On the covariance of the d’Alembert equation: the cases of sound and light

The covariance of the d’Alembert equation for acoustic phenomena, described by mechanical waves in one or three spatial dimensions, under Galilean transformations, is demonstrated without the need to abandon the hypothesis that time is absolute in Classical Mechanics. This is true only if and only if the phase velocity of sound depends on the velocity of the observer. On the other hand, it is also shown that the same d’Alembert equation is covariant under Lorentz transformations if and only if the phase velocity of light does not depend on the observer.

Keywords:
d’Alembert equation; Galilean transformations; Lorentz transformations; sound propagation; light propagation

Authorship

Francisco Caruso ^**Endereço de correspondência: francisco.caruso@gmail.com

Centro Brasileiro de Pesquisas Físicas, Coordenação de Física de Altas Energias, 22290-180, Rio de Janeiro, RJ, Brasil.Centro Brasileiro de Pesquisas FísicasBrasilRio de Janeiro, RJ, BrasilCentro Brasileiro de Pesquisas Físicas, Coordenação de Física de Altas Energias, 22290-180, Rio de Janeiro, RJ, Brasil.

https://orcid.org/0000-0002-5481-9762

Vitor Oguri

Universidade do Estado do Rio de Janeiro, Instituto de Física Armando Dias Tavares, 20550-900, Rio de Janeiro, RJ, Brasil.Universidade do Estado do Rio de JaneiroBrasilRio de Janeiro, RJ, BrasilUniversidade do Estado do Rio de Janeiro, Instituto de Física Armando Dias Tavares, 20550-900, Rio de Janeiro, RJ, Brasil.

*Endereço de correspondência: francisco.caruso@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Figures | Formulas

Figures (2)
Formulas (29)

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Figura 1
Sistemas de coordenadas S e

S^{'}

. V é a velocidade de

S^{'}

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Figura 2
Onda plana eletromagnética segundo S.

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(2)

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Figura 1 Sistemas de coordenadas S e $S^{'}$ . V é a velocidade de $S^{'}$ .

Figura 2 Onda plana eletromagnética segundo S.

\frac{1}{v_{s}^{2}} \frac{\partial^{2}}{\partial t^{2}} Ψ (x, t) = \frac{\partial^{2}}{\partial x^{2}} Ψ (x, t)

(1)

Ψ (x, t) = Ψ^{'} (x^{'}, t^{'})

Ψ (x, t) = f (x - v_{s} t)

x = x^{'} + V t^{'}

{\begin{matrix} \frac{\partial}{\partial x^{'}} = \overset{1}{\overset{⏞}{\frac{\partial x}{\partial x^{'}}}} \frac{\partial}{\partial x} \Rightarrow \frac{\partial^{2}}{\partial x^{'}^{2}} = \frac{\partial^{2}}{\partial x^{2}} \\ \frac{\partial}{\partial t^{'}} = \overset{1}{\overset{⏞}{\frac{\partial t}{\partial t^{'}}}} \frac{\partial}{\partial t} + \overset{V}{\overset{⏞}{\frac{\partial x}{\partial t^{'}}}} \frac{\partial}{\partial x} \\ \Rightarrow \frac{\partial^{2}}{\partial t^{'}^{2}} = \frac{\partial^{2}}{\partial t^{2}} + 2 V \frac{\partial^{2}}{\partial t \partial x} + V^{2} \frac{\partial^{2}}{\partial x^{2}} \end{matrix}

{\begin{matrix} \frac{\partial^{2}}{\partial x^{'}^{2}} Ψ^{'} (x^{'}, t^{'}) = \frac{\partial^{2}}{\partial x^{2}} Ψ (x, t) \\ \begin{matrix} \frac{\partial^{2}}{\partial t^{'}^{2}} Ψ^{'} (x^{'}, t^{'}) & = & \frac{\partial^{2}}{\partial t^{2}} Ψ (x, t) + 2 V \frac{\partial^{2}}{\partial x t \partial x} Ψ (x, t) \\ + V^{2} \frac{\partial^{2}}{\partial x^{2}} Ψ (x, t) \end{matrix} \end{matrix}

Ψ (x, t) = f \underset{y}{\underset{⏟}{(x - v_{s} t)}} \Rightarrow {\begin{matrix} \frac{\partial^{2}}{\partial t^{2}} Ψ (x, t) = v_{s}^{2} f^{''} \\ \frac{\partial^{2}}{\partial t \partial x} Ψ (x, t) = - v_{s} f^{''} \\ \frac{\partial^{2}}{\partial x^{2}} Ψ (x, t) = f^{''} \end{matrix}

\begin{array}{l} \frac{\partial^{2}}{\partial {t^{'}}^{2}} Ψ^{'} (x^{'}, t^{'}) = (\underset{{(v_{s} - V)}^{2}}{\underset{︸}{v_{s}^{2} - 2 v_{s} V + V^{2}}}) f^{″} \\ = {v^{'}}_{s}^{2} \frac{\partial^{2}}{\partial {x^{'}}^{2}} Ψ^{'} (x^{'}, t^{'}) \end{array}

\frac{1}{v_{s}^{'}^{2}} \frac{\partial^{2}}{\partial t^{'}^{2}} Ψ^{'} (x^{'}, t^{'}) = \frac{\partial^{2}}{\partial x^{'}^{2}} Ψ^{'} (x^{'}, t^{'})

\begin{array}{l} (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}) Ψ (x, y, z, t) \\ = \nabla^{2} Ψ (x, y, z, t) = \frac{1}{v_{z}^{2}} \frac{\partial^{2}}{\partial t^{2}} Ψ (x, y, z, t) \end{array}

(2)

Ψ (x, y, z, t) = Ψ^{'} (x^{'}, y^{'}, z^{'}, t^{'})

Ψ (x, y, z, t) = A sen (k_{x} x + k_{y} y + k_{z} z - ω t)

x = x^{'}, y = y^{'}, z = z^{'} + V t^{'} e t = t^{'}

\begin{array}{l} \frac{\partial^{2}}{\partial {x^{'}}^{2}} = \frac{\partial^{2}}{\partial x^{2}}, \frac{\partial^{2}}{\partial {y^{'}}^{2}} = \frac{\partial^{2}}{\partial y^{2}}, \frac{\partial^{2}}{\partial {z^{'}}^{2}} = \frac{\partial^{2}}{\partial z^{2}} \\ e \frac{\partial^{2}}{\partial {t^{'}}^{2}} = \frac{\partial^{2}}{\partial t^{2}} + 2 V \frac{\partial^{2}}{\partial t \partial z} + V^{2} \frac{\partial^{2}}{\partial z^{2}} \end{array}

\begin{array}{l} Ψ (x, y, z, t) \\ = A sen k [(\underset{α}{\underset{︸}{s e n φ \cos θ}} x + \underset{β}{\underset{︸}{s e n φ s e n θ}} y + \underset{γ}{\underset{︸}{\cos φ}} z) + v_{s} t] \\ = f (α x + β y + γ z - v_{s} t) \end{array}

\begin{array}{l} \frac{\partial^{2}}{\partial x^{2}} Ψ = α^{2} f^{″}, \frac{\partial}{\partial y^{2}} = β^{2} f^{″}, \frac{\partial}{\partial z^{2}} Ψ = γ^{2} f^{″} \\ e \frac{\partial^{2}}{\partial t \partial z} Ψ = - γ v_{s} f^{″} \end{array}

\nabla^{2} Ψ = \nabla^{'}^{2} Ψ^{'} = (\underset{1}{\underset{⏟}{α^{2} + β^{2} + γ^{2}}}) f^{''}

\begin{array}{l} \frac{\partial^{2}}{\partial {t^{'}}^{2}} Ψ^{'} = (v_{s}^{2} - 2 γ v_{s} V + γ^{2}) f^{″} \\ = (\underset{{(v_{s} - V \cos ϕ)}^{2}}{\underset{︸}{v_{s}^{2} - 2 v_{s} V \cos φ + V^{2} \cos^{2} φ}}) {\nabla^{'}}^{2} Ψ^{'} \end{array}

\nabla^{'}^{2} Ψ^{'} (x^{'}, y^{'}, z^{'}, t^{'}) = \frac{1}{v_{s}^{'}^{2}} \frac{\partial^{2}}{\partial t^{'}^{2}} Ψ^{'} (x^{'}, y^{'}, z^{'}, t^{'})

{\begin{matrix} x^{'} = γ (V) (x - V t) \\ γ = {(1 - V^{2} / c^{2})}^{- 1 / 2} \\ t^{'} = γ (V) (t - V x / c^{2}) \end{matrix}

(3)

{\begin{matrix} \vec{E} = \hat{y} \underset{E_{y} (x, t)}{\underset{⏟}{E_{_{0}} e^{i (k x - ω t)}}} \\ \vec{B} = \hat{z} \underset{B_{z} (x, t)}{\underset{⏟}{B_{_{0}} e^{i (k x - ω t)}}} \end{matrix}

(4)

(\frac{\partial^{2}}{\partial x^{2}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}) (\begin{matrix} E_{y} \\ B_{z} \end{matrix}) = 0

(5)

(\frac{\partial^{2}}{\partial x^{', 2}} - \frac{1}{c^{', 2}} \frac{\partial^{2}}{\partial t^{', 2}}) (\begin{matrix} E_{y}^{'} \\ B_{z}^{'} \end{matrix}) = 0

(6)

{\begin{matrix} \frac{\partial}{\partial x} = γ (\frac{\partial}{\partial x^{'}} - \frac{V}{c^{2}} \frac{\partial}{\partial t^{'}}) \Rightarrow \\ \frac{\partial^{2}}{\partial x^{2}} = γ^{2} [\frac{\partial^{2}}{\partial x^{', 2}} - 2 \frac{V}{c^{2}} \frac{\partial^{2}}{\partial x^{'} \partial t^{'}} + \frac{V^{2}}{c^{4}} \frac{\partial^{2}}{\partial t^{', 2}}] \\ \frac{\partial}{\partial t} = γ (\frac{\partial}{\partial t^{'}} - V \frac{\partial}{\partial x^{'}}) \Rightarrow \\ \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} = γ^{2} [\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{', 2}} - 2 \frac{V}{c^{2}} \frac{\partial^{2}}{\partial x^{'} \partial t^{'}} + \frac{V^{2}}{c^{2}} \frac{\partial^{2}}{\partial x^{', 2}}] \end{matrix}

(7)

{\begin{matrix} E_{y} (x, t) = a E_{y}^{'} (x^{'}, t^{'}) + b B_{z}^{'} (x^{'}, t^{'}) \\ B_{z} (x, t) = d E_{y}^{'} (x^{'}, t^{'}) + e B_{z}^{'} (x^{'}, t^{'}) \end{matrix}

\begin{array}{l} [\frac{\partial^{2}}{\partial {x^{'}}^{2}} - 2 \frac{V}{c^{2}} \frac{\partial^{2}}{\partial x^{'} \partial t^{'}} + \frac{V^{2}}{c^{4}} \frac{\partial^{2}}{\partial {t^{'}}^{2}}] (a E_{y}^{'} + b B_{z}^{'}) + \\ - [\frac{1}{c^{2}} \frac{\partial^{2}}{\partial {t^{'}}^{2}} - 2 \frac{V}{c^{2}} \frac{\partial^{2}}{\partial x^{'} \partial t^{'}} + \frac{V^{2}}{c^{2}} \frac{\partial^{2}}{\partial {x^{'}}^{2}}] (a E_{y}^{'} + b B_{z}^{'}) = 0 \end{array}

\begin{array}{l} a (1 - \frac{V^{2}}{c^{2}}) \frac{\partial^{2} E_{y}^{'}}{\partial {x^{'}}^{2}} + b (1 - \frac{V^{2}}{c^{2}}) \frac{\partial^{2} B_{z}^{'}}{\partial {x^{'}}^{2}} \\ - \frac{a}{c^{2}} (1 - \frac{V^{2}}{c^{2}}) \frac{\partial^{2} E_{y}^{'}}{\partial {t^{'}}^{2}} + \frac{b}{c^{2}} (1 - \frac{V^{2}}{c^{2}}) \frac{\partial^{2} B_{z}^{'}}{\partial {t^{'}}^{2}} = 0 \end{array}

\begin{array}{l} a (\frac{\partial^{2}}{\partial {x^{'}}^{2}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial {t^{'}}^{2}}) E_{y}^{'} + b (\frac{\partial^{2}}{\partial {x^{'}}^{2}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial {t^{'}}^{2}}) B_{z}^{'} = 0 \\ \Rightarrow (\frac{\partial^{2}}{\partial {x^{'}}^{2}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial {t^{'}}^{2}}) (_{B_{z}^{'}}^{E_{y}^{'}}) = 0 \end{array}

f (x - v_{s} t) + g (x + v_{s} t)

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On the covariance of the d’Alembert equation: the cases of sound and light

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[1] *Endereço de correspondência: francisco.caruso@gmail.com