneous) elastic medium as here. The Lagrangian density of this system is thus written as
where \(\frac{\mathrm {d}}{\mathrm {d}\tau }\) is the particle derivative (cf. Footnote 3) expressed with \(\tau \) the natural time of the bead, \(\tau \), and \(\rho _0\) denotes its natural mass density. \(\rho _0\) of a bead of mass \(m_0\) is again written as in Eq. (1) in the proper reference frame, \(\mathcal {R}_0\), of the bead.
The difference between the Lagrangian density of a system with a bead oscillator (24) and the one with a pointlike high density medium (2) naturally lies in the part dedicated to the ‘particle’—i.e., the bead or the concretion. A basic comparison between the two Lagrangian densities emphasises the following points.

The system with a bead has a waveparticle duality nature, which is de facto imposed from the beginning—from the density Lagrangian. In contrast, the system with a concretion is wave monistic.

The system with a bead has two reference pulsations, the one of the elastic medium or wave (\(\Omega _m\)) and the one proper to the bead oscillator (\(\Omega _0\)). Of course, the latter one does not exist in the monistic system with the concretion—and only remains the reference pulsation of the wave, \(\Omega _m\), which also describes the pulsation of the quadratic potential acting on the ‘particle’ concretion.

The bead involves in the Lagrangian density \(\left( \frac{\mathrm {d}\varphi }{\mathrm {d}\tau }\right) ^2\), while the concretion \(c_m^2\,\partial _\mu \varphi \, \partial ^\mu \varphi \) (recall in the Lagrangian density (2) that \(\mathcal {T}=\mu _0\,c_m^2\)). For the sake of simplicity we place ourself in the proper reference frame \(\mathcal {R}_0\) of the ‘particle’, where the subscript 0 means this reference frame. In one side \(\left( \frac{\mathrm {d}\varphi }{\mathrm {d}\tau }\right) ^2=\left( \frac{\partial \varphi }{\partial t_0}\right) ^2\), while on the other side \(c_m^2\,\partial _\mu \varphi \, \partial ^\mu \varphi = \left( \frac{\partial \varphi }{\partial t_0}\right) ^2  c_m^2\,\left( \mathbf {\nabla }_0\varphi \right) ^2\). The concretion behaves thus as the bead, but with one important difference: on the contrary to the bead, the concretion takes into account the wave slope (at its location).^{Footnote 9} This means that the concretion, due to its elastic medium nature, is ‘more’ than a macroscopic point mass as the bead.
The concretion is admittedly ‘more’ than an usual point mass as in macroscopic systems, but it is associated for all practical purposes to a material point. The concretion, as a macroscopic point mass, has indeed well characterised location, velocity, energy and linear momentum (cf. Sect. 2.4) and angular momentum (cf. Sect. 3.4). However, in comparison to the bead, the elasticmedium nature of the concretion provides some specificities as follows.

The concretion can only be piloted by the wave. On the contrary to the bead, the usual equation of motion of the concretion can be ‘cancelled’ (under certain conditions), which leads to the concretion to be only piloted by the wave (cf. Section 2.3).

Conditions for which the ‘particle’ is no longer a wave source are less drastic for the concretion than for the bead (cf. Sect. 2.3). (It suffice for the concretion to be located at a local extremum of the transverse wave in its reference frame, whereas the bead must oscillate in its reference frame at its proper pulsation.) On the contrary to a system with a bead, a system with a concretion can have several stationary states as in equivalent quantum mechanics.
Calculation of the Wave Equation and the Equation of Motion for the Concretion
The equation of motion for the concretion comes from a principle of least action, when the fourposition of the concretion is subjected to a small change, \(\xi ^\alpha \rightarrow \xi ^\alpha + \delta \xi ^\alpha \), while the wave field \(\varphi \) is fixed. This is applied to the Lagrangian of the concretion \(L_c\). From the Lagrangian density (2) and the mass density of the concretion (1), the Lagrangian of the concretion expressed in a reference frame where the concretion has a velocity \(\mathbf {v}\) is written as
where \(\gamma _m=\frac{1}{\sqrt{1v^2/c_m^2}}\). \(L_c\) is here the only Lagrangian which depends on the location of the concretion.
\(\xi ^\alpha \rightarrow \xi ^\alpha + \delta \xi ^\alpha \) leads at first order to \(\varphi (\xi ^\alpha ) \rightarrow \varphi (\xi ^\alpha ) + \partial _\alpha \varphi (\xi ^\alpha ) \, \delta \xi ^\alpha \), \(\partial _\mu \varphi \rightarrow \partial _\mu \varphi + \partial _\mu \left( \partial _\alpha \varphi \,\delta \xi ^\alpha \right) \) and \(\mathrm {d}\tau = \sqrt{\mathrm {d}\xi ^\mu \mathrm {d}\xi _\mu }/c \rightarrow \mathrm {d}\tau + \delta (\mathrm {d}\tau )\) where \(\delta (\mathrm {d}\tau ) = \frac{U_\alpha }{c_m^2}\, \mathrm {d}(\delta \xi ^\alpha )\) and \(U^\alpha =\frac{\mathrm {d}\xi ^\alpha }{\mathrm {d}\tau }\). Recall that \(\tau \) is the proper time of the concretion (so \(\mathrm {d}t = \gamma _m\,\mathrm {d}\tau \)). Thus, the conditions for which the small change \(\delta \xi ^\alpha \) implies at the first order \(\delta (\int L_c\, \mathrm {d}t) = 0\), where boundary values are fixed, are such that
Integrating by parts the first and the third integrals, with fixed end points, and since the small change \(\delta \xi ^\alpha \) is arbitrary, lead to the equation of motion (3).
The wave equation comes from a principle of least action, when the wave field is subjected to a small change, \(\varphi \rightarrow \varphi + \delta \varphi \), while the fourposition of the concretion is fixed.
\(\varphi \rightarrow \varphi + \delta \varphi \) leads to \(\delta (\varphi ^2) = 2\varphi \,\delta \varphi \) and \(\delta (\partial _\mu \varphi \,\partial ^\mu \varphi )=2\partial _\mu (\delta \varphi )\,\partial ^\mu \varphi \). According to Eq. (2) the conditions for which the small change \(\delta \varphi \) implies at first order \(\delta (\int \mathcal {L}\, \mathrm {d}t\, \mathrm {d}^3 \mathbf {r}) = 0\), where boundary values are fixed, are written as
Integrating by parts the term with \(\partial _\mu (\delta \varphi )\), with fixed end points, and since the small change \(\delta \varphi \) is arbitrary, provide the wave equation (4). (Note that the generalised EulerLagrange equation leads to the same result, as expected.)
Calculation of the Wave Potential
Calculations in this Appendix are near for example to the ones in [25]. The particle derivative of the guidance formula (10) is written as
Using Eq. (10) for the velocity and some basic vector calculus identities yield:
The term in brackets in the right side is evaluated from the wave equation (4) without any source wave – since the concretion and the wave are assumed in these calculations to be in symbiosis. By using the complex notation of \(\varphi \) (8) with \(\psi = F\,\mathrm {e}^{\mathrm {i}\,\Phi }\), the real part of Eq. (4) leads to:
in which terms with \((\frac{\partial \, \Phi }{\partial t})^2\) and \(\frac{\partial ^2 \, F}{\partial t^2}\) are neglected in the lowvelocity approximation. Put into Eq. (29) it appears that the concretion moves under the influence of the potential Q given in Eq. (12).
From the Klein–Gordonlike Equation Without Source to the Equivalent Free Schrödinger Equation
The derivation of the free Schrödinger from the Klein–Gordon equation (without source) in the lowvelocity approximation is mentioned by de Broglie (see e.g. [13] §II.7) and is wellknown in the literature (see e.g., [31] §III.5). Here, by using the modulating wave \(\psi \) contained in \(\varphi \) (8), the Klein–Gordonlike equation (4) without source yields
Since this equation holds for any t, we get:
In the lowvelocity approximation, the term \(\frac{\partial ^2 \psi }{\partial t^2}\) is negligible with respect to the other terms.
To convince ourselves of the validity of this approximation, we consider a free concretion in symbiosis with a plane wave \(\varphi \) (cf. also Sect. 3.2). By using \(\varphi = A\, \cos (\mathbf {k}\cdot \mathbf {r}\Omega \,t)\), the Klein–Gordonlike equation (4) leads to \(c_m^2\,k^2=\Omega ^2\Omega _m^2\); and the \(\varphi \)guidance formula (7) leads to \(\mathbf {v}=\frac{c_m^2 \, \mathbf {k}}{\Omega }\). Thus \(\Omega = \frac{1}{\sqrt{1v^2/c_m^2}}\,\Omega _m\). According to Eq. (8), the modulating wave is written as \(\psi = A\,\mathrm {e}^{\mathrm {i}(\mathbf {k}\cdot \mathbf {r}\omega \,t)}\), where \(\omega = \Omega  \Omega _m\). This leads to \(\omega =\frac{1}{2}\frac{v^2}{c_m^2}\,\Omega _m\) in the lowvelocity approximation. Then, the term \(\frac{1}{c_m^2}\frac{\partial ^2 \psi }{\partial t^2}\) is proportional to \(\frac{\Omega _m^2}{c_m^2}\frac{v^4}{c_m^4}\,\psi \), while the two other terms in Eq. (32) are proportional to \(\frac{\Omega _m^2}{c_m^2}\frac{v^2}{c_m^2}\,\psi \).
In the lowvelocity approximation, the modulating wave \(\psi \) is thus governed by the equivalent free Schrödinger equation (17).
Energy and Linear Momentum of the Concretion
As usual, the energy density, \(\rho _e\), and the momentum density, \(\mathbf {g}\), are evaluated from the stress–energy tensor T of the system (\(T^{\mu \nu }=\frac{\partial \,\mathcal {L}}{\partial (\partial _\mu \varphi )}\,\partial ^\nu \varphi  \eta ^{\mu \nu }\,\mathcal {L}\), where \(\eta ^{\mu \nu }\) denotes the Minkowski metric, with the signature \((+,,,)\) adopted throughout this article). The Lagrangian density (2), where \(\mathcal {T} = \mu _0\,c_m^2\), leads to
In the following we use their timeaveraged values during one oscillation (written as \(\langle \cdots \rangle \)). Spatial integration around the location of the concretion provides the energy, \(W_\mathrm {conc}\), and the momentum, \(\mathbf {p}_\mathrm {conc}\), of the concretion. The following expressions are given in a reference frame, \(\mathcal {R}\), where the concretion has the velocity \(\mathbf {v}\).
We evaluate first the case for which the condition (11) is satisfied, in addition to the \(\varphi \)guidance formula. Eqs. (1), (7), (11) and a little bit of algebra yield:
We have notably used \(\langle \frac{1}{\gamma _m^2} (\frac{\partial \varphi }{\partial t}(\mathbf {\xi },t))^2 \, \rangle = \Omega _m^2\,\langle \varphi ^2(\mathbf {\xi },t) \rangle \). Since the amplitude of oscillation of the concretion, \(F_c\), remains constant in time, it follows equations (13).
Now we consider that the concretion has just its velocity given by the \(\varphi \)guidance formula (7). Let \(\Omega \) (where \(\Omega =\Omega _m+\omega \)) the pulsation of the wave \(\varphi \) in \(\mathcal {R}\). In the same manner as above, Eq. (33) leads to
In the lowvelocity approximation, \(\frac{v^2}{c_m^2}\) and \(\frac{\omega }{\Omega _m}\) have the same order of magnitude. (To convince us, consider how a pulsation \(\Omega _m\) in \(\mathcal {R}_0\) becomes in \(\mathcal {R}\), for example by using LorentzPoincaré transformation.) Moreover \(\langle \, (\frac{\partial \varphi }{\partial t}(\mathbf {\xi },t))^2 \, \rangle = \Omega ^2\,\langle \, \varphi ^2(\mathbf {\xi },t) \,\rangle \) – because the magnitude F at the location of the concretion is \(F_c\), constant in time. Taking into account the condition (14), equations (35) become Eqs. (15) and (16) at firstorder approximation in \(\frac{v^2}{c_m^2}\) and \(\frac{\omega }{\Omega _m}\).
Superposition of Eigenstates in the Linear Cavity
We study the superposition of eigenstates for the concretion in the linear cavity (cf. Sect. 3.3). To have a deeper meaning, we deal with the transverse wave, \(\varphi \), rather than the modulating wave, \(\psi \). According to the Klein–Gordonlike equation (4) without source and the boundary conditions, an eigenstate is written as \(\varphi _n(x,t)=A_n\,\sin (K_n\,x)\,\cos (\Omega _n\,t \theta _n)\), where \(K_n = \frac{n\,\pi }{L}\), \(c_m^2\,K_n^2 = \Omega _n^2\Omega _m^2\), \(A_n\) denotes an amplitude of transverse oscillations and \(\theta _n\) a phase shift, irrelevant in this study. (This expression is in agreement with the one of \(\psi \) written in Sect. 3.3 when \(\omega _n(=\Omega _n\Omega _m) \ll \Omega _m\).) For the sake of simplicity we consider two eigenstates, \(n=1\) and 3 (two odd numbers), such that the superposition is written as
where A is an amplitude of transverse oscillations.
In \(x=L/2\), the wave slope \(\mathbf {\nabla }\varphi (x=L/2,t) = 0\) for any t. According to the \(\varphi \)guidance formula (7), the concretion can be thus located at this point and remain here—as for any eigenstate with an odd number n.
Let us now study the energy of the concretion remaining at \(x=L/2\). According to Eq. (35), after averaging over a transverse oscillation period and when \(\omega _n \ll \Omega _m\), the energy of the concretion is
where \(\Delta \omega = \omega _3\omega _1\). Rather than calculating the average value of \(W_\mathrm {conc}\) over time, it appears that the total energy of the concretion, \(W_\mathrm {conc}\), is periodically equal to zero. The analogy with quantum mechanics seems no longer to hold. Furthermore a total energy of the concretion equal to zero does not seem to be realistic, in particular if the ‘particle’ concretion is a simplified representation of a soliton. Then, the toy model suggested in this paper should not be suitable for dealing with the superposition of eigenstates. If more complex considerations demand that the kind of concretion has a total energy near to \(m_0\,c_m^2\), we could expect that same phenomena as for walkers appear: the system could exhibit very short transitions between eigenstates [32].
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Borghesi, C. Equivalent Quantum Equations in a System Inspired by Bouncing Droplets Experiments. Found Phys 47, 933–958 (2017). https://doi.org/10.1007/s1070101701036
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DOI: https://doi.org/10.1007/s1070101701036
Keywords
 Classical system
 Waveparticle duality
 Guidance formula
 Equivalent quantum equations
 Quantum and relativistic energy